# The Solitaire Memory Game

Memory is a popular card game played by people of all ages around the world. Given a set of n pairs of cards laid out face down, a player may in one move turn over two cards one after another. If the cards form a pair, they get collected off the table, else they get turned over again. For the 2–player game, where players alternate in moves and win if they collect more pairs, an optimal strategy is known and well-studied. Here we consider the 1–player solitaire game, where the goal is to need as few moves as possible to collect all cards off the table. We prove that an optimal strategy needs less than 1.75 · n moves in expectation. Furthermore we prove the lower bound that every strategy needs at least 1.5 · n − 1 moves for arbitrary n in expectation. Intensive numerical calculations lead to the new interesting conjecture that an optimal strategy has a competitive ratio of 1.613603 < c < 1.613706. In particular, we study games where already k different cards are known to the player. We prove that an optimal strategy needs at least 1.5n − 0.5k − 1 and at most 2n − k moves in expectation to finish such a game. If one is interested in a strategy that guarantees to finish a solitaire Memory game, then 2n − 1 moves are both necessary and sufficient.

Top- Foerster, Klaus-Tycho
- Wattenhofer, Roger

Category |
Technical Report (Technical Report) |

Divisions |
Communication Technologies |

Subjects |
Theoretische Informatik |

Date |
2013 |

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