The "Games and Mathematics" article in "Rubik's" Magazine (mentioned
yesterday) asks an interestion question: how can you characterize the
random coloring on a cube in order to determine if the cube is 1) solvable
by twisting, or 2) solvable by dismantling and reassembling.
The obvious criteria are 6 colors, 9 of each, 4 on edges, 4 on corners, 1
on a center, no 2 facies of a cubie the same color. For case 2, Keri
claims you need 4 more tests. For instance, he gives test 1: Given 1
corner with colors A, B, and C, let the other 3 colors be a, b, and c.
Then you can't have a capital and small of the same letter on one corner,
and the 8 corners are exactly the 8 combinations. What are the other
tests? Are 4 really necessary? What are the tests for case 1?
By the way, he (from some other article) classifies the 3 unscrambling
methods as follows:
1) Chemical unscrambling: repaint the sides.
2) Physical unscrambling: dismantle and reassemble
3) Mechanical (or mathematical): normal way, by twisting.
--- Stan -------