Sequencer | StrangerCoug's turn

[Plotinus]

Sequencer is a set collecting game that I am designing. It is kind of like scrabble or rummikub except with numbers. Players get a hand of cards with numbers on them. On your turn you can either steal-and-continue somebody else's sequence or play a new sequence of your own. A sequence is a set of numbers that all have something in common with each other, like all being even or all being tetrahedral numbers. It is okay if you can't write a closed form for your sequence as long as you can tell us what the rule is. When a sequence has 7 or more cards in it, it is converted into points and can't be stolen anymore.

Players:

• Not_Mafia
  
  
• Farren
  
  
• Rockhopper
  
  
• Nancy Drew 39
  
  
• StrangerCoug

Replacements:

• Not_Mafia

Previous winners:

• Game 1: Implosion
  
• Game 2:
  Analysis: Not_Mafia & Micc
  
  
• Game 3:
  Dynamics: lilith2013 & skitter30
  
  
• Game 4:
  Exsecant: Sirius9121 & Nancy Drew 39

How to join:

Type: "/in"
You may in after the game starts, I'll add you to the replacements.

Game flow:

• You start with 7 cards
  
• On your turn, play some cards in only one of these ways:
  • Start one new sequence with
    at least 3 cards
    : "I'm playing 3, 9, 18 for divisible by 3." You are welcome to include a closed form if you can like { 3 | n } but you must translate this into English for the rest of the players.
    
  • Continue one sequence you previously started: "I'm adding a 3 to my primes"
    
  • Steal one sequence from somebody else: "I'm adding 4, and 16 to Plotinus' powers of 2"
• When the sequence has at least 7 cards in it, it is finished. You get n points, for finishing it, where n is the number of cards in the sequence, and the sequence is removed from the game.
  
• After your turn I replenish your hand up to 7 cards so you can be thinking about your next move while the other players go.
  
• If you cannot think of anything at all to do on your turn, and it's your very first turn of your very first game, then you may post your hand for others to help you. After your first turn is over, you may not reveal your hand.
  
• If all players pass consecutively the game is over and whoever has the most points wins.
  
• If the deck runs out of cards and the players run out of cards then the game is over.

Invalid sequences and exceptions:

• Most sequences are valid if you can tell us what the rule is. The rule does not have to be number theoretic; cosmetic rules are fine. They can be as complicated or as simple as you like, though for complicated ones it is worth checking the deck to make sure there are at least 7 such numbers. You don't need to collect the numbers of the sequence in order. There are just a few exceptions:
  
• A sequence is a set of numbers that all have something in common with each other. Numbers may not be excluded from sequences they would naturally belong to, for example "primes except 7".
  
• The rules for a sequence must apply to at least 10 unique cards in the deck.
  
• Variants of "all integers", "numbers less than 1001", "random numbers" -- these are all equivalent to "player 1 starts the game with 7 points for no reason"
  
• Repeats: if someone has already put 15 in the divisible by 5's sequence, you can't put another instance of 15 in it. This goes even for sequences that naturally contain repeats. fibonacci only gets one 1, or a naturally repeating sequence like [1, 2, 1, 2, 1, 2, 1, 2] would be disallowed.
  
• The variables used in closed forms must represent integers (ℤ). For example you may not say that 5 is a member of the sequence n² because the square root of 5 is not an integer.
  
• Sequences that are the same or equivalent to a sequence that we already have, whether it is in active play or in the Finished category. For example if we already have the "divisible by two" sequence, we cannot also have the "even numbers" sequence. If we have "divisible by 3" then we cannot also have the "n = 0 mod 3" sequence. Once a sequence has been used it cannot be reused until we start a new game.

Bingos (putting down 7 cards at once):

• Bingos are worth 8 points.
  
• Bingos must be neither too common nor overengineered. An attempt to clarify what this means follows:
  
• Your sequence should match less than or equal to half of the unique cards (55.5) in either the current deck or in the range [1,100], which may be less tedious to verify.
  
• Your sequence shouldn't have more than 2 working parts, for example "the number is
  prime
  mod 7
  " has two parts and is allowed. The
  digit sum
  of
  x²
  is
  less than 5
  and
  x³
  ends in a prime number
  has four parts and is not allowed.
  
• If you are a following a meta rule to generate rules for sequences, then the meta rule shouldn't be able to generate sequences that are bingos for more than 1/64th of hands.
  • An example meta rule that would generate a bingo all of the time is roots of the 7 degree polynomial (x - my first card)(x - my second card)(...)(x - my seventh card) = 0.
    
  • Another meta rule for generating sequences "look up my hand in a large database of sequences" is also not allowed for bingos, so you should be able to show enough of your work that we can see you worked forwards not backwards.)

Activity Requirements:

• This post contains the order the players are in.
  
• If you see that it is your turn, you may go without waiting.
  
• If it has been your turn for 24 hours and you have not gone, I will prod you.
  
• If you don't go within 24 hours of being prodded we'll replace you, but you can stay if you post before I find someone new.
  
• While you're being replaced, to keep things moving, I'll play the leftmost card from your hand to the topmost sequence it can fit into, starting with your team's sequences, or pass if you don't have any such cards.
  
• Let us know if you're going to be V/LA.
  
• While V/LA, I'll nudge you if it's been 48 hours since you've gone and then at the 72 hour mark, I'll go for you, playing your leftmost hand to the topmost sequence it can fit into, starting with your own team's sequences, or passing otherwise.

Spoiler: deck

Let's use this deck to start with, it is biased towards small numbers to make the game a little bit easier. Every number below 100 is present at least once: 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 23, 23, 24, 24, 25, 25, 25, 26, 27, 27, 27, 28, 28, 28, 29, 29, 30, 30, 30, 31, 32, 32, 33, 34, 34, 35, 35, 35, 35, 36, 36, 36, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 45, 46, 47, 48, 49, 49, 49, 50, 50, 51, 52, 53, 54, 55, 55, 55, 56, 56, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 64, 64, 64, 65, 66, 67, 68, 69, 70, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 81, 82, 83, 84, 84, 85, 86, 87, 88, 89, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 100, 120, 125, 128, 165, 216, 220, 256, 343, 512, 512, 729, 1000

StrangerCoug's deck (with 3^n added in) 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 23, 23, 24, 24, 25, 25, 25, 26, 27, 27, 27, 27, 28, 28, 28, 29, 29, 30, 30, 30, 31, 32, 32, 33, 34, 34, 35, 35, 35, 35, 36, 36, 36, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 45, 46, 47, 48, 49, 49, 49, 50, 50, 51, 52, 53, 54, 55, 55, 55, 56, 56, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 64, 64, 64, 65, 66, 67, 68, 69, 70, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 81, 81, 82, 83, 84, 84, 85, 86, 87, 88, 89, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 100, 100, 120, 121, 125, 128, 144, 165, 169, 196, 200, 216, 220, 225, 243, 250, 256, 256, 289, 300, 324, 343, 361, 400, 400, 441, 484, 500, 500, 512, 512, 529, 576, 600, 625, 676, 700, 729, 729, 729, 750, 784, 800, 841, 900, 900, 961, 1000, 1000, 1000

Other options: all the numbers between 1 and 100 once, all the numbers between 1 and 500 once, all of these numbers 4 times each, etc.

Sample Game State:

Completed:

{

3n - 1

}

K has:

• [13, 28, 43] {
  f(n) = 15n + 13
  } When you divide by 15, the remainder is 13
  
• [1, 2, 4, 16, 32, 64] {
  2ⁿ
  } The numbers you get by repeatedly doubling 1

P has 8 points:

• [3, 9, 27] {
  3ⁿ
  } The numbers you get by repeatedly tripling one.
  
• [8, 10, 13, 18, 39] {
  f(0) = 7, f(n) = f(n - 1) + g(n)
  ;
  g(0) = 1, g(1) = 1, g(n) = g(n - 1) + g(n - 2)
  } Each number starting with 7 has the the next number of the fibonacci sequence added to it
  
• [5, 93, 343] {
  n is a preperiodic point of a fixed point in the map f(n) = n² in ℤ₁₀₀
  } When you repeatedly square these numbers and take the remainder after dividing by 100, these numbers converge on a single point which remains itself when squared, but they are not themselves that final fixed point.
  
• [3, 21, 36, 51] {
  3n
  } these numbers can all be divided evenly by three.

I am P and it is my turn. My secret hand is: [

31, 21, 64, 21, 7, 3, 63

]. I would like to complete K's powers of 2 and convert it into points but it already has a 64 so I cannot. I could put 63 into my 3n sequence but then it will only need 2 more before it gets completed. It is easy to steal because roughly a third of the cards in the deck can go into it, and the pattern is obvious. I think it is less risky to put the 7 into my sequence of periodic points. It fits the pattern: 7² = 49 mod 100, 49² = 1 mod 100, 1² = 1 mod 100, fixed point.

I write "adding 7 to the preperiodic points" or something that identifies it uniquely, the mod hands me a new card and the next person goes.

[Plotinus]

I will start when we have 4 players (3 so far!) or tomorrow morning sometime (less than 15 hours from now). I have added some basic activity requirements to the OP. I am hoping that drawing cards at the end of your turn will speed things because you can just go when it is your turn without waiting for me.

I'll continue adding players to the player list even after we start. I think 6 is a reasonable maximum on active players.

I am still playtesting this idea and am open to suggestions once we have been playing long enough to get a feel for it.

[Plotinus]

That's four! Signups are still open but we can get started.

It is McMenno's turn

[Plotinus]

Great! That's enough for now, future /ins will be added to the replacements list. Everybody has been sent their PMs and can start playing.

[Plotinus]

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

It is implosion's turn

[Plotinus]

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has:

• [15, 53, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 1
  } numbers with at least two digits, all of which are odd

DeathRowKitty has:

• [8, 18, 34] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

When adding numbers to a sequence that is not easily checked*, it is required to show that they match the rules for that sequence, as DeathRowKitty has done.

*ie: a sequence that is easily checked can be worked out with a pencil, a paper, an average understanding of elementary arithmetic and elementary algebra, and without any other tools or lookup tables.

