## Beginner’s guide to solving Rubik’s 2X2

Update – Aug 18, 2013: This blog post will teach you how to solve the 2X2  cube using two different methods. If you’d rather read a more introductory text with lots of images that explain all concepts and moves, then you might want to buy my eBook,  titled “Beginner’s guide to solving the 2X2 cube”. In my eBook, you will find only the first of the methods described here, but depicted and analyzed with the help of images and elementary explanations. So, if you feel overwhelmed by this blog post (even though you should not be), you could start by reading my eBook first.

Title: Beginner’s guide to solving Rubik’s 2X2
Alternative title: All your cube are belong to us
Note to self: All the YouTube videos that show how to make the tube spin easier by lubricating it, are doing essentially what I call “They lube the cube and tube the lube”.

And yes, you have to endure these lame (to say the least) jokes of mine in order to read this guide.

Rubik’s 2X2X2 cube (2X2 in short) is solved like Rubik’s 3X3X3 cube (3X3 in short), where in 2X2 we only have corners. In 2X2 there are no edges, so we only need the algorithms and procedures from 3X3 that pertain to corners.

In this guide, I will describe two easy methods for solving the 2X2. “Easy” means that the person who solves the cube does not have to remember a lot of cases and algorithms. So these methods are easy to learn. The downside is that they require a lot of movements. There are other methods that solve the cube using less turns, but they are not easy to memorize. These other methods are for advanced solvers. This is why I characterized this guide as a “beginner’s guide”.

It is fundamental to understand the difference between a layer and a face. A solved layer has correct colors an all sides as well as its face, whereas a solved face may not have correct colors on the sides.

It is also fundamental to understand the difference between turning a layer and orienting the cube. When we turn a layer, the rest of the cube remains still. When we orient the cube, we rotate the whole cube without changing it.

It is also fundamental to understand that a corner can be in its correct position but not with the correct orientation. If a corner is in its correct position and with its correct orientation, then we consider this corner to be completely solved. This last concept will be used in the second method that I will describe.

I will use the standard notation for the movements that need to be done. This notation is described in the excellent and colourful PDF guide for the 3X3 in the rubiks.com web site.

For the following methods we will first solve the white layer with no help and then we will use algorithms to solve the opposite layer (the yellow layer).

First method for solving the 2X2

I learnt this method by reading the PDF guide for the 3X3. In this method, to solve the yellow layer, we first put all yellow squares at the top face and then we rearrange them in their correct positions.

First step: We solve the white layer with no help, then we orient the cube so the solved white layer is at the bottom, facing the floor.

Second step: In this step, we will bring all yellow pieces to the top (without caring about their sides), thus creating a yellow face.
The procedure to do that is described in the PDF guide, in the page titled “SOLVE THE TOP LAYER – GET ALL THE YELLOW ON TOP – 2nd Step: Make all the corners yellow”.
I will describe the procedure here as well:
(Please note that before we begin, we do not need to turn the upper layer in relation to the bottom layer. In other words, the relative orientation of the two layers does not matter.)
Procedure for the second step:
First we look at the top face.

• If all squares are yellow, we do need to execute this procedure and we can proceed to the next step.

• If there are no yellow squares on the top face, we orient the cube so that we have a yellow square on the near-to-us left upper side square. • If there is one yellow square on the top face, we orient the cube so that the yellow square is on the near-to-us left top square. • If there are two yellow squares on the top face, we orient the cube so that we have a yellow square on the front left top side square. After this orientation of the cube, we perform this algorithm: R U Ri U R U U Ri. If, after this algorithm, all top squares are yellow, we proceed to the next step. If not, we repeat the procedure.
Please note that the algorithm R U Ri U R U U Ri leaves the bottom layer intact after it is finished. Thus, during different executions of the procedure, we can turn the top layer in any way we want, although this is unnecessary. So we can orient the cube instead of turning the top layer, but either turning the top layer or orienting the cube will do.

Third step: This is the final step. In this step, we will rearrange the yellow corners so that they are in their correct positions.
The procedure to do that is described in the PDF guide, in the page titled “POSITION THE YELLOW CORNERS CORRECTLY”.
I will describe the procedure here as well:
Procedure for the third step:
First we turn the top layer so that as many corners as possible are in their correct position.

• If all four corners are in the correct positions, then the cube is solved!
• If two corners next to each other are in the correct positions, then we orient the cube so that these corners are in the back and we do the algorithm Ri F Ri B B R Fi Ri B B R R Ui. The cube is solved!
• If two diagonal corners are in their correct positions, we do not have to orient the cube in any particular way, and we do the algorithm Ri F Ri B B R Fi Ri B B R R Ui that was just mentioned. This will transform the upper layer in such a way that we will be able to turn it (the upper layer) so that two corners next to each other will be in the correct positions. So we then orient the cube so that these corners are in the back and we do the algorithm Ri F Ri B B R Fi Ri B B R R Ui one final time. The cube is solved!

