10 cards with identical backs are face down on a table. Each card face matches exactly one of the other card faces. The cards are in a random sequence. A turn consists of choosing 2 cards, simultaneously reversing them so that they are face up, discarding them if they match, and turning them face down if they do not match. The game ends when all cards are discarded.

a) If you have perfect memory, and an efficient strategy, then what is the expected number of turns in the 10 card game?

b) What is the expected number of turns if instead there are 2n cards in the starting tableaux, with each card matching exactly one other?

Assume 2n cards. Because you have perfect memory, you might as well turn all the cards over 2 at a time, which takes n turns, and then remove all the matches, which takes another n turns. However, the game may take less than 2n turns, because of immediate matches, which do not need to be removed in the 2nd n turns, as they are removed during the first n turns at the time of the initial exposure. On any one of the n initial turns, the probability of an immediate match is 1/(2n-1). The expected number of immediate matches is therefore n/(2n-1). The total number of expected turns is therefore 2n - n/(2n-1). If there are 10 cards, then the expected number of turns is 10 - 5/9 = 9 + 4/9

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