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Saturday, December 3, 2011

Beginner's Method For Solving A 3x3 Rubik's Cube (Part 1)

The past couple of weeks, I have been doing research into methods for solving the Rubik's Cube. According to wikipedia, the Rubik's Cube is by far the world's best-selling toy ever. How such a cerebral toy became so popular is quite surprising to me. More than 30 years after the introduction of the first 3x3 cube, we now have cubes in several different dimensions from 2x2 all the way to 7x7. There are also computer simulations of cubes of practically infinite sizes.

The surprising thing I found is that in spite of my studious search of several websites, I could not find a single site that has the solution steps I found in the printed booklet that came packaged with my Rubik's cube. There are solutions that involve just 2 algorithms, and solutions that involve several hundred algorithms. However, even the official website does not have the solution method that I am using for my cube-solving. In fact, the method explained in the official website is quite involved, and includes some algorithms that include up to 13 moves. On the face of it, it seems quite complicated, and difficult to memorize.

So, I decided to put together the solution method I have been using as a two-part post on my blog. The link to the second part of this solution methodology can be found at the bottom of this post. This algorithm is considered a beginner's algorithm (as opposed to some other methods that are used in speed-solving, and need one to memorize a lot more algorithms. In exchange for the increased memorization, you get higher speed and fewer twists and turns from scrambled state to solved state). You are not going to set any speed records using this method. But, you are going to be able to solve the cube reliably using this method. Once you get that, you might be motivated to learn other methods, so that you can improve your speed.

I believe in the crawl-walk-run philosophy. Playing around with a cube so that you can solve one side intuitively is the crawl stage. This method will get you to the walking stage. If you want to run, you can look up dozens of methods on the internet, and choose a method of your liking to continue your cube education.

To begin with, let us define some basic notation. Take a look at the picture of a cube below.
Naming ConventionThe pictures used in the solution are all going to look like this. In this cube, I consider the red-face to be the front of the cube (F). The blue face is the right side of the cube (R) and white is the upper face of the cube (U). In this cube, yellow is opposite the white, so yellow would be the down face of the cube (D). Similarly, orange is opposite the red, so orange would be the back face of the cube (B). And green is opposite the blue, and would be the left face of the cube (L). This notation of naming faces is the same as that used on Wikipedia.

In all my pictures, the cube will always be shown with three faces. The top face will be the upper face of the cube (U), the face to the right will be the right face (R), and the face to the left of the right face will be the front face (F). It is assumed that the solver is actually looking at the cube almost directly at the front face, but with their point-of-view shifted slightly to the right and above the cube, so that the top face and right face are also visible in addition to the front face.

Every move of the cube involves turning a face of the cube either clockwise or counterclockwise. The direction of the move is as it appears to an observer facing that face of the cube head-on. Thus a 90 degree clockwise rotation of the right face of the cube is denoted as R, and it would involve turning the blue face from front to back (so that three red cubes end up on the upper face, 3 white cubes end up on the back face, 3 orange cubes end up on the down face, and 3 yellow cubes end up on the front face).

An anti-clockwise rotation is denoted with an apostrophe after the letter of the face. Thus R' is an anticlockwise rotation of the right face and would move 3 white cubes to the front face of the cube while moving 3 red cubes to the down face of the cube. Similarly, L' would move 3 red cubes to the upper face while moving 3 orange cubes to the down face. The algorithms in this methodology do not involve any face except the front (F), upper (U), down (D), right (R) and left (L), so make sure you are absolutely sure what clockwise and anticlockwise rotations of each of these faces would involve. You don't have to worry about rotating any of the middle layers or the back layer of the cube if you stick with this method.

