485 - A New Upper Bound for Morpion Solitaire

Henryk Michalewski, Andrzej Nagórko, Jakub Pawlewicz

Abstract: In previous research an upper bound of 705 was proved on the number of moves in the 5T variant of the Morpion Solitaire game. We show a new upper bound of 485 moves. This is achieved in the following way: we encode Morpion 5T rules as a linear program and solve 126912 instances of this program on special octagonal boards. In order to show correctness of this method we analyze rules of the game and use a concept of a potential of a given position. By solving continuous-valued relaxations of linear programs on these boards, we obtain an upper bound of 586 moves. Further analysis of original, not relaxed, mixed-integer programs leads to an improvement of this bound to 485 moves. However, this is achieved at a significantly higher computational cost.