Helicopter Cube

The Helicopter Cube is a Rubik's Cube-like puzzle invented by Adam G. Cowan in 2005 and built in 2006.

However, if some jumbling moves were made, even if the puzzle was subsequently returned to cube shape, it may not be possible to solve it using only 180° twists.

The reason for this is that using only 180° twists, each face center piece can only be permuted within a 6-member cycle, often referred to as its orbit.

However, jumbling moves are able to permute face center pieces between different orbits, thus leaving the puzzle in a state that cannot be solved by 180° twists alone.

Assume that Helicopter Cube is scrambled without jumbling moves (i.e. mixed with only 180 degree twists).

Seven of the corners can be independently rotated, and the orientation of the eighth depends on the other seven, giving 8!×37 combinations.

Assuming that the four centres of each colour are indistinguishable, the number of permutations is reduced to 24!/(4!6) arrangements.

This gives a total number of permutations of The expanded number is 11928787020628077600000 (approximately 11929 trillion or 12 trilliard on the long scale or 12 sextillion on the short scale)[8] To count non-cube positions, we need to count all the possible shapes (ignoring the colours).

Matt Galla has done a full analysis, and wrote up his results in this post on TwistyPuzzles Forum.

The index is also the number of ways any particular shape with that symmetry can be oriented in space (including reflections).