Step 1: Step 1 of this method is entirely intuitive. I don't want to burden you with algorithms for this step. All you want to do is go from an entirely scrambled cube to selecting a top face, and solving the 4 edges of the chosen top face (the edge pieces of a cube have two colors on them). Play around with the cube until you can do this without any problems from any start point. Notice that each edge piece is aligned such that the top face is of the chosen top face color, and the color on the side corresponds to the color of the center tile on that face. This is called solving the top cross.
Scrambled cube
Step 1 Before
Top cross
Step 1 After

Step 2: The second step is also intuitive, and just involves solving the 4 corners of the top face. The corner pieces of a cube are the pieces with 3 colors on them. In this second step, you are going to intuitively get the corner pieces involving blue into the correct corners (make sure the colors on these pieces apart from blue line up with the center pieces of the same colors). This solves the top face, and forms short T's down the four sides of the cube. I refer to the combination of steps 1 and 2 as solving one face to a T. With practice, you should be able to do this in about a minute or so starting from scratch (don't worry if it initially takes you several minutes. The important thing is to get familiar with how different cubes move in relation to each other).
Solved one face to a T
Step 2 After

Step 3: At this point, turn the cube over, so that your solved side is on the bottom. Thus, blue becomes the down side of the cube (D). Your aim in this step is to complete solving the second layer of the cube by getting the edge pieces of the colors around blue in their right places and orientations. Thus, after the end of step 3, you will have two layers of yellow, orange, white and red (your colors may be different depending on how your cube is colored. I am referring to the color scheme on my cube in which green is the color opposite the solved blue side).

Now, if any of the second layer edge pieces are already in the correct position and orientation, you don't have to worry about them. But many times, you will find edge pieces that are supposed to go into the second layer in the third layer.

In one configuration (option 1), you will find second layer edge pieces that need to be rotated from the right face to the corner of the front and right faces. That is, the piece needs to be rotated into place anticlockwise.
Edge needs anticlockwise rotation
Step 3 Before, Option 1

Notice how, in the picture above, the white and red piece in the middle top of the right face needs to be moved anticlockwise to the middle of the edge between the front and right faces. In this case, the following sequence of steps will accomplish this: U' F' U F U R U' R'.
Edge rotated anticlockwise
Step 3 After, Option 1

Notice how the white and red face has been moved anticlockwise to its correct location and orientation.

In the other configuration (option 2), the edge piece needs to be moved from the top of the front face to the corner of the front and right faces (so, it requires a clockwise rotation). Notice the red and yellow corner piece in the picture below.
Edge needs clockwise rotation
Step 3 Before, Option 2

Notice how, in the picture above, the yellow and red piece in the middle top of the front face needs to be moved clockwise to the middle of the edge between the front and right faces. In this case, the following sequence of steps will accomplish this: U R U' R' U' F' U F. If you are astute, you will realize that this is essentially the same set of moves as in option 1, just that you start from move 5 of option 1, and cycle around to move 4 after the end. So, you only have to actually memorize one algorithm for step 3.
Edge rotated clockwise
Step 3 After, Option 2

Notice how the yellow and red face has been moved clockwise to its correct location and orientation.

Now, one complication that you might encounter when you are solving this second layer is that the edge piece you want in a particular position may be in that position, but may be in the wrong orientation. In other words, for example, the yellow and red edge piece may be at the corner of the yellow and red faces, but may be placed such that the yellow side of that edge piece is on the red face and the red side of it is on the yellow face. In that case, there is no simple algorithm to reorient the piece in place. You have to push some other edge piece into its position using one of the algorithms above, and this will pop that edge piece to the upper layer, from where it can be moved down to its correct position and orientation using one of the algorithms above.

At the end of step 3, you will have a cube that has one side fully solved (blue in my case), and 4 sides solved two-thirds of the way (two layers each of red, yellow, orange and white in my case). Your cube will look like the picture below, assuming you have the same color scheme as me. You don't see the orange and yellow sides, but they are also solved two-thirds of the way just like the white and red sides in the picture below. The green side is still a mess while the blue side (which is at the bottom) is fully solved (make sure you have not made any mistakes in applying the step 3 algorithms by turning the cube over and making sure that the blue side is indeed solved. If it is not, re-solve it, and then apply the step 3 algorithms more carefully).
Cube after step 3
Step 3 After

At this point, you are about halfway done. Continue on to the next step in part 2 of this solution guide.

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