Corners : Three letters ( Italics used for letters representing cubies to avoid confusion with letters representing moves)
The Rubik's Cube has a fairly simple structure. A standard 3 × 3 × 3 Rubik's Cube is made up of 26 cubies 25 , including 12 edges, 8 corners and 6 centres (and an empty core). An edge cubie has 2 faces, a corner cubie has 3 faces while a centre cubie only has 1 face. The colour of the centre cubie determines the colour of the face in the solved state because the arrangement of the 6 centre cubies is fixed. 26
Before we further investigate the different permutations of the Rubik's Cube, we first need to understand the idea of “groups”. A 'group' is formed by a set of elements, G , with a binary operation • (the group law of G) that combines any two elements in G , x and y (written as x • y or xy ) to form another element) 27 . The set of elements and the operation must satisfy the following group axioms 28 : 1. Closure - the result of the operation x • y is also an element in G 2. Associativity - ( x • y ) • z = x • (y • z ) 3. Identity element - There is an element e in G, such that x • e = e • x = e 4. Inverse element - For each element x in G, there is an element y such that x • y = y • x = e We can apply group theory on the Rubik's Cube. Each cube permutation can be represented by a group element. 29 If two sequences of moves cause the same permutation but consist of different 'moves', they are still the same element. (E.g. LLL=L') The group operation of the Rubik's Cube group is the concatenation of cube moves 30 . The Rubik's Group obeys all 4 group axioms, for example, the identity element ' e ' of the Rubik's Group is the 'empty' move'. We can work out the total number of permutations of the Rubik's Cube, or in other words, the total number of elements in the Rubik's Group. There are 12 edge cubies, each of which with two possible orientations, giving 12! × 2 12 arrangements. Also, there are 8 corner cubies each with 3 possible orientations, 25 Cubie (no date) Available at: http://en.wiktionary.org/wiki/cubie (Accessed: 19 August 2014) 26 Algorithms to solve Rubik’s cube (2011) Available at: http://www.cs.swarthmore.edu/~knerr/helps/rcube.html (Accessed: 19 August 2014) 27 Group (mathematics) (2014) in Wikipedia. Available at: http://en.wikipedia.org/wiki/Group_(mathematics)#Definition_and_illustration (Accessed: 19 August 2014) 28 Herstein, I. N. (1975) in Topics in algebra. New York: John Wiley, p. 27 29 MIT (2009) The Mathematics of the Rubik’s Cube. Available at: http://web.mit.edu/sp.268/www/rubik.pdf (Accessed: 19 August 2014)
30 Rubik’s Cube group (2014) in Wikipedia. Available at: http://en.wikipedia.org/wiki/Rubik’s_Cube_group (Accessed: 19 August 2014)
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