Pyramorphix

Despite its appearance, the puzzle is in fact equivalent to a shape modification of the original 3x3x3 Rubik's Cube. Its 4 corner pieces on the corners and 4 corner pieces on the face centers together are equivalent to the 8 corner pieces of the Rubik's Cube, its 6 edge pieces are equivalent to the face centers of the Rubik's Cube, and its non-center face pieces are equivalent to the edge pieces of the Rubik's Cube. Thus, the same methods used to solve the Rubik's Cube may be used to solve the Master Pyramorphix, with a few minor differences: the center pieces are sensitive to orientation because they have two colors, unlike the usual coloring scheme used for the Rubik's Cube, and the face centers are not sensitive to orientation (however when in the "wrong" orientation parity errors may occur). In effect, it behaves as a Rubik's Cube with a non-standard coloring scheme where center piece orientation matters, and the orientation of 4 of the 8 corner pieces do not, technically, matter.

Unlike the Square One, another shape-changing puzzle, the most straightforward solutions of the Master Pyramorphix do not involve first restoring the tetrahedral shape of the puzzle and then restoring the colors; most of the algorithms carried over from the 3x3x3 Rubik's Cube translate to shape-changing permutations of the Master Pyramorphix. Some methods, such as the equivalent of Philip Marshall's "Ultimate Solution", show a gradual progression in shape as the solution progresses; first the non-center face pieces are put into place, resulting in a partial restoration of the tetrahedral shape except at the face centers and corners, and then the complete restoration of tetrahedral shape as the face centers and corners are solved.

There are four corners and four face centers. These may be interchanged with each other in 8! different ways. There are 3⁷ ways for these pieces to be oriented, since the orientation of the last piece depends on the preceding seven, and the texture of the stickers makes the face center orientation visible. There are twelve non-central face pieces. These can be flipped in 2¹¹ ways and there are 12!/2 ways to arrange them. The three pieces of a given color are distinguishable due to the texture of the stickers. There are six edge pieces which are fixed in position relative to one another, each of which has four possible orientations. If the puzzle is solved apart from these pieces, the number of edge twists will always be even, making 4⁶/2 possibilities for these pieces.

${\displaystyle {8!\times 3^{7}\times 12!\times 2^{9}\times 4^{6}}\approx 8.86\times 10^{22}}$

The full number is 88580102706155225088000.

However, if the stickers were smooth the number of combinations would be reduced. There would be 3⁴ ways for the corners to be oriented, but the face centers would not have visible orientations. The three non-central face pieces of a given color would be indistinguishable. Since there are six ways to arrange the three pieces of the same color and there are four colors, there would be 2¹¹×12!/6⁴ possibilities for these pieces.

${\displaystyle {\frac {8!\times 3^{4}\times 12!\times 2^{10}\times 4^{6}}{6^{4}}}\approx 5.06\times 10^{18}}$

The full number is 5062877383753728000.