Sequencer - Needs 1 replacement

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Plotinus
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Post Post #0  (ISO)  » Sun Sep 29, 2019 12:09 pm

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:
  • NotAJumbleofNumbers, analyst
  • StrangerCoug, algebraist
  • Micc, analyst
  • Jackal711 replaces Yimmy, algebraist

Replacements:

Previous winners:
  • Implosion

How to join:
Type: "/in"
You may in after the game starts, I'll either add you to the replacements or just start you immediately -- the stealing mechanic should make it easy enough to catch up in points

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, you may pass
  • If all players pass consecutively the game is over and whoever has the most points wins.
  • If the deck runs out of cards I will ask if people want me to reshuffle it and keep going or start a new game. We can also discuss shuffling in a new deck.

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".
  • 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):

  • 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.)

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 skip you.
  • If you're skipped twice in a row I'll put you as inactive and autoskipped until you return. If someone is waiting to pay they will be given a new hand
  • Let us know if you're going to be V/LA or want to go inactive.
  • If you are inactive you must /re-in and be put at the end of the waiting list.

    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

    (with 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.
Last edited by Plotinus on Thu Dec 26, 2019 3:03 am, edited 103 times in total.
Modding checklists | Sequencer is in Game 2 and needs 1 replacement | Turtle is Over | Cellular Automata is Over

implosion
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Post Post #1  (ISO)  » Sun Sep 29, 2019 12:31 pm

Sure.

Plotinus
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Post Post #2  (ISO)  » Sun Sep 29, 2019 1:10 pm

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.
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Felissan
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Post Post #3  (ISO)  » Sun Sep 29, 2019 1:41 pm

/in
"Dammit Felissan, making someone lose the game is NOT NICE" - DeathRowKitty 2016
"Also, the me in your signature just made the me in this thread lose the game and I'm not sure how to feel about this." - DeathRowKitty 2018
I believe in you, Miss Zirconium! <3

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Plotinus
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Post Post #4  (ISO)  » Sun Sep 29, 2019 1:46 pm

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

It is McMenno's turn
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NotMySpamAccount
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Post Post #5  (ISO)  » Sun Sep 29, 2019 1:52 pm

did someone say recreational math? /in
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StrangerCoug
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Post Post #6  (ISO)  » Sun Sep 29, 2019 1:59 pm

Sure, I'll try this out.

/in
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Plotinus
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Post Post #7  (ISO)  » Sun Sep 29, 2019 2:25 pm

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.
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McMenno
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Post Post #8  (ISO)  » Sun Sep 29, 2019 2:55 pm

{7n}: start with [14, 35, 343]

yeah not great
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Plotinus
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Post Post #9  (ISO)  » Sun Sep 29, 2019 2:59 pm

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

It is implosion's turn
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implosion
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Post Post #10  (ISO)  » Sun Sep 29, 2019 5:02 pm

Numbers with at least two digits, all of which are odd: [15, 53, 91]

DeathRowKitty
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Post Post #11  (ISO)  » Sun Sep 29, 2019 7:49 pm

Eventually this game is going to turn into people giving ridiculous sequence definitions or creating Scrabble-related sequences in some misguided attempt to fit the theme of this game. In fact, why don't I get us started on that?

[8, 18, 34]

My sequence is numbers that are the sum of the Scrabble point values of the (not-necessarily distinct) letters in the American English* spelling of numbers in the deck for this game.

8 is the number of points for 3 (three)
18 is the number of points for 22 (twenty-two)
34 is the number of points for 512 (five hundred twelve)



*American English by convention doesn't include "and" in numbers (e.g. 101 would be written as "one hundred one" rather than "one hundred and one")
kero kero

Plotinus
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Post Post #12  (ISO)  » Mon Sep 30, 2019 2:03 am

McMenno has:
  • [14, 35, 343] { 7n } divisible by 7
Implosion has:
  • [15, 53, 91] { n = a010d + a110d-1 + ... + ad100 with ai ≡ 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
Last edited by Plotinus on Mon Sep 30, 2019 2:56 am, edited 2 times in total.
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implosion
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Post Post #13  (ISO)  » Mon Sep 30, 2019 2:17 am

To clarify my sequence, "all of which are odd" was intended to mean "all of the digits are odd" (and it was intended that there could be more than two digits, but looking at the deck, that distinction is irrelevant).

Plotinus
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Post Post #14  (ISO)  » Mon Sep 30, 2019 2:56 am

Thanks for clarifying. I've edited my description of it and written a new closed form for it
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Plotinus
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Post Post #15  (ISO)  » Mon Sep 30, 2019 3:29 am

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.
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Plotinus
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Post Post #16  (ISO)  » Mon Sep 30, 2019 3:48 am

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.
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Post Post #17  (ISO)  » Mon Sep 30, 2019 4:01 am

I play 4, 20 and 36, as numbers that can be written as 16n + 4.
"Dammit Felissan, making someone lose the game is NOT NICE" - DeathRowKitty 2016
"Also, the me in your signature just made the me in this thread lose the game and I'm not sure how to feel about this." - DeathRowKitty 2018
I believe in you, Miss Zirconium! <3

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Plotinus
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Post Post #18  (ISO)  » Mon Sep 30, 2019 6:23 am

McMenno has:
  • [14, 35, 343] { 7n } divisible by 7
Implosion has:
  • [15, 53, 91] { n = a010d + a110d-1 + ... + ad100 with ai ≡ 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
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NotMySpamAccount
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Post Post #19  (ISO)  » Mon Sep 30, 2019 10:24 am

Add 73 and 13 to Implosion's odd digited 2 digits.

Play 2, 4, 32, as powers of 2.

Add 68 to Felissan's {16n+4}.

Add 14 to DRK's scrabble sequence (sixteen = 1+1+8+1+1+1+1=14)

That's all 7.
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Post Post #20  (ISO)  » Mon Sep 30, 2019 11:02 am

you can only do one thing a turn
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Plotinus
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Post Post #21  (ISO)  » Mon Sep 30, 2019 11:06 am

In post 0, Plotinus wrote:
  • 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.
Modding checklists | Sequencer is in Game 2 and needs 1 replacement | Turtle is Over | Cellular Automata is Over

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Post Post #22  (ISO)  » Mon Sep 30, 2019 10:23 pm

Oh ok. I should really get out of the habit of treating "or" as a computer would.

Play 2, 4, 32, as powers of 2.
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Plotinus
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Post Post #23  (ISO)  » Tue Oct 01, 2019 12:51 am

McMenno has:
  • [14, 35, 343] { 7n } divisible by 7
Implosion has:
  • [15, 53, 91] { n = a010d + a110d-1 + ... + ad100 with ai ≡ 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
Modding checklists | Sequencer is in Game 2 and needs 1 replacement | Turtle is Over | Cellular Automata is Over

StrangerCoug
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Post Post #24  (ISO)  » Tue Oct 01, 2019 10:20 am

Play [17, 19, 43] as prime numbers
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