A Simple Solution forthe Rubik’s Cube

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BY J. SANCHEZ

Worldwide popularized in the 80’s, theRubik’s cube is one of the most in-

teresting mathematical puzzles you can find.Maybe this success is due to its almost in-tuitive rules to start playing, although thesolution is not always clear. And this is whythe Rubik’s cube is a beautiful mathemati-cal problem: it’s a hard problem where theunderstanding of the problem is not part ofits complexity. As soon as you have in yourhands the cube, you will start with the refle-xive part, where you are trying to figure outhow to arrange the colors.

The first reflex when we are trying to solvethe cube, is to put in place a face, even ifthe borders don’t have the right dispositionof colors, then we experience some level ofsatisfaction with this little sample of orderbetween all the chaos. After that we start tocomplete the face, but with the right colorsin its crown, our first small victory.

It took about one month to the inventor ofRubik’s cube to complete the puzzle. Origi-nally he created the cube without the coloredfaces. As a architecture, he was interested inthe study of the three dimensional structures,and this mechanism was a really nice tool tobe taught. The idea was to ask his students todescribe the internal mechanism that make

the little cubes rotate almost freely withoutfalling out.

Some time after that, a friend of Erno Ru-bik, suggested him to color the faces and ma-ke of the cube a marketable puzzle. And itwas like that, the fever for the Rubik’s cubestarted, first in Budapest, then in London,Paris, New York and the rest of the world.

On this notes i will give the details of asolution found by myself. Even if there

are a lot of techniques for solve the cube, theparticularity of this solution is that it’s intui-tive and simple. You don’t need to memorizea 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 likesolving a Sudoku. But I don’t think my solu-tion is the one you want if your objective is

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to solve the cube in lest than a minute.

Before start with the description let’s makea sketch of the technique. As I’ve mentionedbefore, to complete a face is the first step toa solution, so we take as the starting point acomplete ordered face. If you don’t know howto make a face then, take your time and try todo it, I’m sure you can. Like that you will befamiliarized with the movements of the cube.The second part, is to align all the eight cor-ners. With one face already made, we onlyneed to arrange the others four corners. Forthat we will need two kind of movements.The first move is used to align the four cor-ners with the already aligned corners, even ifthe colors have some rotations. The second isused to orientate this four corners.

So, in the last part we have one com-plete face and all the eight corners in theright position. We only need to completethe rest of the faces. In this stage, weonly need one kind of movement. Thismovement will permutate opposite internalcubes and leaves without change the restof the cube. Probably you will need toapply this movement several times, and af-ter some time, you will finally solve the cube.

Now let’s begin with the required nota-tions. First we set a point of view for

the cube, and from there we identify the mo-vements.

The axes of the cube gives us 6 possiblesmovements, which are the external face rota-tions.

With each rotation we have tree possibi-lities: 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 in-ternal face rotations.

All this notations will be used to describeour three kinds of basic movements for thecube’s solution.

The first movement is noted Σ and it is ap-plied with the already made face on top.

This face will be called the reference face.The idea of Σ is to maintain without chan-ges the reference face and to switch only twocorners in the bottom face. Here, we don’tcare about the other cubes different from thereference face and corners.

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The action of Σ over the four corners atthe bottom is described in the next image.

We can see that the two corners areswitched, and the other two are in the sameplace, one with a rotation and the otherwithout changes.

Almost similar to the first movement, thesecond movement, Ω, will leaves wit-

hout changes the reference face and will ro-tate the four corners at the bottom face.

Now the description of how the corners ro-tate.

At the end of the movement, one of thebottom corners will stay exactly the same,

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and the others will have a rotation.

The final kind of movement, Φ, will onlyswitch two pairs of internal cubes wit-

hout affecting the rest of the cube.

We can use the letters R, G, W , B, O andY for the face colors. It will be helpful to setup how and which internal cubes we want toswitch.

Once this three movements are unders-tood, we can start to solve the cube.

It is clear that maybe we need a little prac-tice to handle Σ, Ω and Φ, but that only

take a few minutes. Now, the steps for thesolution.

