Pyraminx

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Pyraminx in its solved state

The Pyraminx (aka Pyramix) is a tetrahedron-shaped puzzle similar to the Rubik's Cube. It was invented and patented by Uwe Meffert, who continues to sell it in his toy shop, Meffert's.

1 Description
2 Optimal Solutions
3 See also
4 External links

[edit] Description

Pyraminx in the middle of a twist

The Pyraminx is a puzzle in the shape of a tetrahedron, divided into 4 axial pieces, 6 edge pieces, and 4 trivial tips. It can be twisted along its cuts to permute its pieces. The axial pieces are octahedral in shape, although this is not immediately obvious, and can only rotate around the axis they are attached to. The 6 edge pieces can be freely permuted. The trivial tips are so called because they can be twisted independently of all other pieces, making them trivial to place in solved position.

Scrambled Pyraminx

The purpose of the Pyraminx is to scramble the colors, and then restore them to their original configuration.

The 4 trivial tips can be trivially rotated to line up with the axial piece which they are respectively attached to; and the axial pieces are also easily rotated so that their colors line up with each other. This leaves only the 6 edge pieces as a real challenge to the puzzle. They can be solved by repeatedly applying two 4-twist sequences, which are mirror-image versions of each other. These sequences permute 3 edge pieces at a time, and change their orientation differently, so that a combination of both sequences is sufficient to solve the puzzle. However, more efficient solutions (requiring a smaller total number of twists) are generally available (see below).

The twist of any axial piece is independent of the other three, as is the case with the tips. The six edges can be placed in 6!/2 positions and flipped in 2⁵ ways, accounting for parity. Multiplying this by the 3⁸ factor for the axial pieces gives 75,582,720 possible positions. However, setting the trivial tips to the right positions reduces the possibilities to 933,120 and setting the axial pieces as well reduces the figure to only 11,520, making this a rather simple puzzle to solve.

[edit] Optimal Solutions

The maximum number of twists required to solve the Pyraminx is 11. There are 933,120 different positions (disregarding the trivial tips, and rotation), a number that is sufficiently small to allow a computer search for optimal solutions. The table below summarizes the result of such a search, stating the number p of positions that require n twists to solve the Pyraminx: