Optimal strategy in the childrens game Memory
Erik Alfthan · 2007
Two mathematical games are contructed from the children’s game memory. One game, named Terminating memory is constructed as a two player game with rules as close to the children’s game as possible. The most significant change is made in order to make the game terminate. It turns out that there are non-trivial elements of strategy in Terminating memory. Depending on the expected number of turn overs, i.e. the number of times the lead is lost, the strategy seems to be to try to force the opponent to reach a known losing position which is when the last turn over occurs. However, this could not be proven generally, but is computed for all games with less than 200 pairs. A second game of memory that complies with the rules of combinatorial games was therefore contructed, in order to determine which elements are important to the previous game, Terminating memory. This game, Combinatorial memory was generally solved as game equivalent to a sequence of weighted misere nim games. A hypothesis of implications of this to Terminating memory was presented. It is suggested that the strategy will depend on whether there are expected to be odd or even number of nim games left, of which the last game is probably the largest. Both player are trying to reach a position where they will get the last collect sequence. This is consistant with the main conjecture of terminating memory. A general way to compute whether odd or even number of remaining nim games is most likely is needed to make this result useful.