Usually though, it is played in an electronic form as a computer game. Mahjong solitaire can be played using genuine tiles and a special wooden frame for set-up. Īn analysis of ten million games with the default layout, "the turtle", found that about 3 percent of the turtles cannot be solved even when looking below tiles is allowed. In this case, however, it is NP-complete to decide whether all tiles can be removed. The perfect-information version of this puzzle is where the player knows, before the game starts, the position of every tile. It has been proven that it is PSPACE-hard to approximate the maximum probability of removing all tiles within a factor of n ϵ, assuming that there are arbitrarily many quadruples of matching tiles and that the hidden tiles are uniformly distributed. Playing Mahjong solitaire optimally in the sense to maximize the probability of removing all tiles is PSPACE-complete, and the game is NP-complete if looking below tiles is allowed.
The game is won when all pairs of tiles have been removed from the board, and lost if the remaining tiles contain no exposed pairs. The goal is to match open pairs of identical tiles and remove them from the board, exposing the tiles under them for play. A tile is said to be open or exposed if it can be moved either left or right without disturbing other tiles. The 144 tiles are arranged in a four-layer pattern with their faces upwards. P.S You can also use memoization to avoid visiting the same state twice.An untouched turtle formation, with exposed "free" tiles highlighted in blue Do not try to remove any other pairs in this state.Ĭ)If neither a) nor b) applies, then you will have to try all possibilities(like in a naive version).Īs you can see, these heuristics allow you to avoid branching if either a) or b) applies to the current state, so they can speed up the search(especially at the end of the game, when only a few tiles are left and almost all of them are connected). I claim that if solution exists, it can obtained by removing this pair first.ī)If all tiles of one type can be matched now(even if there are more than 2 of them), match them. Do not try to remove any other pairs in this state. However, there are a few heuristic to speed up the search:Ī)If there are only two tiles of one type and they can be removed, remove them. Otherwise, you can use, let's say, depth first search to find the pairs of matching tiles.Ī naive solution would be to try to match every pair, remove it and keep searching(and do backtracking if this branch fails). If no tiles are left, the solution is found. 1)Let's use backtracking with some heuristics(I don't know how fast it would actually work).Ģ)Let's assume that the the current state is S(state is defined by the tiles which are left).
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