From: Dik.Winter@cwi.nl
To: Cube-Lovers@ai.mit.edu
Subject: Re: MEGAMINX
> I reckon swapping just one twin pair is not possible in
> a complete solution, but that swapping any even number of twins may be
> (unproven), and so there are 512 solutions, each of which would be
> distinct if 12 different colours had been used on the original puzzle.
It is not so difficult to prove. Just as with the cube, also for the
dodecahedron it is easy to see that whenever you turn a face, the
parity of the edge and corner permutations remain the same. So a
single swap of two edges is not possible, that is an odd permutation
and would also require an odd permutation of the corner. However,
interchanging two pairs is possible. Actually any even permutation
of the edges is possible with the corners in place. This is because
there are simple procedures that rotate a triple of edges, leaving the
corners in place. Actually these procedures can be extremely similar
to those used for the cube. Anyhow, this proves it.
dik
[ Moderator's note: Lest a reader misunderstand, let me note that the
parity situation is different between the cube and megaminx. On the
cube an odd permutation of edges is achievable provided the corner
permutation is also odd. On the megaminx, neither the corner
permutation nor the edge permutation can ever be odd. ]
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