1. Complete a face of your favorite color,for example red.

2. Look at the bottom corners, and tryto align the corners with the top face cor-ners using Σ. We don’t care if at the end,the bottom corners have some rotation, weonly need, for example, that below the cubeY −R−B we can find the cube Y −O−B.

3. Now it’s time to align the corners. Forthat we only use Ω. Recall that in Ω, is thebottom left corner which will be stay static.The idea is to arrive to a position where onlyone cube at the bottom is aligned. Then, pla-ce the Rubik’s cube in a way that this cornercube occupies the place of the cube that Ωmaintains static, and apply two times Ω. Youwill see that now, all the corners are align.

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4. In this part we only use Φ to permute theinternal cubes. But, maybe here the questionis: how exactly we use Φ? In order to answerthis question we will use two examples.

With the next disposition, we only applyone time Φ before have the Rubik’s cube sol-ved.

But the power of Φ is showed with a situa-tion like this one.

Here, in order to solve the Rubik’s cube wewant to make the switch,

But, before apply Φ we need to align thecubes. This is make with the next sequenceof movements.

Them, we apply Φ, and we have to co-me back to the original position. For that weapply in the inverse sens all the movementswe use to align the cubes.

So, the important thing when we use Φ topermutate internal cubes is to remember theintermediary movements used to align the cu-bes we wanted to change, and don’t lost trackof the movements when we have to come backto the original position. For that i suggest towright down the movement used to align thecubes. But, be careful, if you make a mistakein your way back, probably you will need tostart over.

This method takes a little reflection whenyou have to choose the cubes to switch atthe final part, and also when you try tofigure out how to align the cubes to applyΦ, but, this is why i say this part is like aSudoku. Is not complicated, but it may taketime and concentration.

One of the interesting things about thismethod is that it don’t have the arti-

ficial flavor of championship solutions, whereyou have to memorize a big list of sequencesof movements. And it can be used to solveother variants of the Rubik’s Cube.

For example, the variant 2 × 2 × 2 is sol-ved only using Σ and Ω and you can useexactly the same method with the Rubik’sCube Mirror. In this one, the cube only haveone color, but the little cubes”have a diffe-rents forms.

In the case of bigger Rubik’s cubes, like7× 7× 7, we only need to find a new versionof Φ.

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But maybe we need new variants of Σ, Ωand Φ in order to solve more insane Rubik’scube, like the four dimensional version.

In a more mathematical aspect of the Ru-bik’s cube, this toy is an example of what

is called, a group. Remember that a groupis just a set of objects, together with an bi-nary operation. This operation, usually wri-ting with product notation ∗, must have aunity and each element in the group musthave a inverse.

A group can be described from its genera-tors. For example, consider a mechanic clockwhich only gives the hours. This clock can bedescribed with a group with twelve elements0, . . . , 11, one for each hour. The operationin the natural sum of hours. But maybe weonly need to use 1 and the operation to des-cribe the other elements of the group. Forexample 4 = 1 ∗ 1 ∗ 1 ∗ 1, or we can use thenotation 4 = 14. The neutral element 0 canbe seen as 0 = 112.

In a description of the Rubik’s cubegroup, we use the generators to do it. First,set a fixed position for the Rubik’s cu-be. The elements of the group will be allthe possible finite movements. But this mo-vements are completely described by so-me basic movements that we will notex1, x2, y1, y2, z1, z2, α, β, γ.

The neutral element of the group is noted1, and represents the no changes movement.

For the multiplication we use the notationαx1, which means that we first apply x1 andthem α. And them we can see some relationslike x41 = 1.

So, the Rubik’s cube group is seen asthe free group over the set of generatorsx1, x2, y1, y2, z1, z2, α, β, γ quotient by someplausible relations. Which are these rela-tions? x41 = 1, x42 = 1, y41 = 1, z42 = 1 andα4 = 1, are part of the list.

This simple look to the world of theRubik’s cube shows us how a good idea

is created and how people take this ideaand start to create all kinds of new results.Maybe some variants of the Rubik’s cubeare incredibly complex, but they are possiblethanks to the very first cube.F

For more information about the subject:

Borrow or buy a Rubik’s Cube.

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