It is Felissan's turn

[Plotinus]

Thanks for clarifying. I've edited my description of it and written a new closed form for it

[Plotinus]

I have thought some more about bingos. It may happen that you are able to generate a sequence that matches all of the cards in your hand. The rules for such a bingo will be more strict than the rules for a general sequence.

• Your sequence should match less than or equal to half of the unique cards (55.5) in either the current deck or in the range [1,100], which may be less tedious to verify.
  
• If you are a following a meta rule to generate rules for sequences, then the meta rule shouldn't be able to generate sequences that are bingos for more than 1% of hands.
  • An example meta rule that would generate a bingo all of the time is roots of the 7 degree polynomial (x - my first card)(x - my second card)(...)(x - my seventh card) = 0.
    
  • Another meta rule for generating sequences "look up my hand in a large database of sequences" is also not allowed for bingos, so you should be able to show enough of your work that we can see you worked forwards not backwards.)

This means the chances of starting with a particular bingo should be at most: (ℙ(unique_hand)) * 1 / 2⁷.

These rules only apply to putting down 7 cards at once. If you want to put down the first few cards of a sequence that applies to 90% of the deck and hope nobody finishes it before it is your turn again, that is your decision.

[Plotinus]

Also, about using databases: I think "I know this sequence exists, and expect that a sizeable portion of the deck matches it, probably including some of my cards, especially these cards which look about right for it, but it is tedious to figure it out by hand, so i will look up the sequence by name" is probably within the limits of acceptability, but "i will look up the numbers in my hand and see what sequences exists for it" is not.

[Plotinus]

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has:

• [15, 53, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 1
  } numbers with at least two digits, all of which are odd

DeathRowKitty has:

• [8, 18, 34] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16

It is NotMySpamAccount's turn

[Plotinus]

• On your turn, play some cards:
  • Either start a new sequence with
    at least 3 cards
    : "I'm playing 3, 9, 18 for divisible by 3." You are welcome to include a closed form if you can like { 3 | n } but you must translate this into English for the rest of the players.
    
  • Or steal somebody else's / continue one of your own: "I'm adding 4, and 16 to Plotinus' powers of 2"

I intended "either / or", and the singular possessive "else's" to mean "you can only do one of these things" per turn: either start one new sequence or steal one existing sequence. I will state this more explicitly in the rules.

NotMySpamAccount may reconsider their move.

[Plotinus]

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has:

• [15, 53, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 1
  } numbers with at least two digits, all of which are odd

DeathRowKitty has:

• [8, 18, 34] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16

NotMySpamAccount has:

• [2, 4, 32] {
  2ⁿ
  } powers of two

It is StrangerCoug's turn

[Plotinus]

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has:

• [15, 53, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 1
  } numbers with at least two digits, all of which are odd

DeathRowKitty has:

• [8, 18, 34] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16

NotMySpamAccount has:

• [2, 4, 32] {
  2ⁿ
  } powers of two

StrangerCoug has:

• [17, 19, 43] primes

It is McMenno's turn

[Plotinus]

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has:

• [15, 53, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 1
  } numbers with at least two digits, all of which are odd

DeathRowKitty has:

• [8, 18, 34] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16

NotMySpamAccount has:

• [2, 4, 32] {
  2ⁿ
  } powers of two

StrangerCoug has:

• [17, 19, 43] primes

It is implosion's turn

[Plotinus]

where's the 6 in the OP?

[Plotinus]

Great. I don't want to do too much reinterpreting of sequences, and the original player has the final say on what the rules for the sequence are. Please speak up if I misinterpret your rules. I think it is okay to work together to make the starting move a sequence be valid or make the closed form be error free.

It would have been better if I pointed out the problem with the six initially and offered to let McMenno come up with a change himself. I'll try to do that in the future, because I think I did overstep here. Sorry about that.

Finished:

{

n² ± [0, 2]

} numbers within 2 of a perfect square

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has 7 points and:

• [15, 53, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd

DeathRowKitty has:

• [8, 18, 34] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16

NotMySpamAccount has:

• [2, 4, 32] {
  2ⁿ
  } powers of two

StrangerCoug has:

• [17, 19, 43] primes

It is DeathRowKitty's turn

[Plotinus]

Finished:

{

n² ± [0, 2]

} numbers within 2 of a perfect square

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has 7 points and:

• [15, 53, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd

DeathRowKitty has:

• [8, 18, 34] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16

NotMySpamAccount has:

• [2, 4, 32] {
  2ⁿ
  } powers of two

StrangerCoug has:

• [17, 19, 43] primes

It is Felissan's turn

[Plotinus]

Felissan has been prodded. It will be NotMySpamAccount's turn in (expired on 2019-10-04 18:50:00) or as soon as Felissan posts, whichever comes first.

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has 7 points and:

DeathRowKitty has:

• [8, 18, 34] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [2, 4, 32, 256] {
  2ⁿ
  } powers of two

NotMySpamAccount has:

StrangerCoug has 7 points and :

• [17, 19, 43] primes

It is McMenno's turn

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has 14 points and:

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [2, 4, 32, 256] {
  2ⁿ
  } powers of two

NotMySpamAccount has:

StrangerCoug has 7 points and :

• [17, 19, 43] primes

It is Felissan's turn

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has 14 points and:

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [4, 20, 36] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

NotMySpamAccount has:

StrangerCoug has 7 points and :

• [17, 19, 43] primes

It is NotMySpamAccount's turn

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has 14 points and:

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

StrangerCoug has 7 points and :

• [17, 19, 43] primes

It is StrangerCoug's turn

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has 14 points and:

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

StrangerCoug has 7 points and :

• [17, 19, 43] primes
  
• [1, 10, 15] {
  n*(n-1)/2)
  }: triangular numbers

It is McMenno's turn

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [4, 16, 81] {
  n²
  } squares

Implosion has 14 points and:

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

StrangerCoug has 7 points and :

• [17, 19, 43] primes
  
• [1, 10, 15] {
  n*(n-1)/2)
  }: triangular numbers

It is implosion's turn

[Plotinus]

implosion has been prodded. It will be DeathRowKitty's turn in (expired on 2019-10-09 19:20:00) or as soon as implosion posts, whichever comes first.

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [4, 16, 81] {
  n²
  } squares

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

StrangerCoug has 7 points:

• [17, 19, 43] primes
  
• [1, 10, 15] {
  n*(n-1)/2)
  }: triangular numbers

It is DeathRowKittys turn

[Plotinus]

Welcome back!

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [4, 16, 81] {
  n²
  } squares

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [56, 57, 64] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

Felissan has:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

StrangerCoug has 7 points:

• [17, 19, 43] primes
  
• [1, 10, 15] {
  n*(n-1)/2)
  }: triangular numbers

It is Felissan's turn

[Plotinus]

Felissan has been prodded. It will be NotMySpamAccount's turn in (expired on 2019-10-11 18:40:00) or as soon as Felissan posts, whichever happens sooner.

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [4, 16, 81] {
  n²
  } squares

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [56, 57, 64] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

Felissan has:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

StrangerCoug has 7 points:

• [17, 19, 43] primes
  
• [1, 10, 15] {
  n*(n-1)/2)
  }: triangular numbers

It is NotMySpamAccount's turn

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [56, 57, 64] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

Felissan has:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares

StrangerCoug has 7 points:

• [11, 17, 19, 71, 43] primes
  
• [1, 10, 15] {
  n*(n-1)/2)
  }: triangular numbers

There are 117 cards left in the deck. It is McMenno's turn

[Plotinus]

McMenno has been prodded. It will be implosion's turn in (expired on 2019-10-14 08:47:00), or as soon as McMenno posts, whichever happens first.

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [56, 57, 64, 76] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares

StrangerCoug has 7 points:

• [11, 17, 19, 71, 43] primes
  
• [1, 10, 15] {
  n*(n-1)/2)
  }: triangular numbers

There are 112 cards left in the deck. It is DeathRowKitty's turn

[Plotinus]

Deathrowkitty has been prodded. It will be Felisan's turn in (expired on 2019-10-16 08:18:00) or as soon as DeathRowKitty posts, whichever happens first.

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [56, 57, 64, 76] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 7 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

NotMySpamAccount has:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares

StrangerCoug has 7 points and:

• [11, 17, 19, 71, 43] primes
  
• [1, 10, 15] {
  n*(n-1)/2)
  }: triangular numbers

There are 105 cards left in the deck. It is NotMySpamAccount's turn

[Plotinus]

NotMySpamAccount has been prodded. It will be StrangerCoug's turn in (expired on 2019-10-17 13:42:04) or as soon as NotMySpamAccount goes, whichever happens first.

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• primes

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [56, 57, 64, 76] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 7 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers

There are 102 cards left in the deck. It is McMenno's turn

[Plotinus]

NotMySpamAccount has requested replacement. If it is their turn before a replacement is found, they will be skipped.

[Plotinus]

popsofctown replaces NotMySpamAccount! Yay!

[Plotinus]

McMenno has been prodded. It will be implosion's turn in (expired on 2019-10-19 01:42:06) or as soon as McMenno goes, whichever happens first.

[Plotinus]

implosion has been prodded. It will be DeathRowKity's turn in (expired on 2019-10-21 07:56:00) or as soon as implosion goes, whichever happens first.

[Plotinus]

Spoiler: Finished sequences:

• {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• primes

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [5, 6, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [56, 57, 64, 76] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 9 points and :

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 7 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers

There are 98 cards left in the deck. It is DeathRowKitty's turn

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7
  
• [56, 57, 64, 76] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 16 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 7 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers

There are 94 cards left in the deck. It is Felissan's turn

edit: I've added the numbers back into the completed sequences spoiler. I was removing them from completed sequences because they took up too much room on discord and made it hard to tell which sequences can still be added to, but this problem doesn't exist on the forum and it's nice to have them there.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 16 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers

There are 91 cards left in the deck. It is popsofctown's turn

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 16 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite numbers such that its greatest non-trivial divisor minus one of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers

There are 88 cards left in the deck. It is StrangerCoug's turn

[Plotinus]

How do you feel about this wording "composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number"?

