Problem of the Week #11: Monday Mar. 20th, 2023 I swear it's pure coincidence that the game we're talking about right after St. Patrick's Day is called the "Irish" Snap. I didn't even know its correct name until I went to write this up yesterday. As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
Irish Snap is a multiplayer card game using the standard deck of 52 playing cards where each player takes turns laying down one face up card each while counting. I.e. the first person says “ace” while laying down the first card; then the second person says “two” while laying down the next card. This process is repeated until the card that has just been laid down matches the number the person just called. When such a match happens, all players race to slap a hand on top of the pile of cards, and the slowest person has to take the pile into their hand. The first person to empty their hand wins.
Now suppose that instead of playing with 52 cards of four different suits, we play Irish Snap with a simpler deck of just 5 cards numbered from 1 to 5 with no suits. Also suppose for the purpose of this problem that we leave the cards in the pile without anyone taking them, and that the game stops when we finish counting one round from 1 to 5 (or 1 to n with a deck of n cards) with all cards in the pile. (Pretend it’s a cooperative game where everyone wins if everyone manages to slap the pile at the right times.)
a). In how many ways can the cards be laid in the pile before a matching number at “four”? For example, one way is to have the first card be 2, the second card be 1, and the third card be 5.
b). In how many ways can all five cards in the deck be laid in the pile without a single matching number?
c). With a deck of n cards numbered from 1 to n with no suits, in how many ways can all n cards in the deck be laid in the pile without a single matching number?
d). Also with the same deck of n cards, in how many ways can cards be laid in a pile before a matching number at “m”, where m is a natural number between 1 and n?
e). In how many ways can the n cards be laid in a pile if in the course of the game there were m matching numbers, where m and n are natural numbers, and m is between 1 and n?
f). Now suppose we are using the standard 52 card deck with four suits. If we ignore the suit differences and only look at the numbers, in how many ways can the entire deck be laid in a pile if in the four rounds of number calling there was no matching numbers?
g). Share your own problem inspired by these scenarios.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.