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## Mathematics in real life II

Another small example I noticed awhile ago and forgot to write up.

Prime numbers, as one of the most fundamental concepts in mathematics, have a way of turning up in unexpected places. For example, the life cycles of some cicadas are either $13$ or $17$ years. It’s thought that this is a response to predation by predators with shorter life cycles; if your life cycle is prime, a predator with any shorter life cycle can’t reliably predate upon you.

A month or so ago I noticed a similar effect happening in the card game BS. In BS, some number of players (usually about four) are dealt the same number of cards from a standard deck without jokers. Beginning with one fixed card, such as the two of clubs, players take turns placing some number of cards face-down in the center. The catch is that the players must claim that they are placing down some number of a specific card; Player 1 must claim that they are placing down twos, Player 2 must claim that they are placing down threes, and so forth until we get to kings and start over. Any time cards are played, another player can accuse the current player of lying. If the accusation is right, the lying player must pick up the pile in the center. If it is wrong, the accusing player must pick up the pile in the center. The goal is to get rid of all of one’s cards.

I’ve been playing this game for years, but I didn’t notice until quite recently that the reason the game terminates in practice is that $13$, the number of types of cards in a standard deck, is prime. If, for example, we stopped playing with aces and only used $12$ types of cards, then a game with $4 | 12$ people need not terminate. Consider a game in which Player 1 has only cards $2, 6, 10$, Player 2 has only cards $3, 7, J$, Player 3 has only cards $4, 8, Q$, and Player 4 has only cards $5, 9, K$, and suppose that Player 1 has to play threes at some point in the game. Then no player can get rid of their cards without lying; since the number of players divides the number of card types, every player will always be asked to play a card they don’t have. Once every player is aware of this, every player can call out every other player’s lies, and it will become impossible to end the game reasonably.

More generally, such situations can occur if $13$ is replaced by a composite number $n$ such that the number of players is at least the smallest prime factor of $n$. This is because people who get rid of their cards will leave the game until the number of players is equal to the smallest prime factor of $n$, at which point the game may stall. But because $13$ is prime, any game played with less than $13$ people has the property that each player will eventually be asked to play a card that they have.