Conway’s groupoid and its relatives

In 1997, John Conway constructed a 6-fold transitive subset M13 of permutations on a set of size 13 for which the subset fixing any given point was isomorphic to the Mathieu group M12. The construction was via a “moving-counter puzzle” on the projective plane PG(2, 3). We discuss consequences and generalisations of Conway’s construction. In particular we explore how various designs and hypergraphs can be used instead of PG(2, 3) to obtain interesting analogues of M13; we refer to these analogues as Conway groupoids. A number of open questions are presented.