Sequencer | StrangerCoug's turn

For completed/abandoned Mish Mash Games.
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Post Post #125 (ISO) » Thu Oct 24, 2019 7:38 am

Post by DeathRowKitty »

I don't know why implosion excluded 19 and 27 in particular but it's perhaps worth noting that there are no 27s remaining in the deck, so I'm not sure he gains much from excluding that particular number.
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Post Post #126 (ISO) » Thu Oct 24, 2019 8:37 am

Post by implosion »

Doing something for no reason at all seems entirely within the spirit of the game to me :p

I just wanted to inject some arbitrariness; there are still plenty of valid numbers in that sequence, so it's not as though excluding those numbers makes it unfairly difficult to steal the sequence from me. (and there was no strategy whatsoever behind the choices, as the fact that one of them is irrelevant shows). If you're worried that I did it because I was excruciatingly close to a bingo and was just trying to make it harder to steal before my next turn I can also tell you that isn't the case.
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Post Post #127 (ISO) » Thu Oct 24, 2019 1:15 pm

Post by Felissan »

I play my whole hand, [5, 7, 15, 19, 34, 45, 64] as numbers that end in a 4, 5, 7 or 9.

I guess my main issue with the game so far is that you don't have enough of an incentive to do anything else than simple, but unimpressive rules like this one if you want to play "optimally". I wonder if there's a way to fix that.
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Post Post #128 (ISO) » Thu Oct 24, 2019 1:43 pm

Post by DeathRowKitty »

Assuming your rule counts as a meta rule, it's disallowed as a bingo by the meta rule rule and I think most such silliness is in fact covered by that rule. The meta rule
those numbers that end in any of the following 4 or fewer digits
captures > 30% of possible sets of 7 numbers, while the meta rule rule caps meta rules at 1%.

Edit: I oopsied, it's actually just under 20% rather than just over 30%, but still way too high for a meta rule.
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Post Post #129 (ISO) » Thu Oct 24, 2019 7:03 pm

Post by 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?
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Post Post #130 (ISO) » Thu Oct 24, 2019 7:16 pm

Post by 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.
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Post Post #131 (ISO) » Thu Oct 24, 2019 7:21 pm

Post by implosion »

"k mod n" is always an allowable bingo under your current rules, *technically* unless n=2; if n=2 then it's actually 1/64 hands that will bingo as either "all even" or "all odd".
In general for "k mod n" the probability that a given hand will bingo for some k is 1/n6, which is well under 1% for n>2.

I actually think the "must hit less than half the cards under 100" and "no meta-rules" rules together do a really good job of stifling bingos that shouldn't happen; for instance i had a hand earlier in the game where i could have "bingoed" with the rule "0 or 1 mod 5", but i felt like I shouldn't. But in reality, the meta-rule "contains only these two congruence classes mod 5" is going to be satisfied by... at *least* (0.4)^5 = 1.024% of hands (this is assuming the first two cards in the hand are different congruence classes, and then stipulating that the remaining cards must all be in those two classes, in reality the probability is slightly higher because the first two cards could be in the same class if you're lucky but it's probably still around the 2% mark). So by the rules as they're now written, that hand is like... on the borderline of being an acceptable bingo, but is probably not. Which is exactly how I felt about having the hand on a gut level.

I think the things in the game that have the potential to be degenerate are:

-Overly-specific 5ish card plays. Consider the sequence "is one of the numbers 3, 6, 14, 19, 55, 57, or 78" when you have those seven numbers in hand, and you choose to play five of them. Extremely unlikely anyone can "snipe" the sequence from you, and you can complete it on your next turn to guarantee exactly 7 points. Is this a good play? Maybe. 7 points over two turns in a way that cannot be touched is... at least alright. But even more degenerate is just never finishing that sequence, and then doing another similar 5-card unsnipable sequence on your next turn.

Basically, there are two things at play here: sequences can be allowed to be as specific as you want, which means they can be degenerate like this. And secondly and IMO more importantly, sequences are allowed to fit as few numbers as you want. This is sort of obviously fine for a sequence like "powers of 2" (which only fits 7 cards between 1-100) but is not fine for "roots of [complicated polynomial designed to have 9 specific numbers as roots]" or the like, *even* if the cards aren't being played as a bingo. I think the arbitrariness of sequences is part of the fun of the game and I'd be a bit loath to restrict how sequences can be specified, but it might be good to add a rule for non-bingoes about, e.g., requiring that they admit at least... 20 cards in the 1-100 range, perhaps. Basically, making it so that such sequences can definitely be stolen. Maybe add a mod-determined exception for sequences that are considered sufficiently meaningful or important, like "powers of 2" or "perfect squares", or even "numbers exactly one more than a perfect square" or even more broad than that.