[Plotinus]

i think i mixed up my plurals/singulars in there somehow but i can't figure out how to untangle it

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 16 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers
  
• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 85 cards left in the deck. It is McMenno's turn.

[Plotinus]

a_i can be 0, so 4 could be written as 0*1000 + 0*100 + 0*10 + 4*1, but i should edit it to show that a_i can't be negative.

How's this?

[Plotinus]

McMenno has been prodded. It will be implosion's turn in (expired on 2019-10-23 19:16:00) or as soon as McMenno goes, whichever happens first.

This is McMenno's second prod in a row. That means that if McMenno doesn't go, he will be marked inactive and autoskipped until he tells us that he is back.

[Plotinus]

McMenno has been marked inactive. It is implosion's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

McMenno is inactive and has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [7, 9, 25, 37] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor

DeathRowKitty has 16 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers
  
• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 81 cards left in the deck. It is DeathRowKitty's turn.

powers of primes usually includes 1, because p⁰ = 1 but the way you wrote it I think 1 is not included because by that logic 1 would have infinitely many prime factors, but let me know if I misunderstood.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers

McMenno is inactive and has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [7, 9, 25, 37] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor

DeathRowKitty has 23 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 78 cards left in the deck. It is Felissan's turn.

Do you guys want me to replace McMenno or do you want to continue with 5 players until he reappears? I know someone who was interested when popsofctown replaced in.

[Plotinus]

Yeah, the meta rule for this bingo is too numerically common to be a proper bingo. I'll let Felissan go again.

That's good feedback though. Let's brainstorm some possible rule changes that we could implement for the next round to make things more fun (or this round I guess, but midgame rule changes are only okay if every single player is enthusiastic about it)

I'm also not wholly satisfied with the way I've specified the rules for bingos. The spirit of what I want is:

• it should be a rare-but-achievable event to get a bingo
  
• a player might have to try to save up for one over the course of a few turns, which would require some luck
  
• "all my cards are even" is a thing you could save up for over the course of several turns and seem to be about the right amount of difficulty
  
• you shouldn't have to do complicated probability calculations to determine whether your hand is an allowable bingo
  
• The numbers should meaningfully have something in common with each other and not be the union of different things.

Without crunching the numbers, I'm worried about whether the meta rule "evenly divisible mod n" meets the 1% rule -- even if it does, "k mod n", another bingo I expect I would allowed, is probably over the line. But I like k mod n.

Maybe "The numbers should meaningfully have something in common with each other" is load-bearing enough to achieve what I want bingos to be?

[Plotinus]

One thing that affects how possible it is to finish a sequence that you have started is how many players there are. If you start a sequence like "Non perfect numbers" (allowable if not a bingo) then there's really no chance it's going to survive 5 other players playing their cards. If only 2 people were playing, there's at least a chance "divisible by 3" could survive long enough for you to finish it yourself but with 6 there's no hope of it.

One variation we could try out in a future game, is have players pair up like in canasta, into teams A, B, and C. you couldn't see your partner's hand but if you played a sequence it'd only have to survive two people before your partner would have a chance at finishing it.

[Plotinus]

I was thinking of "congruent k mod n" as a meta rule because an algorithm for figuring out if you have a bingo is: "are all my numbers in any congruency classes mod 2? mod 3? mod 5? mod 7? mod 11?..." but I think it's an okay meta rule that should be allowed. k mod n (even if k is 1 and n is 2) is meaningful in a way that "roots of this 7th degree polynomial" or "what does the OEIS say about my hand" isn't. I guess we could allow the meta rule "congruent to k mod n for n > 2" and separately grandfather in "even" and "odd" as explicitly allowed.

I think we can get most of what you want by squinting at sequences that combine sets: If you're using something like intersection or exclusion to narrow the range of the set arbitrarily then it still has to apply to at least 20¹ numbers but if your set forms a natural category² all by itself (powers of two) then it doesn't matter if it's a small one. For example your powers of primes last turn narrowed the range from 23 to 21 cards, which seems all right. Narrowing it down to 7 would have been definitely outside of the spirit of the game.

It might be good to rule against making sequences that can never be completed (for example powers of three, in the sample game in the OP, was creating by me in a game with my girlfriend by mistake.)

¹ I'm not sure this is the right number, but I think whatever the number is should be easy to calculate and shouldn't require card counting

²If I have to define natural category more precisely, I'd say a sequence that has a single unifying rule. you could write it without using any of: and, or, xor, union, intersection, or their complements.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers

McMenno is inactive and has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [7, 9, 25, 37] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor

DeathRowKitty has 23 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [5, 7, 64] the nth prime doesn't have any even digits

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 75 cards left in the deck. It is popsofctown's turn.

[Plotinus]

I agree with implosion about the downsides of limiting sequence-reduction to a specific number, it creates an incentive to play in a way that isn't fun.

About scarcity of sequences

Spoiler: Thought experiment about some 8 year olds playing this game

If you imagine a group of eight year olds playing this game, using only sequences that a eight year old has been exposed to, and not thinking through how long their sequences will survive because they're just happy to have noticed the counting by threes numbers, then as soon as the counting by threes numbers are used up, there can't be that sequence again, and the game has gotten harder in a meaningful way -- not too much harder because there's still the counting by fours numbers and so on, but a bit harder. And towards the end they may have used up all of the multiplication table that the deck allows for and have to stretch themselves and independently invent the triangular numbers or the not-divisible-by-anything numbers. When they figure out the "multiples of n + some constant" sequences they'll have a lot more available to them, but those will be harder for them to spot because the won't have those congruency classes memorised, so it balances out a bit.

But if we allow "Mutiples of three except for 6, multiples of three except for 9, multiples of three except for 12..." then we can have as many sequences of "multiples of three" as the deck allows for, and the game doesn't get meaningfully harder towards the end. I don't like that.

Since we're all nerds, we have a lot more sequences available to us that this imaginary group of 8 year olds, but because there are 6 players there are fewer sequences that can make it all the way around table, so there is a scarcity of sequences that you might think are a good idea to play and artificially inflating it with the adult equivalent of "multiples of three except for 6" is tacky.

I am leaning towards disallowing the entire genre of "some sequence minus some arbitrary numbers" but if I do, I would want to retroactively change implosion's sequence to be "numbers with only one prime factor" (which is still "powers of primes except 1" but I guess I feel differently about excluding 1 from a sequence. However, I think implosion would not have played the sequence at all without the < 50 part and it would be unfair to change his sequence that dramatically in a retroactive rule change, so another option would be to refund the cards and let him replace that sequence with another.

Retroactive rule changes require all 5 players to agree to them. How do people feel about the rule "A sequence is a group of numbers that all have something in common with each other. Numbers may not be excluded from sequences they would naturally belong to."

Incentivising sequence theft

I think the "you can complete it in 6 if you stole it and it's currently in front of you" idea is interesting and worth a playtest. I definitely think we should brainstorm around the stealing mechanic to make it worthwhile to contribute to sequences you can't finish right away.

There would be more stealing if we let sequences grow indefinitely and only awarded points at the end, but in practice whoever had the evens or odds would win and that sucks.

[Plotinus]

A less drastic option would be to disallow excluding individual numbers for this game and wait for the next game for the full rule change.

[Plotinus]

popsofctown has been prodded. It will be StrangerCoug's turn in (expired on 2019-10-28 07:59:00) or as soon as popsofctown goes, whichever happens first.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers

McMenno is inactive and has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [7, 9, 25, 37] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor

DeathRowKitty has 23 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [5, 7, 64] the nth prime doesn't have any even digits

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

StrangerCoug has 7 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 72 cards left in the deck. It is Strangercoug's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers

McMenno is inactive:

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [7, 9, 25, 37] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor

DeathRowKitty has 23 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [5, 7, 64] the nth prime doesn't have any even digits

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

StrangerCoug has 7 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [14, 35, 42, 56, 343] {
  7n
  } divisible by 7

There are 70 cards left in the deck. It is implosion's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor

McMenno is inactive:

Implosion has 21 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 30 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [5, 7, 64] the nth prime doesn't have any even digits

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

StrangerCoug has 7 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [14, 35, 42, 56, 343] {
  7n
  } divisible by 7

There are 64 cards left in the deck. It is Felissan's turn.

Completing stolen sequences at 6 cards doesn't have unanimous support for this round and will only be considered for future games.

New rule under consideration:

"A sequence is a set of numbers that all have something in common with each other. Numbers may not be excluded from sequences they would naturally belong to."
No active sequences are in violation of this rule so now I'm even more inclined towards implementing it.

People who support this rule change: DeathRowKitty, implosion, StrangerCoug, popsofctown (putting her here based on 124 even though it was before the discussion), Plotinus
People who haven't weighed in yet:

Felissan

[Plotinus]

p.s. that pun by implosion was amazing

it made my day!

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits

McMenno is inactive:

Implosion has 21 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 30 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [5, 7, 64] the nth prime doesn't have any even digits

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

StrangerCoug has 7 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [14, 35, 42, 56, 343] {
  7n
  } divisible by 7

There are 57 cards left in the deck. It is popsofctown's turn.

New rule under consideration:

"A sequence is a set of numbers that all have something in common with each other. Numbers may not be excluded from sequences they would naturally belong to."

People who support this rule change: DeathRowKitty, implosion, StrangerCoug, popsofctown (putting her here based on 124 even though it was before the discussion), Plotinus
People who haven't weighed in yet: Felissan

[Plotinus]

I've added the new rule. I agree, it was a nice bingo

[Plotinus]

popsofctown has been prodded. It will be StrangerCoug's turn in (expired on 2019-10-30 14:00:00) or as soon as popsofctown goes.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

McMenno is inactive:

Implosion has 21 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 30 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [5, 7, 64] the nth prime doesn't have any even digits

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

StrangerCoug has 14 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 51 cards left in the deck. It is imposion's turn.

The new rule has been added to the OP.

I think I didn't spell this out in the rules but I was thinking that after we all run out of cards to draw, we can continue playing the cards in our hands without drawing new ones, like at the end of Scrabble. At that stage you probably won't be making new sequences because if you can't make a bingo then you know for sure you'll never finish that sequence but you may still be able to finish some pre-existing ones. and then if nobody can take a turn then it's over. Ties can be broken by how many points worth of unfinished sequences you have in front of you.

