Game Theory Problems Using Playing Cards

This is a one player game with the following rules.

• Start with an unlimited quantity of cards, all of which are the queen of diamonds.
• Let n be an integer with n≥2.
• Place n cards face down in a row from left to right.
• On each turn the player selects a face down card, which cannot be the card on the right hand end of the row. The player flips the selected card face up, leaving the card in the same place. The card immediately to the right of the card selected is also flipped over and left in the same place, becoming face up if it was face down, or becoming face down if it was face up.
• The game ends when no legal moves are available.
• When the game ends, the length of the game is the number of turns that the player used, and each turn involves flipping two cards over.
• The game always ends in a finite number of turns, no matter what the value of n is, and no matter what strategy is used.
• Two games are the same if they have the same sequence of moves, and two games are different if they do not have the same sequence of moves.

Problem 1) Let f(n) be the minimum possible length of the game when n cards are used, assuming the game is played until no legal moves are available. Find a formula for f(n) that is valid for each value of n.

Problem 2) Let g(n) be the number of different games that will produce a game of the minimum possible length when n cards are used. Find a formula for g(n) that is valid for each value of n.

Problem 3) Let h(n) be the maximum possible length of the game when n cards are used. Find a formula for h(n) that is valid for each value of n.

Problem 4) What movie used a deck of cards where every card was the queen of diamonds?

Feel free to post comments or solutions. A new blog post with solutions will be provided later.