To a degree the game is self balancing as soon as sequences can be stolen (because you're incentivized to steal from whoever is winning), so it's kind of fine to do whatever from there.
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Post Post #132 (ISO) » Thu Oct 24, 2019 7:23 pm

Post by implosion »

Basically I think that you defined the bingo rules really well and the game just needs something analogous to prevent "bad gameplay" from being a good thing to do for non-bingos.

Another legitimate option is the 1000 blank white cards approach of "players or the mod may veto a sequence if they believe it is outside the spirit of the game", with some kind of majority requirement or the like for veto and the player probably being allowed to redo their turn since different people would have different conceptions.
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Post Post #133 (ISO) » Thu Oct 24, 2019 7:25 pm

Post by implosion »

I guess if you consider "congruent to k mod n" as a meta rule in both k and n then nothing would be a valid bingo under it because 1/64 hands fit into the k mod 2 category; but I think that's the wrong interpretation of the meta rule. I think the right interpretation is to only vary what k is because varying n significantly changes the frequency of hands that will fit it.
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Post Post #134 (ISO) » Thu Oct 24, 2019 8:26 pm

Post by 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.
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Post Post #135 (ISO) » Thu Oct 24, 2019 8:36 pm

Post by implosion »

Plot wrote: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.
Even if you look in every congruence class, so long as you exclude 2, the probability that your numbers are all in the same congruence class mod *any* modulus should be <1%. In fact, it'd be strictly less than the sum from n=3 to infinity of 1/n^6 (less because there's overlap), which wolfram alpha puts at ~0.17%.

So this is an allowable meta rule under your current rules, because it only bingoes less than 1% of hands.
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Post Post #136 (ISO) » Thu Oct 24, 2019 8:48 pm

Post by implosion »

I think moderating this game to its full potential *probably* necessarily involves subjective calls, whether those come in the form of mod intervention or some veto system. For instance, I think if there is a rule that non-bingo hands must allow at least 20 numbers, then you should disallow what i did of arbitrarily excluding numbers (because the "optimal" play might always involve arbitrarily excluding all but 20 numbers from anything you play). But this gets messy because, e.g., if i come up with a rule that hits 25 numbers, and i can come up with some hyperspecific thing that kills some 5 of those numbers, then i can just get down to 20 by applying that filter. But whether such a filter is "natural" or "arbitrary" is necessarily a judgment call.

For instance, exactly 25 numbers up to 100 are congruent to 2 mod 4, so you'd optimally want to exclude 5 of them if you were playing a sequence like that. it's almost definitely fine to say something like "numbers congruent to 2 mod 4 AND that are at least 20" because this sort of is a natural thing to do?

If you're disallowing "arbitrary" disinclusions, then it is obviously not allowable to have a rule of "numbers that are congruent to 2 mod 4 OTHER THAN 14 and 18 and 22 and 74 and 78".

But drawing that line is hard. What about "numbers congruent to 2 mod 4 except for those between 19 and 38 inclusive"? What about "numbers congruent to 2 mod 4 except for those whose second digit is 6"? What about "numbers congruent to 2 mod 4 except for those whose first digit is 3 or 8"? All of these exclude exactly 5/25 of the numbers in the range 1-100, and all of them are kind of reasonable in isolation, but the fact that you can pick any of them that you want makes it kind of minimax-y because you'll *always* be able to get down to exactly 20 possible numbers that fit your sequence.

It's tricky.
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Post Post #137 (ISO) » Thu Oct 24, 2019 8:49 pm

Post by implosion »

(i'm also ranting because i find the theory of how you'd optimize the design of the rules of this game interesting, so don't mind me :p)
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Post Post #138 (ISO) » Fri Oct 25, 2019 9:43 am

Post by DeathRowKitty »

I agree with most of what implosion said.

