Introduction to the Rubik's Cube (Trademark of Ideal Toy Company)

The Rubik Cube is a modern-day mechanical marvel surpassing the ancient wonders of Hellenistic tradition and rivaling John Dee's Scarabaeus of 1547 in the ingeniousness of its internal construction. It is a remarkable puzzle invented by the Hungarian sculptor, designer and architectural engineer, Professor Ernö Rubik of the School for Commercial Artists in Budapest. His intentions were not to create the 20th century's most amazing puzzle, but, rather, to conquer the technical challenge of designing a cube with the given specifications:

It should look like a solid block of twenty-seven small cubes ingeniously linked so that each layer of nine cubes can be rotated about its centre without the whole thing falling apart. Of course the internal construction had to be a little more complicated than just a simple block of twenty-seven cubes. The six centre pieces are attached to the middle by spring-loaded spindles, and the eight corner pieces and twelve edge pieces have plastic flanges which allow the layers to turn but at the same time prevent them from coming apart. (6 centre pieces + 1 middle mechanism [not a real cube] + 8 corners + 12 edges = 27 cubes)

The visible faces of the small cubes are coloured squares and the cube comes packaged with the nine squares on each of the six large faces all the same colour. By turning the layers the patterns formed by these colours can be changed. The object of the puzzle is to find out how to manipulate the cube so that a confused pattern can be restored to the starting pattern.

A natural question is "How many patterns are possible?" The answer is a staggering forty-three million million million (forty-three quintillion), or to be more precise: 43,252,003,274,489,856,000. The number of seconds in a year is about 30 million so even if you could look at a thousand patterns every second it would still take over a thousand million (a billion) years to see them all. According to modern theories of cosmology the universe is only about fifteen or twenty thousand million (fifteen or twenty billion) years old, so you would have to have started looking at the patterns fairly soon after the "big bang'' in order to have seen them all by the present! Even if you restrict yourself to looking at the patterns obtainable by moving just two faces, say red and green, you would still have over seventy-three million to look at!

To the uninitiated it is simply a delightful plaything for children or a perplexing puzzle for adults. But to those who know, deceptively innocent exterior can be transformed into a kaleidoscope of changing patterns, governed by subtle laws of mathematical symmetry. Behind the apparent chaos lies a way into the beautiful theory of groups. Even to fully understand the patterns formed by moving just two faces requires a good grasp of group theory and its relation to finite geometry. Yet none of this knowledge is necessary to solve the puzzle. The solution lies solely in diligent observation of the effects of move sequences on the patterns and use of these sequences as tools to change the cube systematically from chaos to order.

Extracted from:

Don Taylor, Mastering Rubik's Cube(TM): The Solution to the 20th Century's Most Amazing Puzzle. Holt, Rinehart and Winston (An Owl Book), 383 Madison Avenue, New York, NY 10017, (c)1980. Lib. of Cong. Card #81-81556, ISBN: 0-03-059941-5.

And personal knowledge: Robert T. Kelley

A great book on the cube and its mathematical properties is David Singmaster's Notes on Rubik's 'Magic Cube', which, I believe, includes some group theory as applied to the puzzle.

See my Guide to Finding a Solution Yourself
Also see The Rubik Homepage

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©1999 Robert Kelley