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1 A Simple Solution for the Rubik s Cube A post from the blog Just Categories BY J. SÁNCHEZ Worldwide popularized in the 80 s, the Rubik s cube is one of the most interesting mathematical puzzles you can find. Maybe this success is due to its almost intuitive rules to start playing, although the solution is not always clear. And this is why the Rubik s cube is a beautiful mathematical problem: it s a hard problem where the understanding of the problem is not part of its complexity. As soon as you have in your hands the cube, you will start with the reflexive part, where you are trying to figure out how to arrange the colors. The first reflex when we are trying to solve the cube, is to put in place a face, even if the borders don t have the right disposition of colors, then we experience some level of satisfaction with this little sample of order between all the chaos. After that we start to complete the face, but with the right colors in its crown, our first small victory. It took about one month to the inventor of Rubik s cube to complete the puzzle. Originally he created the cube without the colored faces. As a architecture, he was interested in the study of the three dimensional structures, and this mechanism was a really nice tool to be taught. The idea was to ask his students to describe the internal mechanism that make the little cubes rotate almost freely without falling out. Some time after that, a friend of Ernő Rubik, suggested him to color the faces and make of the cube a marketable puzzle. And it was like that, the fever for the Rubik s cube started, first in Budapest, then in London, Paris, New York and the rest of the world. On this notes i will give the details of a solution found by myself. Even if there are a lot of techniques for solve the cube, the particularity of this solution is that it s intuitive and simple. You don t need to memorize a big set of patterns to arrive at the solution, you will need only three kind of movements, and the last part of the method will be like solving a Sudoku. But I don t think my solution is the one you want if your objective is 1

2 to solve the cube in lest than a minute. Before start with the description let s make a sketch of the technique. As I ve mentioned before, to complete a face is the first step to a solution, so we take as the starting point a complete ordered face. If you don t know how to make a face then, take your time and try to do it, I m sure you can. Like that you will be familiarized with the movements of the cube. The second part, is to align all the eight corners. With one face already made, we only need to arrange the others four corners. For that we will need two kind of movements. The first move is used to align the four corners with the already aligned corners, even if the colors have some rotations. The second is used to orientate this four corners. So, in the last part we have one complete face and all the eight corners in the right position. We only need to complete the rest of the faces. In this stage, we only need one kind of movement. This movement will permutate opposite internal cubes and leaves without change the rest of the cube. Probably you will need to apply this movement several times, and after some time, you will finally solve the cube. The axes of the cube gives us 6 possibles movements, which are the external face rotations. With each rotation we have tree possibilities: a 90 rotation, noted by 1, a 180 rotation, noted by 2, and, a 270 rotation, noted by 3. Also we have the 3 possible internal face rotations. ow let s begin with the required notan tions. First we set a point of view for the cube, and from there we identify the movements. All this notations will be used to describe our three kinds of basic movements for the cube s solution. he first movement is noted Σ and it is apt plied with the already made face on top. This face will be called the reference face. The idea of Σ is to maintain without changes the reference face and to switch only two corners in the bottom face. Here, we don t care about the other cubes different from the reference face and corners. 2

3 lmost similar to the first movement, the A second movement, Ω, will leaves without changes the reference face and will rotate the four corners at the bottom face. The action of Σ over the four corners at the bottom is described in the next image. Now the description of how the corners rotate. We can see that the two corners are switched, and the other two are in the same place, one with a rotation and the other without changes. At the end of the movement, one of the bottom corners will stay exactly the same, 3

