Rubik's cube is a cube in which the 26 subcubes on the outside are internally
hinged in such a way that rotation (by a quarter turn in either direction or a half
turn) is possible in any plane of cubes. Each of the six sides is painted a distinct
color, and the goal of the puzzle is to return the cube to a state in which each
side has a single color after it has been randomized by repeated rotations. The puzzle was invented in the 1970s by the Hungarian Ernő
Rubik and sold millions of copies worldwide over the next decade.
The number of possible positions of Rubik's cube is
(Turner and Gold 1985, Schönert). Hoey showed using the Cauchy-Frobenius lemma that there are
positions up to conjugacy by whole-cube symmetries.
The group of operations on Rubik's cube is known as Rubik's group, and the Cayley
graph of that group is called Rubik's graph.
The maximum, over starting positions, of the minimum number of turns required to
solve the cube is the graph diameter of Rubik's
graph, also called God's number. While algorithms
exist for solving a cube from an arbitrary initial position, they are not necessarily
optimal (i.e., requiring a minimum number of turns) and computation of God's
number is very difficult. A lower bound of 20
moves in the worst case had been known since 1995. Rokicki et al. (2010) showed
that no configuration requires more than 20 moves, establishing that God's
number is 20 in the half-turn metric.
The configurations of a
Rubik's cube reachable using only half twists form a Nauru
graph. Wolfram (2022) analyzed of the Rubik's cube via a multiway
graph, the first few steps of which are illustrated above.
For the
cube, Mrzyglocki et al. (2026) give a solving algorithm using at most moves. A move turns
a single slice by a multiple of 90 degrees. The upper
bound is within a factor of
of the known lower bound for almost all configurations.
cube in which the 26 subcubes on the outside are internally
hinged in such a way that rotation (by a quarter turn in either direction or a half
turn) is possible in any plane of cubes. Each of the six sides is painted a distinct
color, and the goal of the puzzle is to return the cube to a state in which each
side has a single color after it has been randomized by repeated rotations. The puzzle was invented in the 1970s by the Hungarian Ernő
Rubik and sold millions of copies worldwide over the next decade.
positions up to conjugacy by whole-cube symmetries.
Rubik's cube reachable using only half twists form a Nauru
graph. Wolfram (2022) analyzed of the 
Rubik's cube via a multiway
graph, the first few steps of which are illustrated above.
cube, Mrzyglocki et al. (2026) give a solving algorithm using at most 
moves. A move turns
a single slice by a multiple of 90 degrees. The upper
bound is within a factor of 
of the known lower bound for almost all configurations.
of Optimal." 7 Nov 2026. 
."