2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76210We 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.19 pages, uses mathrsfsGroup TheoryCombinatorics20B25 (Primary); 05B25, 51E20, 20B20 (Secondary)text