# How to solve Rubik's Cube using mathematical formulas? [closed]

I am trying to solve Rubik's Cube, but it took one month for me when I started. Are there any mathematical formulas, rules and tips for solving Rubik's Cube in less time (i.e. minutes)?

• I think this already has an answer here: puzzling.stackexchange.com/questions/4820/… Commented Apr 5, 2017 at 15:42
• @F1Krazy No, It wasn't useful for me. Commented Apr 5, 2017 at 15:44
• Do you want mathematical approaches, or do you want to increase your speed? It sounds like you want to understand how to solve it logically rather than just by following a method? Is that right? Commented Apr 5, 2017 at 16:02
• @DrXorile, If it is logical or following any method (Initially) , I want to increase my speed (first priority) Commented Apr 5, 2017 at 16:09
• @DrXorile: In case you are interested in a general mathematical solution for all pure permutation puzzles, see first this post and then this post. Commented May 29, 2021 at 18:34

I avoided doing the rubic cube for a while because I thought that you just had to learn a set of moves. But one day I thought it must be possible to figure out the puzzle, and I have become convinced that the real puzzle is not solving the cube but rather figuring out how to solve the cube.

A common approach is to solve it layer by layer. Another approach is to solve the edges and then the corners. I'll look at the former first.

The top layer is relatively easy (make sure it includes the central squares of the four surrounding pieces of course)

The second layer needs a way to insert the 4 edge pieces without messing up the top layer. Here is a logical approach, based on the fact that the bottom layer is already messed up and so it doesn't matter if you mess it up more:

1. Look at the side of the cube so the top layer is the bit you've solved and the bottom is unsolved and the middle right is the one you are trying to solve.
2. You need to find a set of moves that leaves the middle right untouched; replaces the top and middle left with squares from the bottom; makes sure that the bottom layer is still got squares from the bottom layer; and ensures that the piece that fits in the middle right is correctly orientated at the bottom middle.
3. Now turn that face anticlockwise, so that the middle right is correct. The only other pieces you are effecting are pieces that were on the bottom, so don't matter.
4. Now undo the moves you did in 2 and you'll have fixed the middle right without changing the top layer.

Think about this and the logic of number 2 and you'll see why it works. Do this four times. There are slightly easier variants available if the middle left is incorrect, because then you only need to replace the top layer.

Finally, the bottom layer. There are various things that could be wrong. But the key is going to be to find pairs of opposite moves to correct it. This is always possible.

1. Do a sequence of moves to do pair one. This will involve messing up the whole cube, of course, but must end with all the bottom layer still on the bottom, but with a particular pair done. Very similar to your first layer moves.
2. Twist the bottom so that the second pair is lined up.
3. Undo the moves in 1. This will fix the cube to its original state, except that instead of undoing pair one, it will do pair two which is the opposite move to pair one.

With this very general framework (and I'm sure there are others out there) you can always solve it and also know why it's working...

• Thanks for details explanation. I can do top and second layer easily but my problem with Bottom layer. I mess up with many moves but every time I missed with pair of moves to get it done (may be I am missing Undo), I will try and make it possible. Cheers :) Commented Apr 5, 2017 at 16:48