And then we can have an intermission to discuss and finalise rule changes / deck changes. I'm pretty sure I want the next game to have 3 teams of 2 players each, and instead of having "inactive players" I'll just replace people when they disappear, like in mafia.

(If anyone wants McMenno replaced in this game, let me know. I know ErrantParabola was interested in playing.)

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits

McMenno is inactive:

Implosion has 28 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 30 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

StrangerCoug has 14 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 47 cards left in the deck. It is DeathRowKitty's turn.

[Plotinus]

DeathRowKitty has been prodded. It will be Felissan's turn in (expired on 2019-10-31 15:03:39) or as soon as DeathRowKitty goes.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits

McMenno is inactive:

Implosion has 28 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 30 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [3, 54, 80] numbers n for which there exists some positive integer with exactly 2n primitive roots

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

StrangerCoug has 14 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 44 cards left in the deck. It is Felissan's turn.

[Plotinus]

Felissan has been prodded. It will be popsofctown's turn in (expired on 2019-11-02 02:35:57) or as soon as Felissan goes, whichever happens first.

[Plotinus]

popsofctown has been prodded. It will be StrangerCoug's turn in (expired on 2019-11-04 02:52:32) or as soon as popsofctown goes, whichever happens first.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots

McMenno is inactive:

Implosion has 28 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 37 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 14 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

There are 35 cards left in the deck. It is Felissan's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots

McMenno is inactive:

Implosion has 28 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 37 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [3, 39, 165] {
  3n
  } multipes of 3

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 14 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

There are 32 cards left in the deck. It is popsofctown's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots

McMenno is inactive:

Implosion has 28 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 37 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [3, 39, 165] {
  3n
  } multipes of 3

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

There are 31 cards left in the deck. It is StrangerCoug's turn.

[Plotinus]

I don't really like/under/see the point of adding a single entry to someone else's sequence "stealing" the sequence.

Like the rules as is don't give the person the sequence "belongs" to any advantage, right? It is equally allowed to finish your own sequence or steal and finish someone else's sequence. So if you're going to list unfinished sequences and associate them with someone's name I would prefer to them to be associated with the author of the sequence. It's like, just more fun. I want to be angry at DRK about hurting my brain about outshuffles even if someone adds 1 number to it. And I want to see the full portfolio of what a troll implosion is all in one place.


But maybe I just don't understand the rules to this game because DRK has a number of points that's not divisible by 7 and I don't even know how that's possible

Yeah, the stealing mechanic needs some more work. You're right that the way things are now, it's pretty cosmetic whose name the sequence is printed after, since all that matters for points is who has it last.

One thing that might be interesting when we do teams is each team could have a colour and i'd put the numbers in that team's colour, so you could see that [

72, 256, 800

,

2304, 5184

,

16, 729

] was started by team red, stolen by team blue, and completed by team indigo. It's cosmetic but then we could just leave the sequences by the original person's name until they're completed.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots

McMenno is inactive:

Implosion has 28 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 37 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [3, 24, 39, 165] {
  3n
  } multiples of 3

There are 30 cards left in the deck. It is implosion's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

McMenno is inactive:

Implosion has 35 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 37 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [3, 24, 39, 165] {
  3n
  } multiples of 3

There are 26 cards left in the deck. It is DeathRowKitty's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3

McMenno is inactive:

Implosion has 35 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two

popsofctown has 14 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4

There are 23 cards left in the deck. It is Felissan's turn.

[Plotinus]

Since McMenno is probably not coming back, should i add his hand to the bottom of the deck?

[Plotinus]

I hope today goes better for you. It will be popsofctown's turn in (expired on 2019-11-08 08:30:00) or as soon as Felissan goes.

[Plotinus]

I gave her 24 hours from the time I posted, which is 31 hours from the time she last posted, but I don't mind giving her some more time. (expired on 2019-11-08 14:30:00).

[Plotinus]

get well soon!

[Plotinus]

The countdown timers show the time remaining from when you loaded the page, not from when I posted it, so that's probably where the mistake happened.

[Plotinus]

popsofctown has been prodded. It will be StrangerCoug's turn in (expired on 2019-11-09 17:25:50) or as soon as popsofctown goes. I'll shuffle McMenno's cards into the remaining deck before the next time somebody draws a card, unless someone objects before then.

[Plotinus]

Okay, thanks for letting us know. I'll ask errant if she's still interested in replacing in.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two

Implosion has 42 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 21 points and:

popsofctown has 14 points and is being replaced:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

McMenno's hand has been shuffled into the deck. There are 26 cards left in the deck. It is DeathRowKitty's turn.

[Plotinus]

DeathRowKitty has been prodded. It will be Felissan's turn in (expired on 2019-11-11 18:39:03) or as soon as DeathRowKitty goes.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two

Implosion has 42 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210

Felissan has 21 points and:

popsofctown has 14 points and is being replaced:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

There are 23 cards left in the deck. It is Felissan's turn.

[Plotinus]

Sorry to see you go and hope things get better for you.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two

Implosion has 42 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [2, 3, 23, 56] Numbers where each pair of consecutive digits differ by exactly one

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210

Felissan has 21 points and:

popsofctown has 14 points and is being replaced:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [28, 36, 50, 93] numbers with exactly 2 distinct prime factors

There are 19 cards left in the deck. It is implosion's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two
  
• [2, 3, 8, 23, 32, 56, 89] Numbers where each pair of consecutive digits differ by exactly one

Implosion has 49 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210

Felissan has 21 points and:

popsofctown has 14 points and is being replaced:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [28, 36, 50, 93] numbers with exactly 2 distinct prime factors

There are 16 cards left in the deck. It is Deathrowkitty's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two
  
• [2, 3, 8, 23, 32, 56, 89] Numbers where each pair of consecutive digits differ by exactly one

Implosion has 49 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210
  
• [3, 74, 512] numbers n for which there exists no way to make change for a dollar using exactly n coins, each of which may be a penny, nickel, dime, or quarter, without using at least one penny, one nickel, and one dime

Felissan has 21 points and is being replaced:

Yimmy has 14 points:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number

StrangerCoug has 14 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [28, 36, 50, 93] numbers with exactly 2 distinct prime factors

There are 13 cards left in the deck. It is Yimmy's turn, who replaced popsofctown.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two
  
• [2, 3, 8, 23, 32, 56, 89] Numbers where each pair of consecutive digits differ by exactly one

Implosion has 49 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210
  
• [3, 74, 512] numbers n for which there exists no way to make change for a dollar using exactly n coins, each of which may be a penny, nickel, dime, or quarter, without using at least one penny, one nickel, and one dime

Felissan has 21 points and is being replaced:

Yimmy has 14 points:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [2, 8, 20, 512] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2

StrangerCoug has 14 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [28, 36, 50, 93] numbers with exactly 2 distinct prime factors

There are 9 cards left in the deck. It is StrangerCoug's turn.

Yay, welcome Yimmy

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two
  
• [2, 3, 8, 23, 32, 56, 89] Numbers where each pair of consecutive digits differ by exactly one
  
• [1, 3, 5, 35, 74, 90, 512] numbers n for which there exists no way to make change for a dollar using exactly n coins, each of which may be a penny, nickel, dime, or quarter, without using at least one penny, one nickel, and one dime

Implosion has 49 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210

Felissan has 21 points and is being replaced:

Yimmy has 14 points:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [2, 8, 20, 512] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2

StrangerCoug has 21 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [28, 36, 50, 93] numbers with exactly 2 distinct prime factors

There are 5 cards left in the deck. It is implosion's turn.

I don't think I can prove it right now either, but I played around with it long enough to convince myself that I couldn't make change with 35 under those rules. If someone else can find a way to do we can roll it back.

[Plotinus]

I never got new cards after my last turn btw

I moved them to your hand in the mod PT but didn't manage to PM them to you. I've done that now, so you have 48 hours for your move starting from now, not starting from when implosion played, because you didn't have cards before.

I'll update this after breakfast

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two
  
• [2, 3, 8, 23, 32, 56, 89] Numbers where each pair of consecutive digits differ by exactly one
  
• [1, 3, 5, 35, 74, 90, 512] numbers n for which there exists no way to make change for a dollar using exactly n coins, each of which may be a penny, nickel, dime, or quarter, without using at least one penny, one nickel, and one dime

Implosion has 49 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [11, 89, 99]Numbers where any consecutive digits differ by at most 1

DeathRowKitty has 44 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210

Felissan has 21 points and is being replaced:

Yimmy has 14 points:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [2, 8, 20, 512] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2

StrangerCoug has 21 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [28, 36, 50, 93] numbers with exactly 2 distinct prime factors

There are 2 cards left in the deck. It is DeathRowKitty's turn.

Should we cannibalise Felissan's hand too if I don't find a replacement before we run out of cards or just leave it be?

[Plotinus]

DeathRowKitty has been prodded. It will be Yimmy's turn in (expired on 2019-11-16 03:17:05), or as soon as DeathRowKitty goes, whichever happens first.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two
  
• [2, 3, 8, 23, 32, 56, 89] Numbers where each pair of consecutive digits differ by exactly one
  
• [1, 3, 5, 35, 74, 90, 512] numbers n for which there exists no way to make change for a dollar using exactly n coins, each of which may be a penny, nickel, dime, or quarter, without using at least one penny, one nickel, and one dime
  
• [10, 28, 36, 45, 50, 86, 93] numbers with exactly 2 distinct prime factors

Implosion has 49 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [11, 89, 99]Numbers where any consecutive digits differ by at most 1

DeathRowKitty has 51 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210

Yimmy has 14 points:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [2, 8, 20, 512] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2

StrangerCoug has 21 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

There are 6 cards left in the deck. It is Yimmy's turn.

Shuffled Felissan's hand in before giving DeathRowKitty new cards since nobody objected in time. Felissan had 21 points.

