Sequencer | StrangerCoug's turn

Plotinus · Post Post #225 (ISO) » Tue Nov 12, 2019 7:56 am

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.

DeathRowKitty · Post Post #226 (ISO) » Tue Nov 12, 2019 6:40 pm

[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

Plotinus · Post Post #227 (ISO) » Tue Nov 12, 2019 7:33 pm

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.

DeathRowKitty · Post Post #228 (ISO) » Wed Nov 13, 2019 5:03 am

hi new popsofctown!

Yimmy · Post Post #229 (ISO) » Wed Nov 13, 2019 6:36 am

hi guys. im yimmy
math is fun but also hard so don't expect riveting sequences from me as of now
Numbers such that when all individual digits are added together the sum is equal to a power of two: 2, 8, 20, 512

Plotinus · Post Post #230 (ISO) » Wed Nov 13, 2019 7:09 am

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

StrangerCoug · Post Post #231 (ISO) » Wed Nov 13, 2019 8:48 am

Play 1, 5, 35, 90 to finish off the coins one

From what I've tested, you

CAN

make change from a dollar with exactly 35 coins, but the way I found resorts to using all three of dimes, nickels, and pennies (1 quarter, 1 dime, 8 nickels, 25 pennies), and DRK's rule requires excluding at least one of those coins to be ruled out. I have been unable to find a way to do it without using all four denominations, but my check hasn't been completely thorough, so if someone can prove 35 doesn't belong I'll take it back and play something else.

Plotinus · Post Post #232 (ISO) » Wed Nov 13, 2019 9:14 am

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.

DeathRowKitty · Post Post #233 (ISO) » Wed Nov 13, 2019 9:15 am

Yeah, SC's numbers all work. Numbers that could not be put into that sequence are those n with 4 ≤ n ≤ 100 that satisfy at least one of the following 3 properties:
1) n ≡ 1 (mod 3)
2) n ≡ 0 (mod 4)
3) 6 ≤ n ≤ 19

implosion · Post Post #234 (ISO) » Wed Nov 13, 2019 12:49 pm

Similar to, but slightly broader than my previous sequence: numbers where any consecutive digits differ by *at most* 1.

99, 89, 11.

DeathRowKitty · Post Post #235 (ISO) » Wed Nov 13, 2019 2:52 pm

I never got new cards after my last turn btw

Plotinus · Post Post #236 (ISO) » Wed Nov 13, 2019 7:25 pm

In post 235, DeathRowKitty wrote: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 · Post Post #237 (ISO) » Wed Nov 13, 2019 8:07 pm

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?

StrangerCoug · Post Post #238 (ISO) » Thu Nov 14, 2019 7:18 am

In post 237, Plotinus wrote: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?

I'm good with it.

Plotinus · Post Post #239 (ISO) » Thu Nov 14, 2019 8:17 pm

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.

DeathRowKitty · Post Post #240 (ISO) » Fri Nov 15, 2019 4:06 pm

Forgot it was my turn, sorry.

[10,45,86] to numbers with 2 prime factors

Plotinus · Post Post #241 (ISO) » Fri Nov 15, 2019 7:43 pm

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 · Post Post #242 (ISO) » Sat Nov 16, 2019 10:53 pm

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.

Yimmy · Post Post #243 (ISO) » Sun Nov 17, 2019 3:42 pm

Numbers that are prime when the digits are reversed: 2, 14, 17

StrangerCoug · Post Post #244 (ISO) » Sun Nov 17, 2019 5:31 pm

1, 11, 1000 to finish off numbers whose digit sum is a power of 2

implosion · Post Post #245 (ISO) » Sun Nov 17, 2019 5:52 pm

Finish off my last sequence w/ consecutive digit differences with 1,2,8,21

Plotinus · Post Post #246 (ISO) » Sun Nov 17, 2019 7:56 pm

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.

StrangerCoug · Post Post #247 (ISO) » Mon Nov 18, 2019 5:12 am

So I think we've all got the hang of this and can play Game 2 with a slightly bigger deck.

Spoiler: My proposed Game 2 deck

This consists of the following "decks" shuffled together:

The Game 1 deck
Every multiple of 100 less than or equal to 1,000
Every multiple of 250 less than or equal to 1,000
Every square between 100 and 1,000 exclusive (since every square from 1 to 100 inclusive already appears at least twice in the Game 1 deck and √1000 is not an integer)

If a number occurs in multiple decks, it is shuffled in as many times as it occurs total in each deck.

I've been trying (without much effort, admittedly) to "reverse engineer" Plotinus's deck to see how he put it together, but the lowest number to appear once only in the Game 1 deck is 26 and every number above 100 in the Game 1 deck is at least one of a square, a power of two, or a tetrahedral number. For at least some sequences, though, he just puts in the first 10 numbers.

DeathRowKitty · Post Post #248 (ISO) » Mon Nov 18, 2019 6:50 am

pass

also the second number listed for digit-reversed primes should be 14, not 4

Plotinus · Post Post #249 (ISO) » Mon Nov 18, 2019 7:28 am

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.