Permutation Based Logic Puzzles

The Rubik’s Cube and the Gear Cube are two types of logic puzzles that both involve rearranging small pieces until a certain condition is satisfied. This project aimed to find the total number of permutations possible on both puzzles – that is, how many different ways their pieces can be arranged.

For the Rubik’s Cube, counting theory was used to calculate an upper bound for the number of permutations. Principles of group theory were then applied to prove which of these were impossible to achieve by rotating a face of the cube. Finally, all the remaining permutations were proven to be possible by finding moves that allowed for them to be achieved. This proved the true value to be equal to the previously found upper bound.

For the Gear Cube, the analysis centred around the invariants of the cube which arise due to the restrictive rotating mechanism. These were used to find an upper bound of the number of permutations. Finally, the true value was calculated using the computer algebra system, Magma.

The total number of permutations for each puzzle were found to be:
• Rubik’s Cube: 43,252,003,274,489,856,000
• Gear Cube: 41,472.

The disparity can be attributed to the different rotating mechanisms, which places far more restrictions on the movement of the Gear Cube’s pieces.