How to assemble a Rubik's Cube 5x5x5 (part 1)

Back in 2008, a cube of rubik of non-standard sizes fell into my hands. How to collect such a miracle, I had no idea then. At first, my friends and I partially assembled it, having no idea about the assembly algorithm, but then I wanted to learn how to assemble it completely. Through Google, I found some kind of build algorithm, but unfortunately it was incomplete and sinned with inaccuracies. For some time, analyzing the googled algorithm and the classic 3x3x3 cube assembly algorithm, I realized the complete cube assembly algorithm, not only 5x4x5, but 4x4x4 (although I didn’t have such a cube on hand, I wrote a program to simulate such a cube in 3D and checked the algorithm). Everyone who would like to learn how to collect such a cube is welcome to cat.

I. Basic concepts

A 5x5x5 cube consists of several types of small cubes that cannot be interchanged due to the internal structure of the cube:

1. Central
2. Central cross
3. Central corner
4. Lateral middle
5. Lateral intermediate
6. Angular

Each of these categories of small cubes, when turning, passes only into themselves, so the whole cube is arranged. For example, central cubes always remain central and, in addition, their relative position relative to each other always remains unchanged. Those. if in your cube the yellow and white centers are on opposite sides, then anyway, they will always be so located (unless of course you broke the cube). By the way, they determine the color of the face that you will collect around these centers - the yellow face will always be opposite the white one in the assembled cube. Corner cubes also pass only into corner cubes. In addition, in the cube of any size, the corner assembly algorithms are the same. I would like to say a few words about the side intermediate cubes. Take a look at the edge of a large cube - there are two such cubes on it, they have the same colors on the faces. But these cubes themselves are not the same. The fact is that inside they have fixtures that go deep into the cube, and therefore these cubes have mirror symmetry (we can also say that they differ approximately like the right and left triples of vectors). Thus, at the stage of assembly of this category of cubes, it will be strictly determined which one will fall into which place (for more details, see a few sections below).

II. Top face

Choose one of the flowers on the cube that you like best, for example, let it be green. With it we will begin the assembly. Find the face where the center is green and place the cube in space so that this face looks up. Further, for ease of assembly, we will use the conventions.

• F - front face (front)
• B - back face
• L - left side (left)
• R - right side
• U - upper bound (up)
• D - bottom edge (down)

Since we have a 5x5x5 cube, it has inner layers that can also be twisted. I will denote them, respectively, F2, B2, L2, R2, U2, D2 - where X2 means the layer parallel to X and immediately lying behind it. We will rarely turn centers.

Central cross

So, you hold the cube and the green center is in U. First, collect the central cross. Find the cubes you are looking for on the cube (hereinafter we will agree to call the whole structure a cube, and its components, which we see 25 pieces on each face, are cubes, although if you take apart the cube, they will only look like cubes remotely). They are located around the centers F, B, L, R, and D. The first 4 cases are symmetrical. Let the cube be in F. Then rotate F so that it lies in L2 or R2 and rotate L2 and R2, respectively, so that it is at the top. If at the same time part of the cross is already assembled and the cube falls into place of another of the same green cube, turn F before this so that this does not happen. If the cube is in D, turn D so that it stands exactly beneath the place in U where it should be. Then turn the corresponding L2, R2,

Central square

Now we will collect the central square in the upper face - we will also add the central corner ones. Again we get 2 fundamentally different cases - a cube in the side or in the bottom face. For the side face, use the combination K1 (we will use the following notation in the description of the combination - the indication of the face / layer indicates clockwise rotation, if you look at it “in front.” If the apostrophe stands after the name of the face, then rotate it counterclockwise). Also use a combination symmetrical to this and rotate before the combination of U and F so that the desired cubes are in the places shown in the figure. Looking ahead, I’ll say that we will use this technique in the future - before the combination, turn some faces so thatso that this combination moves the cubes we need. Of course, after the combination, everything needs to be returned in the reverse order, although at this stage of assembly this is not even necessary. It’s not worth worrying about this, such “pre-training” will include a maximum turn of two faces. Now it remains to consider the case when the cube is in the lower face. Place the cube in D exactly under the place in U where it should be and use the combination K2. You don’t need to remember it - just execute the algorithm a couple of times and you will understand what it does at all. In general, if you spin the cube and figure out how it works, then you will collect the upper face even without my instructions, using an even faster method - what and where it goes during turns will become obvious. Nevertheless, I will give a rigorous algorithm here how to collect this cube from beginning to end, so let's move on!

