The Rubik's Cube Unravelled...or is that unpieced ?

21 June 2006
Posted by Robert Stanforth.

RubikThe Rubik's Cube is all about symmetry... To make some sense of this bold statement, let us take a closer look at what symmetry really means. When we say that a heart symbol, for example, is symmetric, we are normally referring to its 'mirror' symmetry. This means that when viewing the heart through a mirror, what you see is indistinguishable from the heart itself. We can be more precise about this, and describe the operation - 'the reflection' - that may be applied to the heart without changing its appearance. Reflecting in the heart's vertical centre line, i.e. exchanging its left-hand and right-hand sides, leaves it unchanged.

Sometimes it is useful to keep track of whether the heart has been reflected or not. You could attach coloured labels to the heart; say yellow for the (original) left hand side and blue for the right. Then the reflected heart will appear with blue on the left and yellow on the right. It is important to note, however, that these labels are not a part of the heart itself, for then the heart would no longer have any symmetry. Instead, they are carried along whenever the heart is reflected to allow us to distinguish configurations that the heart itself cannot.

To take a slightly more interesting example, three coloured dots may be attached to the corners of an equilateral triangle. There are six ways in which the triangle may be rotated or reflected (including the 'trivial' operation of leaving it alone), and each results in a different arrangement of the coloured dots.

The sets of symmetry operations that we have considered so far have all been 'rigid operations'. With the Rubik's cube, we may consider a different group of symmetries: those arising from twisting the faces. If we overlook the colours of the stickers, then turning a face by a quarter turn leaves the cube apparently unchanged. The same is true of any arbitrarily long sequence of turns carried out in succession - the net result does not change the shape of the cube, and so any sequence of turns is also a symmetry operation.

Of course the coloured stickers will rapidly become infuriatingly jumbled up. Exactly as in the case of the triangle we may consider the coloured pattern to be an indicator of which particular symmetry operation has been applied to the solved cube to bring it to its current configuration. If we can express this symmetry operation as a particular sequence of quarter-turns, then solving the cube from this configuration is merely a matter of undoing these turns in reverse, one by one.

Expressing this symmetry operation - this encapsulation of the current configuration - as a sequence of individual quarter-turns is infamously difficult. To make things worse, each different configuration will be represented by a different sequence of turns, dispelling the common myth that there is one sequence of turns solving the cube from any position. Instead we seek a recipe: one that, given the current configuration, will construct a sequence of turns for that configuration.

Fortunately there is a central branch of mathematics called 'group theory', which analyses such families of symmetry operations and the effects of combining them. Group theory provides just the right tools for solving the Rubik's Cube, and for working out more of its properties besides. For example, the number of possible distinct configurations of the Rubik's Cube can be calculated as 43252003274489856000, compared with 2 for the heart and 6 for the triangle.

However, this vast number forms only a subset (in the parlance, a 'subgroup') of the configurations that become attainable if you dismantle the cube and reassemble the cubelets. In fact only one in twelve such reassemblies are attainable by the conventional face turning; in other words, if you regroup the cubelets into a random configuration then there is only a one twelfth chance that the cube can be solved without resorting to dismantling it again. To take an analogy with the triangle, it's as if the 'valid' moves consisted only of rotating the triangle. However, once the triangle is reflected, it can never be restored to its original position by rotation alone. It needs to be reflected back again.

And if you are one of those vandals who pulls the stickers off, then group theory has the last laugh. The odds that you will end up with a soluble cube are around one in a million million million !!

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