Score Transformation
Score Change
Score Change (mod 3)
A
0 --> 1 1 --> 2 2 --> 0 0 --> 2 1 --> 0 2 --> 1 0 --> 1 1 --> 2 2 --> 0 0 --> 2 1 --> 0 2 --> 1
+1 / -2
1
B
+2 / -1
2
C
+1 / -2
1
D
+2/-1
2
Total: 0 The truth after k moves implies that the total score is also divisible by 3 after k+1 moves. Therefore, total score change is always divisible by 3 by mathematical induction. If you assume that 7 of the 8 corner cubies are arranged randomly, the remaining corner cubie can only be 'inserted' in a particular orientation to create a 'valid' permutation, so only one-third of the permutations are valid. N/A -6/-3/0/3/6
Edge Orientation proof
The edge-orientation rule will be proved similarly by creating a 'score' system by induction. We position the cube with the yellow face as the 'U' face and the green face as the 'F' face. (Blue = 'B' face, Red = 'L' Face, Orange = 'R' face on standard cube) The scoring system is as follow page:
Initial Position of Cubie (on solved cube) Top/Bottom Layer (containing white/yellow)
Quantity
New Position Condition
Scoring
8
Top Layer
White/Yellow facing upwards
0 if the condition is satisfied, otherwise +1
Second Layer White/Yellow on the L or R face Bottom Layer White/Yellow facing downwards
Second Layer 4
Top Layer
Orange/Red facing Upwards
50
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