Matching expectations

TL;DR

This paper analyzes the expected number of moves, mismatched flips, and flips until matching cards are seen in an optimal memory game with a large number of pairs, providing precise asymptotic formulas.

Contribution

It derives exact asymptotic expectations for key metrics in the optimal memory game, advancing theoretical understanding of the game's probabilistic behavior.

Findings

Expected moves grow linearly with n (~1.61 n)

Expected mismatched flips is ln 2

Expected flips until match is approximately sqrt(pi n)

Abstract

The 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 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 tends to infinity: 1) The expected number of moves is (3 - 2 ln 2)n + 7/8 - 2 ln 2 (approximately 1.61 n), 2) The expected number of times two matching cards are unwittingly flipped over is ln 2, and 3) The expected number of flips until two matching cards have been seen is asymptotically sqrt{pi n}.