Problem of the Week #12: Monday Mar. 27th, 2023 I figured I should try to stick to a topic each month, so this will be the last one on playing cards until I run out of topics and circle back. 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.
Seven is a card game where players take turns extending suits of cards in numerical order, starting from 7 of each suit. It starts with dealing the entire deck equally (or almost equally, if the deck is not divisible by the number of players) to all players. The player who has the 7 of hearts always goes first by putting down the 7 of Hearts. The next player can put down either a 6 or 8 of Hearts, extending the suit, or another 7 to start a new suit. If a player doesn’t have any playable card during a turn, they would need to pass. The first player to put down all the cards in their hand wins the round.
Now for starters, let’s suppose Alice and Bob are playing a game similar to Seven, but with only 7 cards numbered from 1 to 7 with no suit. Let’s also suppose that Alice has been dealt 4 of the 7 cards, and Bob has been dealt 3. Alice starts the game by playing the card numbered 4, so let’s call the game “Four”.
a). Find one distribution of cards (who gets which cards) such that Bob would win the round regardless of how he plays.
b). Find a distribution of cards such that Alice would win the round if she plays with the right strategy, but if she doesn’t play with the right strategy, Bob would win regardless of how Bob plays.
c). Find a distribution of cards such that if any player plays with the right strategy, and the other(s) doesn’t, the one with the right strategy would win.
d). List out all possible distributions of the 7 cards and label each with who would win and if the right strategy is required.
e). Find a card distribution for each of a), b), and c) if they’re playing with one full suit of 13 cards instead of 7.
f). Find a card distribution for each of a), b), and c) if a third person, Charlie, joins the game, and they’re playing with one full suit of 13 cards.
g). List all possible distributions of the 13 cards to the three players and label each with who would win and if the right strategy is required.
h). If three people are playing with a full deck of 52 cards, how many of the card distributions would require a playing strategy to determine who wins the round?
i). Share your own problem inspired by this one.
j). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.