From:

~~~ Subject:

In rec.puzzles article <1990Nov8.182534.18625@agate.berkeley.edu>,

greg@math.berkeley.edu (Greg Kuperberg) writes:

Consider a standard Rubik's cube. Disassemble it and put it back

together at random. Find, with proof, the probability that it can be

solved.

It depends on how you take it apart. If you just pull out the corner

and edge pieces then put them back in without respect to color, the

probability is one in 12 that you will put it back into the right

orbit. I won't bore you with yet another proof of this; if you spent

the last decade in a box see the archives, Singmaster's NOTES ON

RUBIK'S MAGIC CUBE, J. A. Eidswick's article in the March 1986 Math

Monthly, or even Hofstadter's METAMAGICAL THEMAS.

Now if you take the face centers off and scramble them, then there is

only one chance in 60 of getting it right. Of the 720 permutations of

the six face centers, only 24 can be generated by rigid motions of the

cube. But half of these 24 permutations are odd, and leaving the cube

in an unsolvable orbit. If you put the face centers on in the

``standard'' configuration with opposite faces ``differing by yellow''

(i.e., white opposite yellow, red opposite orange, and blue opposite

green), your chances go up to one in four--half the time you will get

an odd permutation, and half the time you will get a mirror-reversed

configuration.

But wait, if you took the face centers off you probably noticed that

the corners and edges don't stay on very well. So, say you scrambled

all three kinds of pieces. You will be able to solve the resulting

cube if you could solve the corner/edge permutation and the face-

center permutation. But if the only thing keeping you from solving

the corner/edge permutation and the face-center permutation is that

both permutation parities were odd, then you will be able to solve the

two of them together. Therefore your chances of success are one in

360 (= (1/12)*(1/60)*2), or one in 24 if you preserved opposite pairs

of face centers.

Now suppose you peeled off the 54 colored stickers and stuck them back

on at random (carefully keeping them out of the reach of children, as

there are rumors the paint contains lead, especially on the cheap

Taiwanese knockoffs), what is the probability of getting a solvable

cube? This question was posed years ago (in Singmaster?) but I

believe it is still open.

Dan Hoey

Hoey@AIC.NRL.Navy.Mil