Recreational Math

Math Puzzles

What are the number of possible positions for a Rubik’s Cube?

Rubik’s Cube was invented in the 1970s by the Hungarian architect, inventor, and mathematician Ernö Rubik (1944-), who also invented a number of other puzzles, including Rubik’s Clock. The cube measures 3 by 3 by 3, with a total of 26 subcubes on the outside. All the subcubes are hinged, making them easy to turn (by a quarter turn in either direction) in any of the planes on the cube. Initially, each of the six sides are painted a certain color; the object is to move the cube planes in a random way, then return the cube so that each side has a single color again.

What are the possible number of positions of a Rubik’s cube? Mathematicians need to use factorials (symbolized with the ! sign; for more information about factorials, see “Algebra”) in order to find out the many iterations, as seen in the following equation:


The number of positions turns out to be 43,252,003,274,489,856,000, or more than 43 quintillion turns.


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