Possibly you have seen or even own a Rubik's cube. Hopefully you know the objective is to rotate the sides of the cube to get all the same colored squares on the same side. And you probably are also aware that it is very difficult. There are approximately 42 zillion different states the cube may be in, only one of which is the correct one. Trial and error is not very likely to work here. (Ironically though, many of us seem to try this method anyway. Though you have a better chance at winning the lottery than relying on blind luck here...)

But happily, you don't need to be frustrated any longer. For your convenience (as well as my own) I have posted this solution to Rubik's Cube. At each step, your cube will be in one of a number of states. You simply have to identify the state that applies to your cube at each step, and perform the rotations indicated. If you do so correctly, then the cube will be solved (and you may be revered as a Mathematical Wizard by friends and family).

But before we start, we need some notation. Hold you cube in front of you. The side facing up, we call the

At this point a number of little warnings are justified:

Warning (1):

Warning (2):

Now, back to the notation. Using the names for faces, we now have names for all the 26 cubes that make up Rubik's Cube (we don't really count the center of the cube, because we never see it). We name each cubelet (a cubelet is one of the smaller cubes that makes up the big cube) with the letters of the sides it is a part of. For example, the cubelet in the upper right-hand front corner, be will referred to as

We have names for all the cubelets. Now we need to determine a way to describe how we rotate the faces. What we do is let

Warning (3): When we say rotate a face clockwise (or counterclockwise), that means rotate it in the direction that would be clockwise if you were looking straight at it. But don't lose your current orientation!

Warning (4): You may be wondering how we can let one symbol stand for two different things, i.e. "

Also, to make reading the sequences of movements simpler to follow visually, if you see

To summarize everything in an example, lets say you have the following sequence of moves:

You should read this as "rotate the front side clockwise 90 degrees, then rotate the right side counterclockwise 90 degrees, then rotate the back side 180 degrees. Next, rotate the left side clockwise 90 degrees followed by rotating the down side clockwise 90 degrees. Perform this last step a total of three times."

See how having a notation makes everything compact and easy to read?

So without further ado, onto the solution:

If it is already in the middle layer (and in the wrong position) then you need to "move it out". Change orientation so that the cubelet you want to move to the top is in the

You should now be able to apply one of the following sequences:

## A: | If the F side of the piece that belongs in the FR position is facing up, then turn U until the piece that belongs in FR is in the UL position. Then apply (L U) (U^{2} F^{2})*3 (U' L') | |

## B: | If the F side of the piece that belongs in the FR position is not facing up, then turn U until the piece that belongs in FR is in the BU position. Then apply (B' U) (R^{2} U^{2})*3 (U' B) |

Repeat Step 3 as needed until all middle layer pieces are in place.

An even number of

Your cube should fall into one of the following cases:

## A: | If the UB and UF edges are incorrectly oriented, then apply
(B L U) (L' U' B') | |

## B: | If the UB and UL edges are incorrectly oriented, then apply
(B U L) (U' L' B') | |

## C: | If all 4 edges are wrong, then: (i) apply step 4a (ii) change orientation by rotating entire cube 90 degrees in such a way that F becomes L, L becomes B, etc.
(iii) apply step 4a again |

## A: | If they are all in place, then skip this step. | |

## B: | If one edge is in place, then turn then whole cube so that this is at the UL position. To cycle the other three:
(i) clockwise, apply (R^{2} D') (U^{2} R' L F^{2} R L') (D R^{2}) (R^{2} D') (R' L F^{2} R L' U^{2}) (D R^{2}) | |

## C: | If no edges are in place, then turn the whole cube so that you need to exchange UF with UR and UL with UB. Then apply
(R^{2} D^{2} B^{2} D) (L^{2} F^{2})*3 (D' B^{2} D^{2} R^{2}) |

## A: | If they're all correct, skip this step. | |

## B: | If one corner is placed correctly (meaning it is in the right position; it does not matter if the sides of the cubelet are facing the right way), then you will need to cycle the other three.
Turn the cube to move the correctly positioned corner to URF, then:
(i) to cycle corners clockwise, apply (L') (U R U' R') (L) (R U R' U')
(U R U' R') (L') (R U R' U') (L) | |

## C: | If there are no corners placed correctly, then:
(i) To exchange adjacent corners ( ULF and URF will switch positions; ULB and URB will switch positions), apply (B) (L U L' U')*3 (B')ULF and URB will switch positions; ULB and URF will switch positions), apply (R' B^{2}) (F R F' R')*3 (B^{2} R) |

## A: | to spin URF clockwise, apply
(D' F D F')*2 | |

## B: | to spin URF counterclockwise, apply
(F D' F' D)*2 |

Turn the

That's it! Your cube should now be restored to its original order. Enjoy your cube!