[Plotinus]

Yimmy has been prodded. It will be StrangerCoug's turn in (expired on 2019-11-18 05:52:55) or as soon as Yimmy goes, whichever happens first.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two
  
• [2, 3, 8, 23, 32, 56, 89] Numbers where each pair of consecutive digits differ by exactly one
  
• [1, 3, 5, 35, 74, 90, 512] numbers n for which there exists no way to make change for a dollar using exactly n coins, each of which may be a penny, nickel, dime, or quarter, without using at least one penny, one nickel, and one dime
  
• [10, 28, 36, 45, 50, 86, 93] numbers with exactly 2 distinct prime factors
  
• [1, 2, 8, 11, 20, 512, 1000] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2
  
• [1, 2, 8, 11, 21, 89, 99]Numbers where any consecutive digits differ by at most 1

Implosion has 56 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 51 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52
  
• [1, 31, 61] numbers relatively prime to 210

Yimmy has 14 points:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [2, 14, 17] numbers that are prime when their digits are reversed

StrangerCoug has 28 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

There are 0 cards left in the deck. It is DeathRowKitty's turn.

StrangerCoug was the last player who could draw any cards. You can continue trying to complete sequences with the cards you have in your hands or you can pass. The game will end when all 4 players pass in a row. If there is a tie between players it'll be broken by how many cards you have in front of you in unfinished sequences.

[Plotinus]

I've edited the second number of the digit-reversed primes, thanks for pointing that out.

The original deck was constructed from the first 100 numbers + the first 10 numbers of the easiest sequences I could think of, until it felt like there were enough cards.

I propose adding 1, 3, 9, 27, 81, 243, 729 to StrangerCoug's deck, or at the very least adding 243 in, because it is sad that you can only put 6 numbers in 3^n.

It is Yimmy's turn.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers
  
• [1, 4, 9, 16, 25, 64, 81] {
  n²
  } squares
  
• [7, 8, 9, 25, 37, 47, 49] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor
  
• [12, 15, 19, 34, 35, 45, 49] 2 digit numbers with increasing digits
  
• [7, 14, 35, 42, 56, 63, 343] {
  7n
  } divisible by 7
  
• [8, 18, 29, 79, 87, 92, 96] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9
  
• [3, 4, 5, 7, 21, 22, 64] the nth prime doesn't have any even digits
  
• [1, 3, 6, 26, 54, 80, 220] numbers n for which there exists some positive integer with exactly 2n primitive roots
  
• [4, 13, 18, 27, 72, 88, 97] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square
  
• [3, 21, 24, 33, 39, 84, 165] {
  3n
  } multiples of 3
  
• [1, 2, 4, 16, 32, 64, 256] {
  2ⁿ
  } powers of two
  
• [2, 3, 8, 23, 32, 56, 89] Numbers where each pair of consecutive digits differ by exactly one
  
• [1, 3, 5, 35, 74, 90, 512] numbers n for which there exists no way to make change for a dollar using exactly n coins, each of which may be a penny, nickel, dime, or quarter, without using at least one penny, one nickel, and one dime
  
• [10, 28, 36, 45, 50, 86, 93] numbers with exactly 2 distinct prime factors
  
• [1, 2, 8, 11, 20, 512, 1000] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2
  
• [1, 2, 8, 11, 21, 89, 99]Numbers where any consecutive digits differ by at most 1
  
• [12, 15, 38, 41, 82, 84, 94] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Implosion has 56 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 51 points and:

• [1, 31, 61] numbers relatively prime to 210

Yimmy has 21 points:

• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number
  
• [30, 40, 55, 75] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [2, 14, 17] numbers that are prime when their digits are reversed

StrangerCoug has 28 points and:

• [5, 7, 10, 14] {
  n*(n-1)/2) + 4
  } triangular numbers + 4
  
• [4, 20, 36, 52, 68] {
  16n + 4
  } remainder is 4 when dividing by 16

There are 0 cards left in the deck. It is StrangerCoug's turn. The game will continue until all 4 people pass in a row.

[Plotinus]

WAIT this game isn't Sequence?

i wonder if i could /in now...

we're starting a new round soon with a new deck and slightly different rules, and we'd be glad to have you. It is probably not Sequence because it is a game I made up.

It is implosion's turn

[Plotinus]

It is DeathRowKitty's Turn

[Plotinus]

we can brainstorm a more interesting name for it perhaps.

[Plotinus]

It's Yimmy's turn.

[Plotinus]

congrats, implosion!

In round 2 we will play with StrangerCoug's deck and we'll play in teams of 2 players. What other changes do we want to make to the prototype for this round?

[Plotinus]

That's three players so far, I think we'll start when we have four.

[Plotinus]

I've PMed DeathRowKitty to ask if she's interested in round two, but I haven't heard back yet.

I'm going to be away next Thursday-Saturday but hopefully it won't disrupt things too much.

[Plotinus]

DeathRowKitty says she's sitting this one out, too. Recruit your friends, we'll start when we have 4 people, which is enough to try out the teams thing.

[Plotinus]

That's probably as many players as we're going to get for this round, and it's enough for us to have teams.

In [41]: import random
In [42]: players = ['Yimmy', 'StrangerCoug', 'Notajumbleofnumbers', 'micc']
In [43]: random.shuffle(players)
In [44]: players
Out[44]: ['Notajumbleofnumbers', 'StrangerCoug', 'micc', 'Yimmy']

Notajumbleofnumbers and Micc will be on team

Analysis

and StrangerCoug and Yimmy will be on team

Algebra

. You can change your team names / colours if you find something you both like. Points will be shared between team members. It will be Notajumbleofnumbers' turn in a minute, after I shuffle the deck. We're using StrangerCoug's deck this turn. Since flaking affects your teammates, people who don't respond to prods will be replaced instead of skipped. I think that's all the things that are different from last game.

[Plotinus]

Analysis (NotAJumbleofNumbers, Micc) has:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6

Algebra (StrangerCoug, Yimmy) has:

It is

StrangerCoug's

turn

[Plotinus]

Glad to have you, Jackal!

Also, for the newer players, the reason you draw cards at the end of your turn is so that you don't have to wait to go if I'm asleep or something. You can also submit your turn early by PM if you're anticipating being away, or submit several possible turns like "add a, b, c to sequence x if it hasn't been completed yet otherwise add f, g, and h to sequence y if possible otherwise make a new sequence d, g, b"

[Plotinus]

1, 3, 5: single-digit numbers

Oh, and

mod: None of Team Analysis's numbers are divisible by 7.

Fixed. I was lazy and quoted the first post of game 1 for the formatting and then failed to edit McMenno's opening sequence thoroughly enough.

Analysis (NotAJumbleofNumbers, Micc) has:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2 }

Algebra (StrangerCoug, Yimmy) has:

• [1, 3, 5] {
  n < 10
  } single digit numbers

It is

Yimmy's

turn

[Plotinus]

No private communication, though I think it'd be interesting to try out a version that had private communication, maybe next game? I won't punish people if it happened before right now because you didn't know.

For now, you can try to strategise with your partner in the public thread, like in Canasta.


I made the game to practice/play with math anyway but I was thinking between the games about whether I should keep trying to put the closed forms in -- it's a way to make the rule clear and doing it has cleared up some misunderstandings of the verbal form in the game 1, but I can only do it for sequences where I, personally, have enough math to produce a closed form, and I think part of DeathRowKitty's strategy last game was trying to make sequences that it was easy [for her, anyway] to check whether a number was in the sequence but hard for somebody who didn't know the closed form, like her φ(φ(n)) thing last time.

I wouldn't want to rule that players have to provide closed forms themselves, it'd exclude everybody who doesn't know how to write one and every sequence that can only be easily described in words. What do you think?

[Plotinus]

Yimmy has been prodded and has (expired on 2019-12-12 08:00:00) to post before I ask Jackal11 if he's still interested in replacing. [A bit of extra time because I want to go to bed on time tomorrow even if I didn't tonight]

The prod schedule I used for last game was prod 24 hours after the last player took their turn, then skip after another 24 hours. Since we're doing replacements instead of skipping, is this still a good schedule? would prod at 48, replace after another 24 be better? I was thinking of doing that schedule for V/LAs, but I could do something more generous for V/LAs if we did 48/24 for nomal prods.

I would rather not replace anyone who is excited about the game just because they had a busy couple days.

[Plotinus]

Analysis (NotAJumbleofNumbers, Micc) has:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2 }

Algebra (StrangerCoug, Yimmy) has:

• [1, 3, 5] {
  n < 10
  } single digit numbers
  
• [2, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n, a_i ≥ 0, d > 0
  } numbers that are divisible by their first digit

It is

NotAJumbleofNumbers's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n, a_i ≥ 0, d > 0
  } numbers that are divisible by their first digit

Analysis (NotAJumbleofNumbers, Micc) has 8 points:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k | a_i ≥ 0, d > 0, k ∈ ℤ
  } sum of the digits is a power of 2 }

Algebra (StrangerCoug, Yimmy) has:

• [1, 3, 5] {
  n < 10
  } single digit numbers

It is

StrangerCougs

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit

Analysis (NotAJumbleofNumbers, Micc) has 8 points:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }

Algebra (StrangerCoug, Yimmy) has:

• [1, 3, 5] {
  n < 10
  } single digit numbers
  
• [16, 20, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime }

It is

Micc's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit

Analysis (NotAJumbleofNumbers, Micc) has 8 points:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [15, 32, 66] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two

Algebra (StrangerCoug, Yimmy) has:

• [1, 3, 5] {
  n < 10
  } single digit numbers
  
• [16, 20, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime }

It is

Yimmy's

turn

whoops, thanks for catching that for me, Micc.

[Plotinus]

Yimmy has been prodded and has (expired on 2019-12-14 05:01:16) to go before I start looking for a replacement.

[Plotinus]

Yimmy has requested replacement and Jackal711 has confirmed he's still interested, so it is now Jackal711's turn.

