You can find a lot of solutions for the Rubik's cube on the net but very few websites take the time to explain you the principles behind solving the Rubik's cube. This is a bit unfortunate because solving the cube by simply following a "recipe" will not help you with other similar puzzles.
In this article, I will tell you more about some general principles that can help you solve the Rubik's cube but also other similar twisting puzzles. Have a look.
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I hate this principle as it has little to do with the puzzle itself but I must admit that it is an important one. When you get a puzzle, it is of course nice to start playing around with it and see what you can achieve, etc... but if the puzzle is too difficult to solve purely intuitively, then you should start planing.
Here, for example, you should understand that solving one face, then another face and after that a third face is NOT a good approach. This is where planing should help you. If you read my previous article about Rubik's cube fundamentals, you know that the cube has edges and corners, so a possible plan could be to solve the edges first and then the corners. Or the corners first and then the edges, etc...
I am impressed by Jessica Fridrich when she says she started solving the cube before even having one! 
2. Saving and Restoring
This is an important principle. Imagine the following situation:Credit: Adragast
(this image and other cube images of this article have been generated with http://ruwix.com)
To put the last yellow corner on the top, you simply need to perform the move R (turning the right layer once clockwise when looking at it). The problem is that you are "destroying" two pieces that were solved:
So how to do it? Before performing this R move, you need to "save" the 2 pieces that were going to be misplaced somewhere and you can restore them afterwards:Credit: Adragast
Performing the wanted move
Restoring the "saved" pieces
Please note that this is not the only way to solve the initial situation (nor the most natural one). But it illustrates the concept of:
- checking what you are misplacing by doing what you want
- saving these misplaced pieces somewhere where they are safe
- doing the move you wanted
- restoring what you saved
3. Building Somewhere Else
This is a useful trick for many puzzles. When you see that your freedom is restricted somewhere, you may try to build things together somewhere else and only when they are nice and like you want, do you place them at the proper position.
Using the same example:Credit: Adragast
This time, instead of thinking "I will put this corner on the top", you will think: "I will put together the corner with the solved 2 pieces to form a line". How to do this? Use move D' to move the corner on the side, get the two yellow pieces down with R', move the corner back with D and your line is built! Just use R to place it on the top.
In images:Credit: Adragast
Moving the corner on the side
Coming down with two yellow pieces to create a 3-piece line
Creating the line
And the last move is obvious. I like to illustrate this particular sequence of moves as someone taking an elevator. I see the bottom yellow corner in the initial position as a person who wants to go up. He steps aside to avoid being trampled down by the elevator, the elevator comes, he takes the elevator and the elevator goes up.
This elevator illustration is generally easy to grasp but please remember also the principle in general (building things together somewhere and placing them altogether).
4. Destroying and Recreating
Something else that can help you solve this puzzle is to "destroy" something in a certain way and "recreate" it another way.
We have just seen 2 different ways to solve a corner from the bottom layer. Do the reverse of the first way to move the corner away and apply the second way to solve it again.
The result? What was solved is solved back but as you use one way to misplace and another way to solve back, the rest of the cube is modified.
Congratulations, you have just created an algorithm! What you need is to understand what it does to the other pieces of the cube and when you can use it (through planing, remember point 1 of this article).
In this example, the resulting algorithm is not very helpful but my aim here is mainly to show you some principles, not a complete solution.
This is the most complicated concept here but also the most important one.
To understand this concept, imagine the top layer and the rest of the cube as two disassociated systems. Imagine that you have found a sequence of moves that flips a corner on the top layer but scrambles otherwise the cube.
What to do with such a sequence of moves? Well, you can notice that by applying the inverse of this sequence you are solving back the rest of the cube (but flipping back the corner also). So, if you manage in between to swap corners, you will manage to create an algorithm that flips two corners while leaving the other pieces in place.
Here is the algorithm:
- apply the sequence: it flips one corner and messes up the rest of the cube
- move the positions of the corners (on the top layer without disrupting the rest of the cube)
- apply the inverse of the sequence: it will restore the rest of the cube and flips the corner which is now where the first corner was
- move back the position of the corners
With maths notation, you can write it as X Y X-1 Y -1 (X being your sequence of moves, Y being the move to exchange the corners, X -1 being the inverse of X, etc...).
It took me forever to understand commutators and without the explanation found on this site I may never have managed it. Don't focus too much about it now, but feel free to come back several times as this is really a key concept.
I don't pretend that these principles make it easy to solve the Rubik's cube but they definitively help if you want to solve the cube by yourself or are simply curious about how it is possible to solve the cube with principles and without using algorithms learnt by heart. It can also help you with other similar puzzles.
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