How can you apply math in solving the rubiks cube?

Question

I understand you can use abstract algebra. Please explain the abstract algebra and its use in solving the rubiks cube

Answer 1

http://akbar.marlboro.edu/~mahoney/courses/Spr00/rubik.html

Answer 2

A group is a set "S" and a binary operation "*" over the elements of "S" that satisfy a few simple rules:
- for all a and b in S, a * b is a unique element of S.
- (a * b) * c = a * (b * c)
- there is an element e of S such that a * e = e * a = a
- for every element a of S, there is an element a_inverse such that a * a_inverse = a_inverse * a = e.

For Rubik's Cube, the set "S" is the set of all changes to the cube. "*" is concatenation; apply change A and then apply change B. The do-nothing "change" is the identity element.

The key to solving the cube is the formula A * B * A_inverse * B_inverse applied in a certain way.

First, note that for any sequence of changes A * B * .. * Y * Z, you can always back them out and get to where you started by doing Z_inverse * Y_inverse .. B_inverse * A_inverse.

Now, hold the cube and cover up the bottom two levels, with only the top 3x3 level showing. Imagine some move that flips one of the four middle-edge pieces over but otherwise leaves the top level unchanged. (This move will temporarily screw up the lower two levels, but we are not looking at them, remember? :-)) You can always undo this move, end up right back where you started, and have that middle-edge piece back the way it was. BUT!! What happens if you "cheat on" the cube, and rotate the top level 90 degrees before doing that "undo" sequence? Please think for a moment about this very simple trick.

You will have flipped TWO of the four top middle-edge pieces over, and left the rest of the cube entirely unchanged!

The first edge-flipping move is "A", and the 90-degree rotation is "B". So, A * B * A_inverse * B_inverse does our two-edge flip.

Now, figure out a way to rotate one of your four top corners by 120 degrees, again leaving the rest of the top unchanged. The same trick as above will allow you to rotate two corners by 120 degrees, in opposite directions.

Finally, say you do a move that replaces a top-row piece with one from the lower two rows. You can always undo this sequence, and restore things to the way they were when you started. But, let's say again you "cheat" the cube and rotate the top first. When you do the "undo" operation, what happens?
- The top "keeps" the piece from below that was originally transferred up.
- The top piece that was sent down below comes back up,but into a different top slot.
- The top "gives back" to the bottom a different piece than the one that the bottom had originally given.

So, we swap around three pieces, and leave the entire rest of the cube unchanged.

You now know everything you need to know about solving the Rubik's Cube, modulo the easy tasks of changing the top level as needed. And, one other minor observation is required. So far, we only know how to rotate pairs of pieces that are both on the top level. If we have a pair that are not together on a level, well, do one rotate so that they temporarily are on the same level, fix them, and undo that rotate. I.e., A * B * A_inverse.

Answer 3

You don't actually use abstract algebra to solve a Rubik's cube, but abstract algebra can be used to explain some of the stuff.

I first started solving cubes when I was 15; long before I had any group theory.

When solving a Rubik's cube (at least what I do) is to start off by placing pieces where you want them. Then you need to keep those pieces there (or make sure they go back after a "move") and put other pieces where you want them. Starting out, the moves are very easy, since you don't have to worry about many pieces, but as you get more set, you need to use more complicated moves.

This a great cube page with a lot of awesome videos:

http://www.speedcubing.com/chris/

Answer 4

i can do it under 30 sec no lie and under 30 is not bad its really bad. under ten is good not great. didn't use math to solve it. i dont even use math to solve a 11x11. i dont even use thinking. but if u were to do it with no knowledge of how to solve it than its gonna take alot of math.

Answer 5

You need to use group theory. It is far too complex to describe here. But basically it shows how repeated applications of a pattern of moves will cycle the orientations of the little cubes in predictable patterns.

Answer 6

OMG........It is so funny you asked this. Back in 80, when Rubik's first came out, my sister and brother-in-law were visiting from Australia. He took my cube, and worked on it all night. In the morning we found him, among mathmatical equations and diagrams, asleep at the dining room table(sitting straight up), with the cube still in his hands. If their is one please let me know so I can tell him.ROTF