Solving Mahjong Solitaire boards with peeking

We first prove that solving Mahjong Solitaire boards with peeking is NPcomplete, even if one only allows isolated stacks of the forms aab and abb. We subsequently show that layouts of isolated stacks of heights one and two can always be solved with peeking, and that doing so is in P, as well as finding an optimal algorithm for such layouts without peeking. Next, we describe a practical algorithm for solving Mahjong Solitaire boards with peeking, which is simple and fast. The algorithm uses an effective pruning criterion and a heuristic to find and prioritize critical groups. The ideas of the algorithm can also be applied to solving ShisenSho with peeking. Mahjong Solitaire is a game which is played with the 144 tiles of the Chinese game Mahjong. The tiles are distributed in 36 groups of four tiles each. In the beginning of the game, the tiles are stacked randomly in a predefined pattern, called the layout. The so-called turtle layout is used the most and therefore called the default layout as well. After stacking the tiles, the object is to remove all tiles under certain rules. These rules are as follows. • A tile is playable, if and only if no other tile is lying upon it, not even partially, and either its left side or its right side does not touch any other tile. • Only playable tiles may be played, but solely in pairs of tiles of the same group. Thus removing all tiles takes 72 removals of pairs of similar tiles. During game play, one cannot see tiles which are completely below other tiles. Sometimes a tile can be seen partially, namely where it is not covered by an other tile.