User:Edwin Shade/An Algorithmic Analysis Of The 3x3x3

The 3x3x3 cube, (known as the Rubik's cube), has many possible algorithms, and this page seeks to list data about those algorithms, (hence the title). Traditional move notation is being used, all else is my invention.

Firstly, what is the cycle length of all unique one move algorithms ? There are two such algorithms, and their cycle lengths are below. Keep in mind \([A]\) indicates the cycle length of an algorithm A. (Of course there are 18 different algorithms if you are counting other faces, but I am only including the move not isomorphic to each other, which will ease the amount of algorithms derived for many-moved algorithms, as for each additional move that is added there may be 18 times as many algorithms.

One Move Algorithms
[R]=4 [R2]=2

Two Move Algorithms
By this point there are technically 324 algorithms, but we only need to consider six of them due to the fact there are really only six unique algorithms.

[RU]=105 [RU']=63 [RU2]=30 [RL]=4 [RL']=4 [RL2]=4

Three Move Algorithms
[RUD]=90 [RUD']=180 [RUD2]=168 [RUF]=80 [RUF']=60 [RUF2]=36 [RUB]=84 [RUB']= [RUB2]=9 [RUL]= [RUL']= [RUL2]= [RUR]= [RUR']= [RUR2]= [RU'D]= [RU'D']= [RU'D2]= [RU'F]= [RU'F']= [RU'F2]= [RU'B]= [RU'B']= [RU'B2]= [RU'L]= [RU'L']= [RU'L2]= [RU'R]= [RU'R']= [RU'R2]= [RU2D]= [RU2D']= [RU2D2]= [RU2F]= [RU2F']= [RU2F2]= [RU2B]= [RU2B']= [RU2B2]= [RU2L]= [RU2L']= [RU2L2]= [RU2R]= [RU2R']= [RU2R2]= [RLU]= [RLU']= [RLU2]= [RLD]= [RLD']= [RLD2]= [RLF]= [RLF']= [RLF2]= [RLB]= [RLB']= [RLB2]= [RLR]= [RLR']= [RLR2]= [RL'U]= [RL'U']= [RL'U2]= [RL'B]= [RL'B']= [RL'B2]= [RL'F]= [RL'F']= [RL'F2]= [RL'B]= [RL'B']= [RL'B2]= [RL'R]= [RL'R']= [RL'R2]= [RL2U]= [RL2U']= [RL2U2]= [RL2D]= [RL2D']= [RL2D2]= [RL2F]= [RL2F']= [RL2F2]= [RL2B]= [RL2B']= [RL2B2]= [RL2R]= [RL2R']= [RL2R2]=

Four Move Algorithms
[RUDU]= [RUDU']= [RUDU2]= [RUDF]= [RUDF']= [RUDF2]= [RUDB]= [RUDB']= [RUDB2]= [RUDL]= [RUDL']= [RUDL2]= [RUDR]= [RUDR']= [RUDR2]= [RUD'U]= [RUD'U']= [RUD'U2]= [RUD'F]= [RUD'F']= [RUD'F2]= [RUD'B]= [RUD'B']= [RUD'B2]= [RUD'L]= [RUD'L']= [RUD'L2]= [RUD'R]= [RUD'R']= [RUD'R2]= [RUD2U]= [RUD2U']= [RUD2U2]= [RUD2F]= [RUD2F']= [RUD2F2]= [RUD2B]= [RUD2B']= [RUD2B2]= [RUD2L]= [RUD2L']= [RUD2L2]= [RUD2R]= [RUD2R']= [RUD2R2]= [RUFU]= [RUFU']= [RUFU2]= [RUFD]= [RUFD']= [RUFD2]= [RUFB]= [RUFB']= [RUFB2]= [RUFL]= [RUFL']= [RUFL2]= [RUFR]= [RUFR']= [RUFR2]= [RUF'U]= [RUF'U']= [RUF'U2]= [RUF'D]= [RUF'D']= [RUF'D2]= [RUF'B]= [RUF'B']= [RUF'B2]= [RUF'L]= [RUF'L']= [RUF'L2]= [RUF'R]= [RUF'R']= [RUF'R2]= [RUF2U]= [RUF2U']= [RUF2U2]= [RUF2D]= [RUF2D']= [RUF2D2]= [RUF2B]= [RUF2B']= [RUF2B2]= [RUF2L]= [RUF2L']= [RUF2L2]= [RUF2R]= [RUF2R']= [RUF2R2]=