Edit: let's have this on this page, too

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit

Analysis (NotAJumbleofNumbers, Micc) has 8 points:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [15, 32, 66] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two

Algebra (StrangerCoug, Jackal711) has:

• [1, 3, 5] {
  n < 10
  } single digit numbers
  
• [16, 20, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime }

It is

Jackal711's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit

Analysis (NotAJumbleofNumbers, Micc) has 8 points:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [15, 32, 66] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two

Algebra (StrangerCoug, Jackal711) has:

• [1, 3, 5] {
  n < 10
  } single digit numbers
  
• [16, 20, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime }
  
• [69, 72, 78] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with 3 | Σ_i∈[0,d]a_i; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose sum is a multiple of 3

It is

NotAJumbleofNumbers's

turn

This sequence has another closed form that looks very different from this one, but I wanted to match the verbal form as closely as I could

[Plotinus]

NotAJumbleOfNumbers has been prodded and has another (expired on 2019-12-17 03:53:25) to go before I start looking for a replacement.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit

Analysis (NotAJumbleofNumbers, Micc) has 8 points:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [15, 32, 66] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [16, 32, 20, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime

Algebra (StrangerCoug, Jackal711) has:

• [1, 3, 5] {
  n < 10
  } single digit numbers }
  
• [69, 72, 78] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with 3 | Σ_i∈[0,d]a_i; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose sum is a multiple of 3

It is

StrangerCoug's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two

Analysis (NotAJumbleofNumbers, Micc) has 16 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 32, 20, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime

Algebra (StrangerCoug, Jackal711) has 9 points and:

• [1, 3, 5] {
  n < 10
  } single digit numbers }

It is

Jackal711's

turn

Now that it's done, some other ways of writing the closed form for digit sums are multiples of 3 is

3n

, or

n % 3 == 0

or

3 | n

. I'm replacing the closed form in the completed sequences spoiler to make it easier to see which sequences are taken

[Plotinus]

Jackal has been prodded and has another (expired on 2019-12-19 02:59:21) to move before I start looking for a replacement.

I know some holidays are coming up so I was thinking that starting Monday I'd still prod people as needed but I wouldn't start looking for replacements until Friday or something? And then a similar arrangement between December 30 and January 2.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two

Analysis (NotAJumbleofNumbers, Micc) has 16 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 32, 20, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime

Algebra (StrangerCoug, Jackal711) has 9 points and:

• [1, 3, 5, 6, 9] {
  n < 10
  } single digit numbers }

It is

NotAJumbleofNumbers's

turn

[Plotinus]

Would a longer prod schedule work better for your internet access issues, Jackal? We could switch to 48/24 like a mafia game if that would be easier for everyone

[Plotinus]

Prodded NotAJumbleOfNumbers. They have another (expired on 2019-12-21 04:10:40) to move before I start looking for a replacement.

[Plotinus]

Finally occurred to me to edit the thread title.

Feel free to PM your friends who might be interested

[Plotinus]

yay, welcome! It's now Not_Mafia's turn.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two

Analysis (Not_Mafia, Micc) has 16 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 32, 20, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [13, 35, 300] contains a treble letter when written in Roman numerals

Algebra (StrangerCoug, Jackal711) has 9 points and:

• [1, 3, 5, 6, 9] {
  n < 10
  } single digit numbers }

It is

StrangerCoug'ss

turn

Yay, we're back in session

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two

Analysis (Not_Mafia, Micc) has 16 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [20, 35, 79] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 32, 20, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [13, 35, 300] contains a treble letter when written in Roman numerals

Algebra (StrangerCoug, Jackal711) has 9 points and:

• [1, 3, 5, 6, 9] {
  n < 10
  } single digit numbers }
  
• [90, 97, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square

It is

Micc's

turn

[Plotinus]

Micc has been prodded and has (expired on 2020-02-27 08:05:00) to go before I start looking for a replacement.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }

Analysis (Not_Mafia, Micc) has 25 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [16, 32, 20, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [13, 35, 300] contains a treble letter when written in Roman numerals

Algebra (StrangerCoug, Jackal711) has 9 points and:

• [1, 3, 5, 6, 9] {
  n < 10
  } single digit numbers }
  
• [90, 97, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square

It is

Jackal711's

turn

nice one!

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime

Analysis (Not_Mafia, Micc) has 25 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals

Algebra (StrangerCoug, Jackal711) has 16 points and:

• [1, 3, 5, 6, 9] {
  n < 10
  } single digit numbers }
  
• [90, 97, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square

It is

Not_Mafia's

turn

All four of you have posted now since the replacement. I'm glad we're all still here! It would have been sad if Not_Mafia had replaced in only to find the rest of you missing after all this time.

[Plotinus]

StrangerCoug has submitted his turn by PM:

Play 3, 17, 64, 128, 729 as powers of primes on my turn (the first powers are intended to be allowed).

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }

Analysis (Not_Mafia, Micc) has 32 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals

Algebra (StrangerCoug, Jackal711) has 16 points and:

• [90, 97, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [3, 17, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)

It is

Micc's

turn

[Plotinus]

Jackal711 has submitted his turn by PM:

My next move, assuming Micc's play doesn't steal the sequence, will be to play [5, 16, 25] to score

powers of primes (1st and higher)

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)

Analysis (Not_Mafia, Micc) has 32 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [12, 34, 48] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.

Algebra (StrangerCoug, Jackal711) has 24 points and:

• [90, 97, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square

It is

Not_Mafia's

turn

[Plotinus]

You don't have an 8 but I think this is a typo on a number that you do have, so you can edit your turn if you like

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.

Analysis (Not_Mafia, Micc) has 40 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals

Algebra (StrangerCoug, Jackal711) has 24 points and:

• [90, 97, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square

It is

StrangerCoug's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.

[10, 88, 90, 97, 100, 225, 441] {

n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ

} digit sum is a perfect square

Analysis (Not_Mafia, Micc) has 40 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine

Algebra (StrangerCoug, Jackal711) has 31 points and:

It is

Jackal711's

turn

Fixed the powers of primes sequence, thanks StrangerCoug!

Jackal711 is V/LA until the end of Sunday. I'll prod him before I go to bed on Monday night if he hasn't posted before then.

[Plotinus]

Just got a PM from Jackal!

Play [19, 46, 64, 361] as

numbers whose digit sum equals 10

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.

[10, 88, 90, 97, 100, 225, 441] {

n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ

} digit sum is a perfect square

Analysis (Not_Mafia, Micc) has 40 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine

Algebra (StrangerCoug, Jackal711) has 31 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10

It is

Not_Mafia's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.

[10, 88, 90, 97, 100, 225, 441] {

n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ

} digit sum is a perfect square

Analysis (Not_Mafia, Micc) has 40 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero

Algebra (StrangerCoug, Jackal711) has 31 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10

It is

StrangerCoug's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square

Analysis (Not_Mafia, Micc) has 40 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero

Algebra (StrangerCoug, Jackal711) has 31 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [3, 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers

It is

Micc's

turn

Fixed the powers of primes sequence

No you haven't

I fixed it in one post which wasn't the most recent one so then I quoted the one that still had an error in it a dozen times. But I've fixed all of them now, thanks.

[Plotinus]

There's a few reasons to allow it or disallow it.

In support of

allowing

it, a slight amendment to the rule meets the "less than 1% of hands generate a sequence like this criteria": 3 <= n <= 27.

Spoiler: details

the meta rule for this one is "seven numbers that are within 27 of each other", or "within 25 of each other" if the rule is ammended to "numbers between 3 and 27 inclusive". I wasn't sure how often this would give a bingo over all possible hands, so I tried to figure it out with this python script, which overcounts by a bit because it doesn't mind hands that have duplicate cards, but in practice you can get rid of duplicates pretty easily over the course of a couple turns and it makes the math a lot easier:

Code: Select all

from scipy.special import comb # from https://pypi.org/project/scipy/
import bisect

def countHands(arr, spread):
    totalHands = comb(len(arr), 7)
    low = 0
    high = spread
    counter = 0
    while high < arr[-1]:
        hand = bisect.bisect_right(arr, high) - bisect.bisect_left(arr, low)
        if hand >= 7:
            counter += comb(hand, 7)
        high += 1
        low += 1
        if high % 100 == 0:
            print(f'{high}: {counter}')
    return counter/totalHands

deck = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 23, 23, 24, 24, 25, 25, 25, 26, 27, 27, 27, 27, 28, 28, 28, 29, 29, 30, 30, 30, 31, 32, 32, 33, 34, 34, 35, 35, 35, 35, 36, 36, 36, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 45, 46, 47, 48, 49, 49, 49, 50, 50, 51, 52, 53, 54, 55, 55, 55, 56, 56, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 64, 64, 64, 65, 66, 67, 68, 69, 70, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 81, 81, 82, 83, 84, 84, 85, 86, 87, 88, 89, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 100, 100, 120, 121, 125, 128, 144, 165, 169, 196, 200, 216, 220, 225, 243, 250, 256, 256, 289, 300, 324, 343, 361, 400, 400, 441, 484, 500, 500, 512, 512, 529, 576, 600, 625, 676, 700, 729, 729, 729, 750, 784, 800, 841, 900, 900, 961, 1000, 1000, 1000]

print(countHands(deck, 27))

So it steps through the deck, first counting the combinations of the numbers being in the range 1 to 27, then 2 to 28, then 3 to 29, etc, and then it adds those all up and divides by the total number of ways of drawing 7 cards from this deck.

The results for 27 is 0.01248515104786979 which is over the 1% line. The result for 25 is 0.008847709943621824, which is below it. 25 turns out to be the maximum spread that keeps us under 1%.

In support of

disallowing

it, there's the other rule: "A sequence is a set of numbers that all have something in common with each other. Numbers may not be excluded from sequences they would naturally belong to, for example "primes except 7".

In support of

allowing

it, I did previously allow single digit numbers which is "numbers that are less than 10", so there is some precedent for excluding numbers from the ends of sequences, and we've previously ruled in favour of letting sequences exclude 1 from the end, for example powers of primes excluded p⁰ = 1.

In support of

disallowing

it, having only a single digit in base 10, fits the "something in common with each other" criteria much more clearly than "between 3 and 27" does. And there's a lot of precedent in general that disallowing 1 from things as a special case: 1 is disallowed from the series of primes by mathematical convention, just because it makes a bunch of things easier, even though it is divisible by only 1 and itself. It used to be included but people wound up having to say "the primes except for 1" too often and they got tired of it and kicked it out of the primes.

