I know something about permutations... yet I don't even know how to start figuring out the permutations for the Rubik Cube.
Ok, this is what Wiki says (this info can also be found in Rubik's site):
A normal (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! ways to arrange the corner cubies.
Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 possibilities. There are 12!/2 ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 possibilities: 8! x 37 x 12! x 210 = 4.33x1019
I can probably never come up with that on my own.
Casually searching for info about math on the Rubik Cube, I've found this class!
http://match.stanford.edu/bump/rubik.html.
I wish I can sign up for this class. Look folks, this is abstract algebra in action!
The PDF files are beautifully written in LaTex and they look very interesting! (as long as I don't have to be tested on it in a classroom setting). I wish I understand it all. I love it when advanced math is attached to something concrete (such as the Rubik cube).
2 comments:
I took my Abstract Algebra class and did miserably in it. Didn't understand anything after a third of the class and eventually struggled to get to the end since i need the credit for graduation. The professor gave us grace and I was guessing he gave everyone A's, since the next day after the final he was throwing frisbee on the green. We never got our finals back, but who cares. Yeah, I wish I can fully understand those things some day.
I get a much bigger number if not considering any constraint in the cube:
each side have 8 "free grids", there're 6 sides in the cube, to put one color into the free grids I have:
8*6C8 permutations
after putting all 6 colors into the grids:
8*6C8 * 8*5C8 * ...
= (8*6)!/(8!)^6
= 10^33
way too big...
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