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How many combinations are there in a Pyraminx?

The Pyraminx is another WCA puzzle that has a pyramid shape, it is triangle based and much simpler to solve than the regular 3x3x3, this is probably a lot to do with the far lower number of possible combinations, there are 75,582,720 possible combinations in a pyraminx, however, each of the tips can be rotated 3 ways and since there are 4 tips, the number of tip combinations is 3^4 = 81, since tips are trivial to solve, we can discount them and that provides us with a real number of combinations of 933,120.

This can be calculated as the number of 'corners' is 4 and they can be twisted 3 ways and cannot be moved relative to each other so the number of combinations of those 4 corners is 3^4=81. The number of edges is 6 and they can be flipped in two ways each so the number of possible flip combinations is 2^6=64, the pieces can be positioned in one of 6 different locations so the number of combinations of orientations is 6! = 720, however, just like on a 3x3x3, only one edge being flipped or only two edges swapped round is not possible so we must divide the result by 4 so we have ((2^6)/4)x720=11520.

The total number comes from multiplying these two results together so we have 11520x81=933120 possible pyraminx combinations, all of which the top solvers can do in under 2 seconds.

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