This method is thoroughly described in the two pages of the PDF guide and we only have to learn two algorithms R U Ri U R U U Ri and Ri F Ri B B R Fi Ri B B R R Ui. This method requires minimum thinking and examination of the cube.

Second method for solving the 2X2

I learnt this method by watching YouTube videos, although I did not find them completely correct. Here I present my own variation of the method. In this method, to solve the yellow layer, we first put all corners in their correct position without caring if their yellow square is at the top, and then we perform an algorithm that has the end effect of rotating each corner in place to bring the yellow squares on top. So, conceptually (but not procedurally or algorithmically), we go the opposite way than the first method. (In the first method, we first put the yellow squares on top and then we put the corners in their place. In this method, we put the corners in their place first, and then we put each corner’s yellow square on top.)

First step: We solve the white layer with no help, then we orient the cube so the solved white layer is at the bottom, facing the floor.

Second step: In this step, we look at the top layer and we turn it in such a way that the maximum number of corners are in their place (but we do not care if their yellow square is on top or not). (Please note that either two of four corners will be in their correct place if we turn the top layer appropriately in relation to the bottom layer. If you find that only one corner is in the right place, continue turning the top layer until two corners are in their place. In other words, in this variation of mine, it is incorrect to continue with only one corner in the right place.)

• If four corners are in their place, we continue to the next step.
• If two corners next to each other are in the right place, then we orient the cube so the two correct corners are in the back and in order to bring all corners in their place, we perform the algorithm U R Ui Li U Ri Ui L. This algorithm is quite “symmetrical” and easy to remember once you perform it even once. After this algorithm, we turn the top layer so that all corners are in their correct place and we continue to the next step.
• If two corners diagonally are in their right place, we orient the cube so that either one of these corners is in the upper right front and we perform the algorithm U R Ui Li U Ri Ui L. This will make the top layer obtain two corners next to each other and in their right place. Next we orient the cube so the two correct corners are in the back and in order to bring all corners in their place, we perform the algorithm U R Ui Li U Ri Ui L once more. After that, we turn the top layer so that all corners are in their correct place and we continue to the next step.

Third step: In this step, we will perform a procedure that will have the end effect of rotating each corner in place to bring the yellow squares on top.
First, we examine the top layer to find if at least one corner is in the correct place and also has the yellow square on top (thus effectively a corner completely solved).

• If all corners are completely solved, then the cube is solved!
• If at least one such corner exists (if any, there will be either one or two, not three and with four we have the complete solution), we orient the cube so that any of the completely solved corners is in the upper left front and an unsolved corner is in the upper right front.
• If no completely solved corners exist, then it does not matter how we orient the cube (in other words, it does not matter which unsolved corner will be in the upper left front).

We then perform the algorithm Ri Di R D as many times as needed to get the upper right front corner completely solved. Then we turn the top layer clockwise (we do a U). If the corner that is now in the upper right front is solved, we perform a U again and continue. If it is unsolved, we again perform the Ri Di R D algorithm as many times as needed to get this upper right front corner completely solved. And so on, until all top layer corners are completely solved. Then, we might need to turn the upper layer to get it aligned with the bottom layer and then the cube will be solved!
(The procedure that we use in this last step is also demonstrated (albeit a little differently, but it does not matter) in the second part of the excellent YouTube video Rubik’s Cube Questions Once Again Answered).

For this method, again we only have to learn two algorithms: U R Ui Li U Ri Ui L and Ri Di R D. These algorithms are easier to learn than the ones in the first method. The problem with this method is that in the beginning of the second step we need to examine the cube to find and position the upper layer for the maximum number of correct in-place (but not necessarily with the yellow square facing up) corners and this may take time observing and turning and matching. Also the second algorithm (Ri Di R D) may have to be performed a lot of times in the third step, something that may be tedious.

Which method should you use?

Well, you can use any one of these two methods. Practice and decide for yourself. I prefer the first method, because it does not require a lot of thinking and examination, although the algorithms are a little bit harder to memorize (but not that much).

Actually, the best thing to do would be to decide after the first step. The first step is the same in both methods. After the first step is done, it may so happen that all yellow squares are on top. Then it may be wise to choose the first method, since its second step is already done. Or it may so happen that after the first step, two corners are completely oriented and solved and even better, the other two corners are in their correct place. Then it may be wise to choose the second method, since its second step is already done and also half of its third step.

Explanation of how I created the variation of the second method

This second method is a variation that I created (but I am sure that others have thought of it also). My variation pertains to the way I solve the second step. I began with the algorithm U R Ui Li U Ri Ui L and the knowledge that what it does is that it acts on the top layer in a way that it leaves the right front upper corner intact and rotates the other three upper corners counter-clockwise. That knowledge was enough for me to use this algorithm (and this algorithm only) to correctly position all upper corners in the second step of the method.

When we begin the second step, it is obvious that we can choose any one of the upper corners and position it in its correct place. But if we choose and position one corner at random, chances are that the other three corners will not be in their correct positions. After all, the final goal of the second step is that: to correctly position all corners and we are in the beginning of this step.

