Colors on the Rubix Cube
Date: 01/15/2005 at 12:52:16 From: Jay Subject: How many color combinations are there in a Rubix cube? How many color combinations are there in a Rubix cube? Each of the six faces is a different color, and there are nine cubes of that color on each side. I thought that six to the fifty-fourth power was it because there are six sides on a cube and nine different colors. But then I started to think about the rows and columns and each large face combinations. My mind was beginning to boggle so I thought you would help me get started at least with the correct formula procedure--then I would do the actual math. Thank you very much.
Date: 01/18/2005 at 11:43:34 From: Doctor Vogler Subject: Re: How many color combinations are there in a rubix cube? Hi Jay, Thanks for writing to Dr. Math. Your suggestion of 6^54 is a good guess, but you might notice that it also includes the possibility of all 54 faces being red. That can't happen. Look closely at your Rubik's Cube and you'll find that those 54 colors lie on 8 corner pieces which have 3 colors each, 12 edge pieces which have 2 colors each, and 6 center pieces which have 1 color each. Then you might also notice that the center pieces don't move. All of these observations will also help you to learn how to solve the cube. Now you might notice that you can't swap corner pieces with edge pieces, or other odd combinations, but you can rearrange the corner pieces, and you can rearrange the edge pieces. (The corner pieces don't move, recall, except by rotating the whole cube.) You can also rotate the corner pieces into three different positions, and you can flip over the edge pieces. So all of this might lead you to believe that there are 12! ways to permute the corners, 8! ways to permute the edges, 3^8 ways to rotate the corners, 2^12 ways to flip over the edges, for a grand total of 12! * 8! * 3^8 * 2^12 total positions of the Cube. But you should first ask if all of these positions are possible. In fact, they aren't. First you should check that the last corner's rotation will always be determined by the other 11 rotations, and the last edge's position (how it's flipped) will always be determined by the other 7. Furthermore, half of the 12! permutations of the corners are what mathematicians call "odd" permutations, and half of them are even. Same with the 8!. It turns out that if the corners are in an odd permutation, then so are the edges, and vice-versa. So that means that we really have 12! * 8! * 3^7 * 2^10 total positions of the Rubik's Cube. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
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