Man, I don't know. What do you guys think? Does allowing sequences of this type make the game more fun or less fun? I guess I'm leaning slightly against but I'll listen to other viewpoints.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square

Analysis (Not_Mafia, Micc) has 40 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [12, 20, 21, 27] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2

Algebra (StrangerCoug, Jackal711) has 31 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [3, 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers

It is

Jackal711's

turn

Sorry, Micc! Jackal711 is V/LA until Sunday night. I'll prod Jackal on Monday night if he hasn't posted by then.

[Plotinus]

Prodded Jackal, he has another (expired on 2020-03-03 15:54:06) or so to go before I start looking for a replacement.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square

Analysis (Not_Mafia, Micc) has 40 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [12, 20, 21, 27] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2

Algebra (StrangerCoug, Jackal711) has 31 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [1, 2, 3, 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers

It is

Not_Mafia's

turn

Get well soon, Jackal!

[Plotinus]

Prodded Not_Mafia, who has another (expired on 2020-03-05 04:12:53) before I start looking for a replacement.

[Plotinus]

StrangerCoug has submitted his hand by PM:

84, 56, 3, 29, 93, 5, 70: Numbers whose digital root is prime

Of the 139 numbers that appear at least once in the deck, 57 (less than half of them) have prime digital roots: 2, 3, 5, 7, 11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 39, 41, 43, 47, 48, 50, 52, 56, 57, 59, 61, 65, 66, 68, 70, 74, 75, 77, 79, 83, 84, 86, 88, 92, 93, 95, 97, 120, 128, 165, 169, 196, 200, 250, 300, 484, 500, 529, 700, 750, and 961.

It can be shown that finding whether a number has a prime digital root is equivalent to finding whether it leaves a prime remainder when divided by 9; I leave that to the article to save space here.

I like this, very cool. 109 of the 272 cards in the deck, 40% meet this criteria, and 44% of the numbers <= 100 meet the criteria, so it's allowed by either of those rules. And I can't think of any broader meta rules that would be a problem ("does anything interesting happens when you examine various congruency classes" has been previously determined to be all right).

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime

Analysis (Not_Mafia, Micc) has 47 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [12, 20, 21, 27] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2

Algebra (StrangerCoug, Jackal711) has 38 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10

It is

Micc's

turn

[Plotinus]

Prodded Micc. He has (expired on 2020-03-06 04:38:12) to post before I start looking for a replacement.

[Plotinus]

Micc has submitted his turn by PM

22, 3, 8, 65, 51, 64, 12 are all divisible by their last digit.

I believe this works (just barely) as a bingo because only 50 of the 100 cards 1-100 and 69 of the 140 cards in the deck meet the criteria.

Spoiler: Divisible by last digit

1 24 55 85 225
2 25 61 88 243
3 31 62 91 324
4 32 63 92 361
5 33 64 93 441
6 35 65 95 484
7 36 66 96 512
8 41 71 99 576
9 42 72 121 625
11 44 75 125 729
12 45 77 128 784
15 48 81 144 841
21 51 82 165 961
22 52 84 216

This looks good to me, and it's interesting because naively i'd expect it to be less than about half, but you get all the one digit numbers, all the numbers ending in one, two, five, and then all the numbers account for only 11 other options.

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit

Analysis (Not_Mafia, Micc) has 54 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [12, 20, 21, 27] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2

Algebra (StrangerCoug, Jackal711) has 38 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10

It is

Jackal711's

turn

[Plotinus]

Prodded Jackal a bit late, sorry. He has another (expired on 2020-03-08 13:13:58) to go before I start looking for a replacement.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit

Analysis (Not_Mafia, Micc) has 54 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [12, 20, 21, 27] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2

Algebra (StrangerCoug, Jackal711) has 38 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [15, 50, 75] {
  5n
  } numbers that are evenly divisible by 5

It is

Not_Mafia's

turn

[Plotinus]

Evenly divisible means without a remainder, so all multiples of five.

[Plotinus]

StrangerCoug has submitted his turn by PM:

Play 100, 50, 35, and 250 to complete the multiples of 5 if it's still there on my turn.

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5

Analysis (Not_Mafia, Micc) has 61 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero

Algebra (StrangerCoug, Jackal711) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10

It is

Micc's

turn

[Plotinus]

Prodded Micc. He has another (expired on 2020-03-11 09:44:47) to go before I start looking for a replacement

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5

Analysis (Not_Mafia, Micc) has 61 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, Jackal711) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10

It is

Jackal711's

turn

[Plotinus]

prodded Jackal711, who has another (expired on 2020-03-12 13:47:55) to go before I start looking for a replacement

[Plotinus]

Jackal has requested replacement due to internet issues. He's hoping to rejoin us in round 3. The scummy this game got created some interest in the game so I have a couple people lined up. We won't have to wait 2 months this time.

[Plotinus]

skitter30 is replacing jackal711! 324 doesn't seem to obey the rule of that sequence as I understand it, so skitter30 can change their move or re-explain the rule.

[Plotinus]

Unfortunately we've had an equivalent sequence already, in the completed sequences spoiler:

[6, 21, 27, 42, 69, 72, 78, 165, 576] { 3n } numbers whose sum is a multiple of 3


You can go again, it's okay

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5

Analysis (Not_Mafia, Micc) has 61 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, skitter30) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [18, 99, 324] {
  k² | n; k ∈ ℤ
  } numbers with a factor that is a square number

It is

Not_Mafia's

turn

That did it! It's hard jumping in midstream, thanks for joining us.

[Plotinus]

For the sake of interestingness, let's limit skitter's sequence to factors greater than one.

Strangercoug has submitted his turn by PM

6, 15, 44: Numbers divisible by each of its digits

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5

Analysis (Not_Mafia, Micc) has 61 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [13, 28, 35, 300] contains a treble letter when written in Roman numerals
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, skitter30) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [18, 99, 324] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [6, 15, 44] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits

Cards left in deck: 112

It is

Micc's

turn

[Plotinus]

I had two submissions by PM from other players who wanted to complete that sequence. Good job, Micc!

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals

Analysis (Not_Mafia, Micc) has 69 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, skitter30) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [18, 99, 324] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [6, 15, 44] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits

Cards left in deck: 107

It is

skitter30's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals

Analysis (Not_Mafia, Micc) has 69 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, skitter30) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [18, 99, 324] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [6, 15, 44] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [17, 33, 73] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8

Cards left in deck: 105

It is

Not_Mafia's

turn

Play entire hand on "numbers with a factor that is a square number", if that's not a valid move then add 28 to the roman numerals

98, 28, 343, 63, 289 to complete triple letters when written in roman numerals.

28 is a duplicate here

Fixed, I've removed a point and the 28 from the list. I've already given Micc his new cards but I'll reshuffle that 28 back into the deck so it can be drawn again.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits

Analysis (Not_Mafia, Micc) has 76 points and:

• [16, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, skitter30) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [18, 99, 324] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [17, 33, 73] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8

Cards left in deck: 101

It is

StrangerCoug's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits

Analysis (Not_Mafia, Micc) has 76 points and:

• [16, 56, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, skitter30) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [18, 99, 324] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [17, 33, 73] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8

Cards left in deck: 100

It is

Micc's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number

Analysis (Not_Mafia, Micc) has 84 points and:

• [16, 56, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, skitter30) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8

Cards left in deck: 95

It is

skitter30's

turn

[Plotinus]

You're right, I did. I remember looking carefully at both of those sequences and trying to pick the right one and still got it wrong. Fixed it, I hope.

We recently discussed numbers less than 27 and ruled against it for being uninteresting / not having something meaningful in common. Does framing "n <= 31" in terms of days of the year make it interesting enough to rule this one differently?

I think some day of the year sequences could be interesting, like intercalary days, prime days of prime months, palindromic dates, [nationality] name days for names starting with [letter]. Months with 31 days would have a stronger case for being interesting than January but would also be easier to steal.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number

Analysis (Not_Mafia, Micc) has 84 points and:

• [16, 56, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600, 60] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [25, 49, 100] {
  n²
  } square numbers

Algebra (StrangerCoug, skitter30) has 45 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8

Cards left in deck: 94

It is

Not_Mafia's

turn

[Plotinus]

Strangercoug has submitted his turn by PM, ironically beating somebody else to the punch:

Complete the squares with 256, 400, 529, and 841 if I'm not unlucky enough to be beaten to the punch again.

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers

Analysis (Not_Mafia, Micc) has 84 points and:

• [16, 56, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600, 60] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [1, 15, 17, 18, 1000] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one

Algebra (StrangerCoug, skitter30) has 52 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8

Cards left in deck: 81

It is

Micc's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600, 60] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero

Algebra (StrangerCoug, skitter30) has 52 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8

Cards left in deck: 78

It is

skitter30's

turn

There's already a 1 in the sequence, so you keep that card. Since this still completes the sequence you just get 8 points instead of 9

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [10, 40, 600, 60] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero

Algebra (StrangerCoug, skitter30) has 52 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13

Cards left in deck: 78

It is

Not_Mafia's

turn

[Plotinus]

You don't have a 1. You can resubmit your turn

[Plotinus]

fixed

[Plotinus]

StrangerCoug has submitted his turn by PM:

Play 20, 800, 1000 to finish the numbers ending in 0.

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13

Cards left in deck: 74

It is

Micc's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [83, 89, 400] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13

Cards left in deck: 71

It is

skitter30's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 196, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [83, 89, 400] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13

Cards left in deck: 70

It is

Not_Mafia's

turn

[Plotinus]

Not_Mafia has been prodded and has (expired on 2020-03-21 06:21:23) to post before I start looking for a replacement

[Plotinus]

Subject: Sequencer | skitter30's Turn

83, 89, and 400 for their first digit being a power of 2.

1, 41, 45, 484 to finish this off.

