Log24

Rick’s Tricky Six Puzzle:
S₅ Sits Specially in S₆
by Alex Fink and Richard Guy

Abstract. Rick Wilson identified a sliding block puzzle, the Tricky Six puzzle, in which a uniquely small fraction of the possible scrambled arrangements of the six moving pieces can be restored to the solved state. The permutations one can perform form the abstract group S₅, the symmetric group on five letters, but surprisingly they aren’t any of the “obvious” copies of S₅ in S₆ that fix a single point and allow the other five to be permuted arbitrarily. This special S₅ comes from the outer automorphism of S₆, a remarkable group-theoretic map whose presence is felt in several combinatorial objects. We track down this outer automorphism in the Tricky Six puzzle as well as the projective plane of order 4, the Hoffman-Singleton graph, the Steiner system S(5,6,12), and a couple of error-correcting codes.

Meanwhile:

Click to enlarge.

Background:

A pair of matronly women
gave readings of
bad mathematical poetry
on April 28 at

the MAA’s Carriage House
Conference Center in
Washington, D.C.