John H. Conway introduced this sliding tile puzzle, which bears the same relation to the Mathieu group as Sam Lloyd's famous 15 puzzle bears to the alternating group . ( was one of the first sporadic simple groups to be discovered.) Initially, tiles numbered 1–12 are placed on the same-numbered points of a projective plane of order three, with the point left uncovered. Hovering over a tile reveals the four points in the unique line of the projective plane that connects it to the uncovered point. Clicking the tile executes the sole legal move of the game, which is a double transposition: the clicked tile slides in to the uncovered space, and the other two points on the line exchange positions. The object, of course, is to restore the tiles to their initial order from a scrambled position.
To use the Solve button, make sure the position (at the top of the circle) is open. Clicking the Solve button gives a sequence of numbers that refer to the light-gray point numbers around the outside of the circle. Click those positions in the order indicated, and the puzzle will be returned to its initial state.
The M(13) in the title refers to the pseudogroup of permutations of the 13 tiles (where we think of the open space as simply another tile bearing the number 13) that can be achieved by legal moves in the game. The subset of permutations that leave the point uncovered form a group of 95,040 elements ismorphic to the Mathieu group .
The solver is based on the algorithm described in the following paper (which would be impractical for a human puzzler to use):
J. H. Conway, N. D. Elkies, and J. L. Martin, "The Mathieu Group and Its Pseudo Group Extension ," Experimental Mathematics, 15(2), 2006 pp. 223–236. [PDF]