As far as bingos are concerned, I agree that the current rules are pretty solid and agree that some degree of subjectivity helps to plug up any gaps. The statement in the rules that you should be thinking forwards rather than backwards for bingos I think covers well what kind of subjectivity you'd want - if a bingo sequence seems like it was specifically constructed awkwardly around one's hand, it had better be good or it's really no fun. Ultimately, Plotinus is a good mod who I trust not to make any sort of egregious calls, so it works out fine.
Not sure how noticeable it was, but the bingo rule actually changed mid-game in response to Plotinus judging a sequence as acceptable that the original reading of the rule probably disallowed. Both the original rule and current rule really aren't perfect (the original rule, while probably more "accurate", was also too cumbersome to expect people to deal with properly), but it's just quibbling over edge cases and trying to rules lawyer the exact line rather than allowing for mod judgment calls would make the game less fun.



For non-bingos, I'm perhaps a bit hesitant about introducing much in the way of rules. I agree that implosion's against-the-spirit-of-the-game-and-also-just-super-unfun examples shouldn't be acceptable, but I'd want to be really careful about where the line is drawn. For example, I thought Felissan was hinting in her comment for this sequence that matches exactly 7 numbers in the deck that she had the only 165 and that no one else should bother trying to complete it. At this point we know that's not actually the case, but I thought it was good strategy to put down a sequence whose completion requires a card that only you have and that it's fair to have that as a perk of having a unique card. It might be unfun with a different starting distribution that causes the situation to arise more often, but I think that, with the current deck, it's fairly unobjectionable. Perhaps I'll feel differently when I see how the end of this game plays out and more cards become effectively unique. Who knows.

Personally, I think the biggest flaw in gameplay is that it's rarely good to add to a sequence without completing it. Every time you do so, you're making a sequence easier for someone else to snipe without making it any easier for yourself. I think the variation Plotinus mentioned in which players pair (or maybe even triple) up would suffer far less from this problem, both because there are fewer people to snipe before it's "your" turn again and because you are in fact making it easier for your teammates to complete the sequences. For the current version of the game, one possible solution (that I don't think I like very much, but will mention anyway) is to say that you can finish a sequence with 6 cards if it's not your first time adding to it. I think 6 may be too few cards to complete a sequence that has to start out with at least 3 cards, which is why I don't think I like it as a solution, but I do think that giving more incentive to add to sequences would be really beneficial to gameplay. Also, having a ton of active sequences that have to be checked on every turn is a bit tedious.
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Post Post #139 (ISO) » Fri Oct 25, 2019 10:48 am

Post by DeathRowKitty »

My 6 card suggestion is probably better if it only applies to the person who currently owns the sequence. Not sure why I didn't think of that when I was writing my post.
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Post Post #140 (ISO) » Fri Oct 25, 2019 10:50 am

Post by popsofctown »

I'm so lost
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"Well, I..."
"What? Nothing? You would make the prince suffer over... nothing?"
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Post Post #141 (ISO) » Fri Oct 25, 2019 11:30 am

Post by Felissan »

I play [5, 7, 64] as numbers such that their square root, rounded down, contains a 2.
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Post Post #142 (ISO) » Fri Oct 25, 2019 11:49 am

Post by popsofctown »

Five's negative square root rounds down to negative three, which doesn't have a two :p :evil:
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Post Post #143 (ISO) » Fri Oct 25, 2019 12:23 pm

Post by Felissan »

... wait a second, I got it mixed up with another, much uglier, sequence I was thinking of for those numbers, and 64 obviously doesn't belong here at all. :facepalm:
Back to the drawing board for a second.
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Post Post #144 (ISO) » Fri Oct 25, 2019 12:27 pm

Post by Felissan »

I play the same numbers (5, 7, 64) as numbers such that the nth prime number doesn't have an even digit.
The respective prime numbers are 11, 17 and 311, which satisfy the condition.
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Post Post #145 (ISO) » Fri Oct 25, 2019 6:37 pm

Post by 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 = a010d + a110d-1 + ... + ad100 with ai ≡ 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 , 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)/3m and (n-2)/3m 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] {
    pk | pk < 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 = a010d + a110d-1 + ... + ad100 with Σi∈[0,d]ai = k^2 | ai ≥ 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.
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Post Post #146 (ISO) » Fri Oct 25, 2019 8:25 pm

Post by 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.
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Post Post #147 (ISO) » Fri Oct 25, 2019 10:33 pm

Post by 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.
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Post Post #148 (ISO) » Sat Oct 26, 2019 7:59 pm

Post by 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.
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Post Post #149 (ISO) » Sun Oct 27, 2019 4:55 am

Post by popsofctown »

14 7 and 10

Sequence is triangular numbers +4

Vla effort yo
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