What to Expect in a Game of Memory

AbstractThe game of memory is played with a deck of n pairs of cards. The cards in each pair are identical. The deck is shuffled and the cards laid face down. A move consists of flipping over first one card and then another. The cards are removed from play if they match. Otherwise, they are flipped back over and the next move commences. A game ends when all pairs have been matched. We determine that, when the game is played optimally, as n → ∞: • The expected number of moves is (3 − 2 ln 2)n + 7/8 − 2 ln 2 ≈ 1.61n.• The expected number of times two matching cards are unwittingly flipped over is ln 2.• The expected number of flips until two matching cards have been seen is