only 180-degree turns. These subgroups are the principle underlying the computer cubing methods by Thistlethwaite and Kociemba, which solve the cube by further reducing it to another subgroup. Solutions Singmaster notation Many 3×3×3 Rubik's Cube enthusiasts use a notation developed by David Singmaster to denote a sequence of moves, referred to as "Singmaster notation". Its relative nature allows algorithms to be written in such a way that they can be applied regardless of which side is designated the top or how the colours are organised on a particular cube. F (Front): the side currently facing the solver B (Back): the side opposite the front U (Up): the side above or on top of the front side D (Down): the side opposite the top, underneath the Cube L (Left): the side directly to the left of the front R (Right): the side directly to the right of the front f (Front two layers): the side facing the solver and the corresponding middle layer b (Back two layers): the side opposite the front and the corresponding middle layer u (Up two layers): the top side and the corresponding middle layer d (Down two layers): the bottom layer and the corresponding middle layer l (Left two layers): the side to the left of the front and the corresponding middle layer r (Right two layers): the side to the right of the front and the corresponding middle layer x (rotate): rotate the entire Cube on R y (rotate): rotate the entire Cube on U z (rotate): rotate the entire Cube on F When a prime symbol ( ′ ) follows a letter, it denotes an anticlockwise face turn; while a letter without a prime symbol denotes a clockwise turn. These directions are as one is looking at the specified face. A letter followed by a 2 (occasionally a superscript 2) denotes two turns, or a 180-degree turn. R is right side clockwise, but R′ is right side anticlockwise. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes, corresponding to R, U, and F turns respectively. When x, y, or z is primed, it is an indication that the cube must be rotated in the opposite direction. When x, y, or z is squared, the cube must be rotated 180 degrees. The most common deviation from Singmaster notation, and in fact the