One of the greatest toys ever invented, the Rubik’s Cube celebrates its 30th anniversary in 2010. With more than 300 million cubes sold worldwide, it is still one of the most popular puzzles in the world. The Hungarian professor and architect Ernő Rubik, the inventor of the cube, continues to challenge the world: last year he introduced “Rubik’s 360“, another fascinating puzzle-like toy that is becoming increasingly popular.

Here is a photo of four great puzzles: Rubik’s Cube (3x3x3), Rubik’s Revenge (4x4x4), Professor’s Cube (5x5x5), and Rubik’s 360:

The Rubik’s Cube, as well as its larger cousins, reveals mathematical structures of amazing complexity. The number of possible configurations of the classic (3x3x3) cube is extremely large: 43,252,003,274,489,856,000. To get an idea how big this number is, imagine that you are playing with the cube by moving exactly one of its faces (front, back, up, down, left, right) every second. Assuming that you get a new configuration after each move, you will need no less than 1,371,512,026,715 years to go through all the possible configurations. This is indeed a very long time: about 97 times the age of the Universe!

It is remarkable that this amazing complexity is accomplished by using only the six basic moves: F, B, U, D, L, R, where F = move the front face clockwise, B = move the back face clockwise, and so on. Each basic move has order 4 (for example, if you move the right face clockwise four times, you will return to the original configuration). Mathematically, the complexity of the configurations of the Rubik’s cube is captured by the abstract object called the Rubik’s Cube Group. The elements of this group are finite ordered collections (words) of basic moves (for example, RBF or UBBDLLL). Two words are identified if, when applied to the solved cube, they produce the same configuration. For example, the word RRRRR, regarded as an element of the Rubik’s cube group, is equal to the word R. There are many other identities, a less obvious one being LRRRFFLLLRUULRRRFF=UURRRLFFRLLLUURRRL.

In spite of this colossal number of configurations, the diameter of the Rubik’s cube group is no larger that 29, meaning that, for any configuration, there exists a sequence of basic moves of length less than or equal to 29 which solves the cube. Many speedcubers can solve the cube in about 40-50 moves, allowing them to finish their solves record times. The best cubers in the world can solve the cube in about 10 seconds. The world record, held by Erik Akkersdijk is mind-boggling: 7.08 seconds!

How can such a simple toy hide so much complexity? One can get an idea by looking at the way the cube is built: six centers are fixed together and form the core of the cube, and 8 corner pieces and 12 edge pieces are attached to the core. Here is a photo of the a disassembled Rubik’s cube, showing the core, the 8 corner pieces and the 12 edge pieces:

Each corner piece has 8 possible locations in space, hence there are 8! possible placements for the corners. Furthermore, each corner can be oriented in three possible ways (if the colors on the corner are red, white, and blue, there is exactly one orientation of the corner piece with red up, one with white up, and a third one with blue up). Seven corner pieces can be oriented in any way, but the orientation of the last corner is determined by the orientations of the other 7. Similarly, there are exactly 12 locations in space where one can place the 12 edge pieces hence 12! ways. There is a small subtlety though, stemming from the fact that the corners placement of the corners influences the possible placement of the edges. The permutation which gives the placement of the 8 corner pieces must have the same parity as the permutation that gives the placement of the 12 edges. Hence, once we choose one of the 8! ways to place the corner pieces, and, once the corners are placed, there are “only” 12!/2 distinct ways of placing the edges. One can see that each edge piece has two orientations and 11 of them can be oriented in any way we want.

Hence we can conclude that the number of possible configurations is , which is equal to 43,252,003,274,489,856,000 (please use a calculator if you want to check this!).

The class “Mathematics of Rubik’s Cube” was offered by the Department of Mathematics and Statistics during Winter Study 2010. The students who took the class learned how to solve the cube and were introduced to the mathematical theory of abstract groups. At the end of January we organized a Rubik’s Cube Competition; here is a photo from the final, with Greg White and Lauren Goldstein-Kral:

This year cubers around the world celebrate 30 years since the Rubik’s Cube was introduced on the market. You can celebrate too: just pick up your old cube (or get a new shiny one), scramble it and try to solve it. Even if you don’t succeed on the first try, you will have a lot of fun! Happy cubing!

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