Packing Puzzle

Solver for a packing puzzle.

Symmetries

The group of symmetries of the cube is isomorphic to the permutation group on 4 symbols. Below is an explicit isomorphism between S4 and a matrix group.

| Permutation | x | y | z | matrix | |-------------|----|----|----|----------------------------------------------------| | 0123 | x | y | z | [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] | | 0132 | -z | -y | -x | [ [ 0, 0, -1 ], [ 0, -1, 0 ], [ -1, 0, 0 ] ] | | 0213 | -x | -z | -y | [ [ -1, 0, 0 ], [ 0, 0, -1 ], [ 0, -1, 0 ] ] | | 0231 | y | z | x | [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] | | 0312 | z | x | y | [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ] | | 0321 | -y | -x | -z | [ [ 0, -1, 0 ], [ -1, 0, 0 ], [ 0, 0, -1 ] ] | | 1023 | z | -y | x | [ [ 0, 0, 1 ], [ 0, -1, 0 ], [ 1, 0, 0 ] ] | | 1032 | -x | y | -z | [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ] | | 1203 | -z | -x | y | [ [ 0, 0, -1 ], [ -1, 0, 0 ], [ 0, 1, 0 ] ] | | 1230 | -y | x | z | [ [ 0, -1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ] | | 1302 | x | z | -y | [ [ 1, 0, 0 ], [ 0, 0, 1 ], [ 0, -1, 0 ] ] | | 1320 | y | -z | -x | [ [ 0, 1, 0 ], [ 0, 0, -1 ], [ -1, 0, 0 ] ] | | 2013 | -y | z | -x | [ [ 0, -1, 0 ], [ 0, 0, 1 ], [ -1, 0, 0 ] ] | | 2031 | x | -z | y | [ [ 1, 0, 0 ], [ 0, 0, -1 ], [ 0, 1, 0 ] ] | | 2103 | y | x | -z | [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, -1 ] ] | | 2130 | z | -x | -y | [ [ 0, 0, 1 ], [ -1, 0, 0 ], [ 0, -1, 0 ] ] | | 2301 | -x | -y | z | [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, 1 ] ] | | 2310 | -z | y | x | [ [ 0, 0, -1 ], [ 0, 1, 0 ], [ 1, 0, 0 ] ] | | 3012 | y | -x | z | [ [ 0, 1, 0 ], [ -1, 0, 0 ], [ 0, 0, 1 ] ] | | 3021 | -z | x | -y | [ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] | | 3102 | -y | -z | x | [ [ 0, -1, 0 ], [ 0, 0, -1 ], [ 1, 0, 0 ] ] | | 3120 | -x | z | y | [ [ -1, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ] ] | | 3201 | z | y | -x | [ [ 0, 0, 1 ], [ 0, 1, 0 ], [ -1, 0, 0 ] ] | | 3210 | x | -y | -z | [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] |