Hold a solved cube in front of you.
All the pieces are correct.
Move one side a quarter twist.
Now some of the pieces are not correct. Eight to be exact.
Turn another side. It doesn't matter which one. It doesn't matter which direction. It doesn't matter if it's a double twist either. Just turn one more side.
Now undo your very first move.
There are still exactly eight pieces that are incorrect.
I could prove to you why this is always the case. But you can take my word for it. This information isn't of much practical value, but it does show one thing: the cube has many mathematical constraints. Rarely can anything about a Rubik's Cube be described as "random".
When I learned how to solve a Rubik's Cube, I memorized sequences moves. Each move was a tool I kept in a mental toolbox. Each tool has a specified function and was used in a specific situation. When the situation arises, I pull out the tool and use it with full confidence that it will do what I intend it to do without questioning how it works or how it was invented. When you begin solving a cube, tools aren't very necessary. So many pieces are in the wrong place, many of which you won't care about until you're nearly done solving the cube so you have so much room to work with. The opening of your solve is easy. Many people can solve one side of a cube, because it's a small task of simply moving pieces out of the way temporarily to put other pieces in place. Some people can continue on and solve two rows, because they have that precious bottom row to use as "fudge room" to position the other pieces. But after that, almost everyone looks towards move sequences as their only tool for solving any further. Rubik's Cube move sequences can be created by a computer program with some sort of search algorithm, but that's no fun. Nor do you learn anything from that.
Hold a solved cube in front of you again.
We are now going to derive the algorithm to shuffle 3 corners.
Move the top once. It doesn't matter which direction. Just move it.
Eight pieces are now wrong. How do we fix this? Well, we just move it back. This is trivial and obvious.
One move can be undone by moving that same face in the opposite direction.
Now here's a very short sequence of moves that you're probably very familiar with.
- Turn the right side counterclockwise.
- Turn the bottom clockwise.
- Turn the right side clockwise.
Suppose you were solving the top row in a traditional manner and there was a corner piece in the FRD position with the top color's sticker facing left. Then you would use this move to put that corner into place on the top row because it does not disrupt anything else on the top row other than that one corner.
This is a handy move, extremely short, and something you were probably using all the time already.
Because you had a solved cube before you made these 3 moves, then the top row should still be entirely solved, except for the one corner in the front-right. The piece that goes in that position is now on the bottom row. Somewhere. There is also some random piece in that corner's place. What if wanted to solve the cube again? That's obvious. We just do the same 3 moves except backwards.
- Right counterclockwise.
- Bottom counterclockwise.
- Right clockwise.
These are the same three moves, except in the opposite order and in the opposite direction. Just like how we reversed one move earlier.
Do these same three moves again and then undo them. Do that over and over. Just to get a feel for it.
Watch the pieces as you do this. Eight wrong. Eight correct. Eight wrong. Eight correct.
Now do just one move. Now undo it. Do that over and over. Eight wrong. Eight correct. Eight wrong. Eight correct.
No, I'm not crazy. Bear with me. This is where the fun comes in.
Cosider this:
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Do the three move sequence outlined above.
There are eight wrong pieces, but only ONE is in the top row. This move will stick that random wrong piece in the top row and moves the correct one somewhere to the bottom.
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Turn the top in a random direction.
You now rotated the the top. Instead of adding 8 wrong pieces, you've only added 7 because one was already wrong. Furthermore, the wrong piece that was in the top row is now in a different wrong place, and a DIFFERENT top row piece is now in the line of fire for...
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Undo the three move sequence.
Ordinarily, this would simply restore the cube by moving the wrong piece in the top row to its original position, and move the top row piece on that bottom back to its original position on the top. But now, there's a different piece in FRT. This different top-row piece goes to FLD and the top-row piece on the bottom now goes to the top row, but in a different slot.
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Undo the random move you did to the top.
Now the rest of the top row (aside from those 2 corners) returns to its original state.
Congratulations. You have just seen the theory behind a 3 corner shuffle.
Many 3 piece shuffle algorithms (for both edge and corners) fall into this formula...
- Move around the cube such that two of the pieces you want to shuffle are on one side and the third piece you want to shuffle is on the opposite side. Let's call these sides side A and side B respectively.
- Move the piece from side B to where one of the pieces from side A was. However, move it in such a way that this was the only piece that is disrupted on side A.
- Twist side A such that the currently undisturbed piece is now in the location of the "bumped" piece.
- Undo your first sequence of moves
- Undo your single twist
These three pieces, whether they are edges or corners, are now shuffled. Using a little preplanning, you can use this to completely control the orientation and the direction of the shuffling.
This also isn't just handy for permuting three corners and edges, but also for orienting them too. Remember how I said the cube is mathematically constrained? It is impossible for one corner to be oriented incorrectly. It is also impossible for one edge to be oriented incorrectly. There always must be at least two. When you're down to the last 3 edge pieces or the last 3 corner pieces to solve, when you transfer one piece from one side to the other before you make switch it with another piece, if you bump a piece that's already oriented correctly, and if your transferred piece is placed correctly, then you can automatically assume the final piece will also be correctly oriented. This is the basis of 3-corner orientation algorithms or 2-edge orientation algorithms. You break the positioning of the 3 corners or two edges and then you restore them, but in a different and controlled way, such that the orientation fixes itself upon resetting them.