Now please note that it is possible to turn the upper layer in such a way that not only one but two corners will be in their correct place. These corners will either be next to each other or diagonally from each other.

But let us choose to correctly position a corner that places no other corner in its correct position. In this case, we would like to orient the cube so that the correct corner is in the right front upper corner and use the algorithm U R Ui Li U Ri Ui L.

We know that the algorithm will leave the correct corner intact and we also know that it will rotate the other three corners counter-clockwise. Unfortunately, the use of this algorithm, no matter how many times we execute it, will not produce every possible orientation of the four corners and it may so happen that the orientation that it will not produce might be the desirable one.

For example, suppose that we look at the upper layer and corner number 4 is in its correct position and corners 1, 2 and 3 are not. By repeatedly applying the algorithm, we obtain the following configurations:

```1 2    2 3    3 1    1 2
->     ->     ->     -> and so on
3 4    1 4    2 4    3 4```

That is, corners 1, 2 and 3 are moving counter-clockwise, whereas corner 4 stays still. The problem here is that not all configurations are obtained. The following three configurations are not obtained (and it may so happen that one of them may be the desirable one):

```2 1       3 2       1 3

3 4       1 4       2 4```

So what are we to do? Should we learn yet another algorithm? Should we discard this algorithm altogether? Is there a way to use this algorithm successfully? Well, yes, as I discovered. If we turn the upper layer appropriately, we will be able to place not one but two corners in their correct place. These correct corners will be either next to each other or diagonally placed from each other. Let us see how we can proceed in each case, with only the use of the U R Ui Li U Ri Ui algorithm, in order to successfully complete the second step of the second method:

First case: The two correct corners are next to each other

If the two correct corners are next to each other, I found that we can use the algorithm U R Ui Li U Ri Ui L only once to place all corners in their correct positions. The solution that I discovered is that we have to orient the cube in a way so that the two correct corners are in the upper back, away from us and then perform the algorithm.

In the following examples, A and B are the correct corners and 1 and 2 are the corners that are not in their correct positions and have to exchange place. The solution that I came up with, as I mentioned already, starts by orienting the cube so that A and B are at the back, and then finishes by performing the algorithm.

```A B    B 1                                       A B
->     success: by turning the layer we have
1 2    A 2                                       2 1```

So the two incorrect corners changed placed and the upper layer is ready to be turned to its correct position relative to the bottom layer. Success! The second step of the method is complete!

What would have happened, if we would not follow my advice of putting A and B in the back? The other three orientations we can start with are:

```B 2    2 A
->     disaster: A and B are no longer adjacent
A 1    B 1

2 1    1 B
->     disaster: A and B exchanged places
B A    2 A

1 A    A 2
->     disaster: A and B are no longer adjacent
2 B    1 B```

So I just gave you all four orientations of the upper layer and the result we get when we apply the algorithm starting from each orientation. I discovered the correct orientation at first not by examining all orientations as I did now, but by understanding the effect of the algorithm. I observed the first orientation, the correct one with the A and B in the back and understood that when we apply the algorithm, corner 1 goes to the place B occupies and also (and here is the important part) A and B move together and without exchanging places. This was the “a-ha moment” for me.

Second case: The two correct corners are in a diagonal

If the two correct corners are diagonally placed, the solution that I discovered is that we have to orient the cube so that one of these correct corners is in the upper right front. Let us again study what happens when we apply the algorithm to the four possible orientations of the layer. A and B are the correct corners and 1 and 2 are the corners that are not correct (i.e. they need to change places).

```A 1    1 2
->     success: 2 and B are now correct and adjacent
2 B    A B

1 B    B A
->     disaster: no corner is now correct
A 2    1 2

B 2    2 1
->     success: 1 and A are now correct and adjacent
1 A    B A

2 A    A B
->     disaster: no corner is now correct
B 1    2 1```

In the first orientation we have one of the correct corners (corner B) in the upper right front. After the algorithm is applied, corner B and corner 2 are correct, whereas corner A and corner 1 are not. We messed up corner A in order to fix corner 2. But now we have two correct adjacent corners (B and 2), so we can apply the algorithm again as I described in the first case.

The same happens for the third orientation, whereas the second and the fourth orientation do not end up nicely after the application of the algorithm. Thus, in this case, we conclude that, before the application of the algorithm, we have to orient the cube so that one of the correct corners is in the upper right front. We then apply the algorithm and after that, we proceed as I described in the first case (where the two correct corners are next to each other).

Final notes

So there you have it: two methods for solving the 2X2. Each method requires knowledge of only two algorithms. This was my first and foremost concern: fewer algorithms and cases to remember (the downside being that we need to perform more turns until the cube is solved).

This guide is also useful for the 3X3. This is how you solve the whole of the 2X2 and how you also solve the corners of the 3X3. So, if you need a beginner’s guide to only help you with the corners of the 3X3, you can use this guide.

Happy cubing and keep in mind that a solved cube is a cute cube (yes, yes, you have to endure this lame joke, too). 