Sequencer | StrangerCoug's turn

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Sure.

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

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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).

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In the mean time I'll just make my move, since it's unaffected by menno's and to keep the game moving:

[5, 9, 10, 27, 48, 66, 98] {

n² + k, -3 < k < 3

} numbers that are within two of a perfect square
(4, 9, 9, 25, 49, 64, 100 are the close squares)

Note that each of the 10 squares 1² through 10² contributes at most 5 numbers to this set between 1 and 100, and there is some overlap, so it contains fewer than 50 of the numbers between 1 and 100.

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I'll take that and complete it with 10, 70, 45, 21.

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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": [30, 42, 65]
30->2*3*5->ten
42->2*3*7->twelve
65->5*13->eighteen

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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):

9, 17, 69, 77.

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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).

5 -> 9 -> 13
6 -> 9 -> 13
50 -> 55 -> 64
125 -> 145 -> 164

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i mean, i'm forced to interpret "preferred factor" as a factor that lives in the sequence here :p

(and i think that definition looks good plot)

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Numbers less than 50 with exactly 1 prime factor, *excluding 27 and 19*: 9, 37, 25, 7.

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In what way?

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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.

Yeah, the way you wrote it is good. My intent was 1 isn't included because it has 0 prime factors.

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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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"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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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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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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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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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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(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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Complete the squares (ha) by adding 1, 9, 64.

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Fine with the first rule change (also happy to retroactively apply it to my sequence, if desired).

Second one i'm eh on adding mid-game but no *strong* objection.

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yeah that bingo is nice.

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I'll finish the nth prime doesn't have any even digits sequence with
3 (5)
4 (7)
21 (73)
22 (79)

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Eh shrug sure.

Numbers where each pair of consecutive digits differs by exactly one: 2, 3, 23, 56.

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finish numbers whose digit sum is a square with 97, 88, 18, 4.

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1, 16, 64 to finish powers of 2

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32, 89, and 8 to finish my consecutive digits sequence

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

99, 89, 11.

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

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pass.

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gg.

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I'll sit out the next game; may play more in a future game though if it's still going.