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 196, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13

Cards left in deck: 66

It is

Micc's

turn

[Plotinus]

Fixed, thanks for reminding me

[Plotinus]

Micc has been prodded and has another (expired on 2020-03-23 02:33:28) to go before I start looking for a replacement

[Plotinus]

skitter has submitted her turn by PM

15, 57, and 21 as numbers that are 3*prime number

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 196, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [30, 58, 87] 2 digit numbers where one digit is prime and the other is not

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number

Cards left in deck: 60

It is

Not_Mafia's

turn

[Plotinus]

[*][1, 15, 17, 18, 1000, 11, 16, 19] {

n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0

} numbers that start with one

We had that one already. You can go again

[Plotinus]

I didn't notice this when you posted earlier, but those are only two numbers and new sequences have at least 3

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 196, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [30, 58, 87] 2 digit numbers where one digit is prime and the other is not
  
• [5, 10, 18] {
  n is odd (mod 9)
  numbers whose digital root is odd

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number

Cards left in deck: 57

It is

StrangerCoug's

turn

That works!

[Plotinus]

Strangercoug, I think you are looking at a previous version of your hand, the most recent hand I sent you had the title "Re: Sequencer | Micc's Turn" instead of "Sequencer Hand" because I forgot to change it. You can go again, and then Micc can confirm if that's still his move.

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 196, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [30, 58, 87] 2 digit numbers where one digit is prime and the other is not
  
• [5, 10, 18] {
  n is odd (mod 9)
  numbers whose digital root is odd

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number
  
• [29, 71 89] {
  n is prime
  } primes

Cards left in deck: 54

It is

Miccs

turn

Micc, is that still your move?

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2

Analysis (Not_Mafia, Micc) has 92 points and:

• [16, 56, 76, 196, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [30, 58, 87] 2 digit numbers where one digit is prime and the other is not
  
• [5, 10, 18] {
  n is odd (mod 9)
  numbers whose digital root is odd
  
• [1, 125, 216] at least one digit is 1

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number
  
• [29, 71 89] {
  n is prime
  } primes

Cards left in deck: 51

It is

skitter30's

turn

[Plotinus]

skitter30 has been prodded and would have (expired on 2020-03-29 16:08:24) to make her move because she's V/LA on Saturdays.

[Plotinus]

Micc has sent his turn by PM:

If possible my next play will be to add 3, 7, 11 and 13 to complete the primes sequence.

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2
  
• [5, 10, 18, 43, 30, 59, 81, 27, 45] {
  n is odd (mod 9)
  numbers whose digital root is odd
  
• [29, 61, 71, 89, 3, 7, 11, 13] {
  n is prime
  } primes

Analysis (Not_Mafia, Micc) has 109 points and:

• [16, 56, 76, 196, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [30, 58, 87] 2 digit numbers where one digit is prime and the other is not
  
• [1, 125, 216] at least one digit is 1

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [1, 17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number

Cards left in deck: 39

It is

skitter30's

turn again!

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2
  
• [5, 10, 18, 43, 30, 59, 81, 27, 45] {
  n is odd (mod 9)
  numbers whose digital root is odd
  
• [29, 61, 71, 89, 3, 7, 11, 13] {
  n is prime
  } primes

Analysis (Not_Mafia, Micc) has 109 points and:

• [16, 56, 76, 196, 676] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [30, 58, 87, 31, 47] 2 digit numbers where one digit is prime and the other is not
  
• [1, 125, 216] at least one digit is 1

Algebra (StrangerCoug, skitter30) has 59 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [1, 17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number

Cards left in deck: 37

It is

Not_Mafia's

turn

[Plotinus]

StrangerCoug has submitted his turn by PM

Finish off the two-digit numbers with one prime and one nonprime digit with 21, 34, and 63.

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2
  
• [5, 10, 18, 43, 30, 59, 81, 27, 45] {
  n is odd (mod 9)
  numbers whose digital root is odd
  
• [29, 61, 71, 89, 3, 7, 11, 13] {
  n is prime
  } primes
  
• [16, 56, 76, 196, 676, 6, 36] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [30, 58, 87, 31, 47, 21, 34, 63] 2 digit numbers where one digit is prime and the other is not

Analysis (Not_Mafia, Micc) has 116 points and:

• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [1, 125, 216] at least one digit is 1

Algebra (StrangerCoug, skitter30) has 67 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [1, 17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number

Cards left in deck: 32

It is

Micc's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2
  
• [5, 10, 18, 43, 30, 59, 81, 27, 45] {
  n is odd (mod 9)
  numbers whose digital root is odd
  
• [29, 61, 71, 89, 3, 7, 11, 13] {
  n is prime
  } primes
  
• [16, 56, 76, 196, 676, 6, 36] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [30, 58, 87, 31, 47, 21, 34, 63] 2 digit numbers where one digit is prime and the other is not

Analysis (Not_Mafia, Micc) has 116 points and:

• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [1, 125, 216] at least one digit is 1
  
• [4, 21, 30] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i | 12
  } sum of the digits is a factor of 12

Algebra (StrangerCoug, skitter30) has 67 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [1, 17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number

Cards left in deck: 29

It is

skitter30's

turn

[Plotinus]

Spoiler: Completed sequences

• [1, 2, 3, 4, 5, 10, 39, 55] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a₀ | n; a_i ≥ 0; d > 0
  } numbers that are divisible by their first digit
  
• [6, 21, 27, 42, 69, 72, 78, 165, 576] {
  3n
  } numbers whose sum is a multiple of 3
  
• [4, 8, 9, 15, 20, 32, 66, 80] {
  n = 2ⁱ ± 2^j
  } numbers that can be written as the sum or difference of two powers of two
  
• [20, 35, 79, 169, 8, 53, 2, 35, 13 ] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 2^k; a_i ≥ 0; d > 0; k ∈ ℤ
  } sum of the digits is a power of 2 }
  
• [16, 20, 25, 32, 94, 120, 200] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i is prime; φ(n) < n - 1; a_i ≥ 0, d > 0; k ∈ ℤ
  } composite numbers whose digit sum is prime
  
• [1, 3, 4, 5, 6, 7, 9] {
  n < 10
  } single digit numbers }
  
• [3, 5, 16, 17, 25, 64, 128, 729] {
  p^k, p is prime, k ≥ 1
  } powers of primes (1st and higher)
  
• [12, 34, 48, 729, 86, 81, 64, 24] numbers that have two or more tall letters (b, d, f, h, k, l, t) when written as a word.
  
• [10, 88, 90, 97, 100, 225, 441] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = m²; a_i ≥ 0, d, m > 0; k ∈ ℤ
  } digit sum is a perfect square
  
• [1, 2, 3, 7, 9 11, 77]{
  str(n)[::-1] == str(n)
  } Palindromic numbers
  
• [84, 56, 3, 29, 93, 5, 70] {
  n % 9 ∈ {2, 3, 5, 7}
  Numbers who digital root is prime
  
• [22, 3, 8, 65, 51, 64, 12] {
  i % (i % 10) == 0 }
  Numbers divisible by their last digit
  
• [12, 20, 21, 27, 512, 52, 24] {
  n = a * 100 + b * 10 + c, with 2 ∈ {a, b, c}
  } Numbers where at least one digit is 2
  
• [15, 50, 75, 100, 50, 35, 250] {
  5n
  } numbers that are evenly divisible by 5
  
• [13, 28, 35, 300, 98 343, 63, 289] contains a treble letter when written in Roman numerals
  
• [6, 15, 44, 1, 36, 55, 4] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i | n
  } Numbers divisible by each of its digits
  
• [18, 99, 324, 500, 8, 45, 81, 700] {
  k² | n; k > 1
  } numbers with a factor that is a square number
  
• [25, 49, 100, 256, 400, 529, 841] {
  n²
  } square numbers
  
• [1, 15, 17, 18, 1000, 11, 16, 19] {
  n = 1 * 10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with one
  
• [10, 40, 600, 60, 20, 800, 1000] {
  n = 10 * a₀*10^d + a₁10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that end with zero
  
• [83, 89, 400, 1, 41, 45, 484] {
  n = 2^x*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d, x ≥ 0
  } first digit is a power of 2
  
• [5, 10, 18, 43, 30, 59, 81, 27, 45] {
  n is odd (mod 9)
  numbers whose digital root is odd
  
• [29, 61, 71, 89, 3, 7, 11, 13] {
  n is prime
  } primes
  
• [16, 56, 76, 196, 676, 6, 36] {
  n ≡ 6 (mod 10)
  } numbers that end in 6
  
• [30, 58, 87, 31, 47, 21, 34, 63] 2 digit numbers where one digit is prime and the other is not
  
• [4, 21, 30, 2, 84, 1, 3] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i | 12
  } sum of the digits is a factor of 12

Analysis (Not_Mafia, Micc) has 116 points and:

• [91, 900, 961] {
  n = 9*10^d + a₀10^d-1 + ... + a_d-110⁰ with a_i, d ≥ 0
  } numbers that start with nine
  
• [1, 125, 216] at least one digit is 1
  
• [7, 8, 42, 1000] { {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_d <= a₀
  } last digit is less than or equal to the first digit

Algebra (StrangerCoug, skitter30) has 74 points and:

• [19, 46, 64, 361] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = 10; a_i ≥ 0, d > 0; k ∈ ℤ
  } numbers whose digit sum equals 10
  
• [1, 17, 33, 73, 729] {
  n ≡ 1 (mod 8)
  } numbers that are congruent to 1 mod 8
  
• [10, 23, 36, 49] {
  n ≡ 10 (mod 13)
  } numbers congruent to 10 mod 13
  
• [15, 57, 21] {
  3 * p, p is prime
  } numbers that are 3 times a prime number

Cards left in deck: 21

It is

StrangerCoug's

turn

I'd like to accept blue numbers because I believe you that the numbers are blue but I think it needs to be a sequence that anyone could theoretically learn to apply the rule themselves. If there's something about the shape of the numbers that tends to make them blue or something about the sound the number makes in a language you speak that makes them blue, like "numbers with 2s or 7s in them" then we can roll back and accept the bingo if less than half the cards are blue.

[Plotinus]

I'm assuming it's synaesthesia, but most people with colour-grapheme synaesthesia see different colours from each other