thus giving 8! × 3 8 arrangements. However, not all of these arrangements are valid. In fact, only one-third of the permutations have valid corner orientations, half of the permutations have valid edge orientations, and a further half have the correct cubie-rearrangement parity (these will be explained later) 31 . This gives 8! .12! . 3 8.2 12 3.2.2 = 43252003274489856000 (4.325 × 10 19 ) valid arrangements.
Corner Orientation Proof
We position the cube such that the yellow face is facing upwards and the white face downwards (a standard cube has yellow and white on opposite faces). For each of the 8 corner cubies, we assign a number 0, 1 or 2 based on the colour facing upwards or downwards .
If the corner cube is on the top layer, the 'score' is determined by the colour facing upwards.
If the corner cube is on the bottom layer, the 'score' is determined by the colour facing downwards.
If the determinant face is yellow or white, the score given is '0'. Other colours are given a score with the following (1221) pattern for each face. We can then
prove by induction that the total score for all 8 cubies is always divisible by 3. Induction Base : In the initial
solved position, the total score is 0, which is divisible by 3. Induction Hypothesis: We assume the total score is divisible
by 3 after k moves. Induction Step: If the k+1 th move is on the yellow or white face, the total score stays the same (orientations of all cubies stay the same). Notice that the effect on the score of the move R is equivalent to (U D' F). Also, F' is the same element as FFF. Therefore, it is sufficient to prove that the move F changes the score by a multiple of 3 (or 0). Without loss of generality, we let the green face be the front face and label the 4 corner cubies A, B, C and D. When we perform a F move:
31 Mathematics of the Rubik’s Cube - Permutation Group (no date) Available at: http://ruwix.com/the-rubiks- cube/mathematics-of-the-rubiks-cube-permutation-group/ (Accessed: 19 August 2014)
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