4 and the others will have a rotation. is clear that maybe we need a little pracitaket tice to handle Σ, Ω and Φ, but that only a few minutes. Now, the steps for the he final kind of movement, Φ, will only T switch two pairs of internal cubes without affecting the rest of the cube. solution. 1. Complete a face of your favorite color, for example red. We can use the letters R, G, W, B, O and Y for the face colors. It will be helpful to set up how and which internal cubes we want to 2. Look at the bottom corners, and try switch. to align the corners with the top face corners using Σ. We don t care if at the end, the bottom corners have some rotation, we only need, for example, that below the cube Y R B we can find the cube Y O B. 3. Now it s time to align the corners. For that we only use Ω. Recall that in Ω, is the bottom left corner which will be stay static. The idea is to arrive to a position where only one cube at the bottom is aligned. Then, place the Rubik s cube in a way that this corner cube occupies the place of the cube that Ω maintains static, and apply two times Ω. You will see that now, all the corners are align. Once this three movements are understood, we can start to solve the cube. 4

5 4. In this part we only use Φ to permute the Them, we apply Φ, and we have to cointernal cubes. But, maybe here the question me back to the original position. For that we is: how exactly we use Φ? In order to answer apply in the inverse sens all the movements this question we will use two examples. we use to align the cubes. With the next disposition, we only apply one time Φ before have the Rubik s cube solved. So, the important thing when we use Φ to permutate internal cubes is to remember the intermediary movements used to align the cubes we wanted to change, and don t lost track of the movements when we have to come back to the original position. For that i suggest to wright down the movement used to align the cubes. But, be careful, if you make a mistake But the power of Φ is showed with a situain your way back, probably you will need to tion like this one. start over. This method takes a little reflection when you have to choose the cubes to switch at the final part, and also when you try to figure out how to align the cubes to apply Φ, but, this is why i say this part is like a Sudoku. Is not complicated, but it may take time and concentration. ne of the interesting things about this O method is that it don t have the artificial flavor of championship solutions, where you have to memorize a big list of sequences of movements. And it can be used to solve other variants of the Rubik s Cube. For example, the variant is solved only using Σ and Ω and you can use exactly the same method with the Rubik s Cube Mirror. In this one, the cube only have one color, but the little c ubes have a differents forms. In the case of bigger Rubik s cubes, like 7 7 7, we only need to find a new version of Φ. Here, in order to solve the Rubik s cube we want to make the switch, But, before apply Φ we need to align the cubes. This is make with the next sequence of movements. 5

6 But maybe we need new variants of Σ, Ω and Φ in order to solve more insane Rubik s cube, like the four dimensional version. In a more mathematical aspect of the Rubik s cube, this toy is an example of what is called, a group. Remember that a group is just a set of objects, together with an binary operation. This operation, usually writing with product notation, must have a unity and each element in the group must have a inverse. The neutral element of the group is noted 1, and represents the no changes movement. For the multiplication we use the notation αx 1, which means that we first apply x 1 and them α. And them we can see some relations like x 4 1 = 1. So, the Rubik s cube group is seen as the free group over the set of generators x 1, x 2, y 1, y 2, z 1, z 2, α, β, γ quotient by some plausible relations. Which are these relations? x 4 1 = 1, x 4 2 = 1, y 4 1 = 1, z 4 2 = 1 and α 4 = 1, are part of the list. This simple look to the world of the Rubik s cube shows us how a good idea is created and how people take this idea and start to create all kinds of new results. Maybe some variants of the Rubik s cube are incredibly complex, but they are possible thanks to the very first cube. For more information about the subject: Borrow or buy a Rubik s Cube. A group can be described from its generators. For example, consider a mechanic clock which only gives the hours. This clock can be described with a group with twelve elements {0,..., 11}, one for each hour. The operation in the natural sum of hours. But maybe we only need to use 1 and the operation to describe the other elements of the group. For example 4 = , or we can use the notation 4 = 1 4. The neutral element 0 can be seen as 0 = In a description of the Rubik s cube group, we use the generators to do it. First, set a fixed position for the Rubik s cube. The elements of the group will be all the possible finite movements. But this movements are completely described by some basic movements that we will note x 1, x 2, y 1, y 2, z 1, z 2, α, β, γ. 6

Grade 7/8 Math Circles. Visual Group Theory

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