Posted by: gupta2 | October 5, 2008

## Finally, I found an easy way to solve a Rubik’s Cube

I have never been able to solve a Rubik’s cube. I’ve tried multiple algorithms but have never been able to successfully implement them. Some of my friend’s, though, have not had them same issue and quickly picked up the craft of solving Rubiks cubes. Well, today I stumbled upon a tool that will level the playing field: http://www.wrongway.org/cube/solve.html .

So the way this tool works is simple. You look at your Rubik’s cube and on the applets cube diagram you mark how the colors fall on your cube. Then the algorithm will go ahead and find every move you will need to make to ensure the perfect cube. The great thing about it is provides diagrams and you can easily just restart the process if you completely mess up. This tool probably won’t get you breaking world records (http://www.youtube.com/watch?v=JqFFXs0RAnI) but it is a start.

Now, let’s get to a more mathematical aspect of a Rubik’s cube. While I was finishing my cube, I wondered how many actual ways one of these cubes could actually be arranged. The answer actually turned out to be pretty simple to figure out using stuff we learned in class. A rubik’s cube is 3x3x3 cube which means it has eight corners and twelve edges. Thus there are are 8! ways to arrange the corner units. The first seven can be arranged independently of one another but the state of the eighth is dependent on those of the first seven. This gives us 37 possibilities. There are 12!/2 ways to arrange the edges and each of the eleven independent edges can be flipped two ways for a total of 211 possibilities. This gives us a grand total of ${8! \times 3^7 \times 12! \times 2^{10}} \approx 4.33 \times 10^{19}$ or 43 quintillion possible positions. That is an astounding number when you consider that a Rubik’s cube is just a 3x3x3, 57 mm box.

If you want to learn more about the Rubik’s cube, I strongly recommend wikipedia (http://en.wikipedia.org/wiki/Rubik’s_Cube), easily the world’s best source for random information.