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I've never seen anything on patterns for the megaminx, with the

sole exception of Kurt Endl's book "Megaminx".

It is long ago I had it in my hands, and I have no books. What I say

is from memory; probably correct. Note that a face turn induces an

even permutation on both the corner and the edge "cubies". So odd

permutations are not possible. On the other hand (if I remember well)

*all* combinations of even permutations are possible.

There are 6 opposite pairs of faces on the megaminx. There are 4 ways

to rotate the centres for each pair to generate a 10 spot. I'll

speculate that there are 6*4 = 24 possible 10-spots.

Right.

I suspect various 12-spots are possible. I have no idea how to

easily permute centre pieces on the megaminx.

Indeed. Every rotation of the center skeleton is possible (if you

consider the remainder fixed...). So there are 12 centers that can

come out at top; for each center at top you have 5 possible positions

of the remainder leading to 60 configurations. Of these 24 are

10-spots, 1 is the solved puzzle, so the remainder (35) is 12-spots.

dik