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Interactive Supplements to Rick's Tricky Six Puzzle
Ricks Tricky Six Puzzle: S5 Sits Specially in S6
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.
Ensley, Doug, "Interactive Supplements to Rick's Tricky Six Puzzle," Loci (March 2009), DOI: 10.4169/loci003295
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