  We study a construction of the Mathieu group $M_{12}$ using a game
reminiscent of Loyd's ``15-puzzle''. The elements of $M_{12}$ are realized as
permutations on~12 of the~13 points of the finite projective plane of order~3.
There is a natural extension to a ``pseudogroup'' $M_{13}$ acting on all~13
points, which exhibits a limited form of sextuple transitivity. Another
corollary of the construction is a metric, akin to that induced by a Cayley
graph, on both $M_{12}$ and $M_{13}$. We develop these results, and extend them
to the double covers and automorphism groups of $M_{12}$ and $M_{13}$, using
the ternary Golay code and $12 \x 12$ Hadamard matrices. In addition, we use
experimental data on the quasi-Cayley metric to gain some insight into the
structure of these groups and pseudogroups.

Group Theory;Combinatorics