Rubik's Cube Possibilities Question?

Question

In the summer of 1974, a Hungarian architect invented a three dimensional object that could rotate about all three axis. He wrote up the details of the cube and obtained a patent in 1975. The cube is now known worldwide as the Rubik's Cube. When you purchase the cube (3x3), it is arranged so that each face is showing a different colour, but after a few turns it seems next to impossible to return to the start. A manufacturer claims there are 8.85801027 x 10^22 possible arrangements, of which only contains only one correct solution. Do you agree with the claim? If yes, show how you obtain that answer; If no, what is the possible arrangements. Show your working properly.

Answer 1

The proof is by counterexample. No other solution arranges the colored squares to be the same on each side. Notice that the cube has 6 squares which cannot move topologically speaking. These are the center squares. They are attached to the moving mechanism and fixed in place. Any amount of rotation does not change the location of these center squares. That means the topological order of which color appears on which face cannot change.

As such, there's no possible solution where you can move a cube to another location and still have it work out. For example, take a the side (middle edge) cube between white and red. You can't put that anywhere else because the white has to match the white side and the red has to match the red side. Ergo, only on possible location. Same holds true for corner pieces. The colors have to match up or you don't have a solution.

Answer 2

A normal (3×3×3) Rubik's Cube can have (8! × 3^(8−1)) × (12! × 2^(12−1))/2 = 43,252,003,274,489,856,000 different positions (permutations), or about 4.3 × 10^19, forty-three quintillion (short scale) or forty-three trillion (long scale). The puzzle is often advertised as having only "billions" of positions, as the larger numbers could be regarded as incomprehensible to many.

In fact, there are (8! × 3^8) × (12! × 2^12) = 519,024,039,293,878,272,000 (about 519 quintillion on the short scale) possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually reachable. This is because there is no sequence of moves that will swap a single pair or rotate a single corner or edge cube. Thus there are twelve possible sets of reachable configurations

And if for any reason, later on, anybody says that I'm cheating because I pasted it from wikipedia, let me tell you that I actually DID the wikipedia update!!

Answer 3

artie cheated. copy-pasted the answer from wikipedia.

http://en.wikipedia.org/wiki/Rubik_cube

from the same source, i understood the asker's number came from the Rubik being allowed to change centers or something. but in normal rubik, where you made no marking, the permutation is 1/2048 of the asker's number. so, not agreeing for reason that happen only because of tampering on the rubik's.

Answer 4

I agree with the claim that there is only one solution, as for the rest?

Answer 5

i can solve it in about 2 minutes.... so I can navigate the quintillion combinations easily enough...looks like you already have a pretty good answer on possibilities so i'll forgo anything else.