Number of Combinations
The Skewb Ultimate has 6 large "edge" pieces and 8 smaller corner pieces. Only even permutations of the larger pieces are possible, giving 6!/2 possible arrangements. Each of them has two possible orientations, although the orientation of the last piece is determined by the orientations of the other pieces, hence giving us a total of 25 possible orientations.
The positions of four of the smaller corner pieces depend on the positions of the other 4 corner pieces, and only even permutations of these positions are possible. Hence the number of arrangements of corner pieces is 4!/2. Each corner piece has 3 possible orientations, although the orientation of the last corner is determined by the orientations of the other corners, so the number of possible corner orientations is 37. However, the orientations of 4 of the corners plus the position of one of the other corners determines the positions of the remaining 3, so the total number of possible combinations of corners is only .
Therefore, the number of possible combinations is:
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