A fundamental property of Rubik's family cubes of all sizes is that any permutation applied a sufficient number of times will result in the cube state returning to that which applied before the first application of the permutation. The permutation (move) cycle length is the minimum number of times the permutation needs to be applied for the new state to correspond with the initial state. The permutation cycle length parameter may have relevance to users who have an interest in cube rules or mathematics. For a given cube style, a cube of any size greater than 3 which is subject to only outer layer rotations will return the same cycle length as that for a size 3 cube. For any given permutation, cycle length may vary according to other variables as indicated in the following table.
Variable | Effect |
Cube size | Cube size can have a major effect on cycle length. |
Initial cube state | For cubes with unmarked centres, the cycle length for a cube with an initial set state may be different to that for an initial scrambled state. This arises because centre cubies can end up in different positions that appear identical. For cubes with marked centres, the cycle length is independent of initial state. The cycle count for a cube with unmarked centres and an initial random state is likely to be the same as that for a cube with marked centres. |
Cube style (unmarked or marked centres) | Cycle count for cubes with unmarked centres can be the same or lower than that for a cube with marked centres. |
Spatial orientation | The state of a cube is not changed if its spatial orientation is changed (e.g. if a hardware cube is turned upside down). The Permutation Cycle Length Determination dialog allows the user to treat either just the original spatial orientation or any of the 24 possible spatial orientations as the end point. For instance the simple move WF will rotate the whole cube a quarter turn about the F face axis. For the options above the result will be 4 and 1 respectively. Adoption of the latter option can increase execution time as there are 24 checks required after each application of the permutation. |