Side Medium Cubes

We have assembled the central 3x3 square in U. Put in place 4 lateral middle cubes. This is done in exactly the same way as in the original 3x3x3 cube. Notice that we are collecting the upper face with the correctly assembled upper strip (as in the figure at the beginning of this large section), so be careful. Let the center of the front face be red. Then find the green-red lateral middle cube. If it is located in the central horizontal layer or the upper face, move it with the necessary rotation to the lower face (if of course it already stands as it should - you don’t need to do anything :))) If you hit any of the “finished” blocks, and this could only happen if the cube was in the middle horizontal layer, then rotate D 90 degrees to either side and return the “damaged” face to its place. Now the desired cube is exactly in D and in U nothing went wrong. Next, turn D so that thisthe cube is in F. If his red color is in F, then just rotate F 180 degrees. If the front is green, then use the combination: DR F'R '. The combination is obvious. Repeat the procedure with L, R and B. Do not forget that the second color on the cube (the first on green) should match the color, respectively. center. What should turn out can be seen in the figure on the right.

Lateral intermediate

Now put in the right place 8 lateral intermediate cubes. For cubes located on the side edges, use combinations of K3 and K4 , as well as symmetrical to them on the right side. Before combinations, of course, turn D2 and U2 so that the combination performs the correct movement. If the cube is in D, then a simple combinationreturn it to the side face - first, make sure that it does not lie, for example, in L, then rotate L in any direction by 90 degrees, then rotate D so that our cube is in L and finally return L to its place. Now use the algorithm described in a few lines above. Of course, put the cubes so that the side color matches the side color of the adjacent side middle cube, because we also collect the upper strip! Here, by the way, you can notice that the visually identical (for example, green-red) cubes are actually different - one of them will appear to the left of the corresponding lateral middle cube, and the other to the right (if you look at the cube in a certain way - the green face is on top) .

Corner cubes

Now there are corner cubes. They are also collected, as in the 3x3x3 cube. We collect in turn. Select a corner cube with one green face that does not stand as it should. If it is in U, then move it to D by rotating the side face containing it 90 degrees, then turn D so that this cube no longer lies in the turned side face and return this face to its place. Even if this cube is in its place, but only oriented incorrectly, still do these actions. Now it all depends on where the green line “looks”. She can look either sideways or down. If down, then rotate D so that it appears under that corner cube in U, where there is still no necessary cube, i.e. the one that will be there in the assembled cube. Next, rotate the side face containingour cube so that its green side looks sideways and is in D, then by turning D remove it from the turned side face and return this face to its place. Now our cube looks sideways and is in the bottom face. It remains only to apply a combination of K5 or symmetrical to it. Make sure that the top strip of the cube is collected. After the described actions, the following should happen (see the picture on the left): the upper face, the upper strip is assembled, and the color of the strip on each face coincides with the color of the center of this face.

III. All lateral central and corner cubes.

As a result, it should turn out, as in the picture on the right (note, now we look at the lower, left and front faces).

Lateral central cubes in the middle horizontal and lower layer

First, put in place the cubes, which should be in the middle horizontal layer. Here we need one K6 combination , which moves the cube from D to the middle horizontal layer (see figure). Here you sometimes have to use a symmetric combination (it depends on the orientation of the colors of the moved cube). If you need to move the cube from the middle horizontal layer to another place in the same layer, you should first move it to the lower layer (simply putting any other cube from D in its place using K6 ).
Now about the bottom layer. The plan will be as follows: first, we do not look at the orientation of the cubes, but simply make them stand in the right places, i.e. one of the colors on each cube should be the color of the bottom face, and the other should be the color of the corresponding side face. Then correctly orient the cubes of the lower face.
Point one. Using the K7 algorithm, put all the side middle cubes in D to the correct positions. The algorithm moves in a circle 3 of 4 necessary cubes. First, make sure that exactly one cube in D stands in its position, and then with this combination put in place all the others. It is claimed that this is enough for assembly :)
Point two. Orient the cubes correctly. It just so happened that only 0, 2 or 4 cubes can be incorrectly oriented (such is the structure of the cube). If 0, that you do not need to do anything for obvious reasons, if 4, then you have to build an algorithm for two twice.
Algorithm for 2. Rotate D so that the cube that needs to be oriented correctly is in F. Then perform a simple operation 4 times: rotate F clockwise, rotate the central horizontal layer clockwise when viewed from the U side. Now rotate D so that the second the cube that needs to be turned is in F and again 4 times do the procedure described above. Next, just return D to its original position. In the end, we get what is in the picture on the right.

Corner cubes in the lower face

At the time of assembly of these cubes, we turn the cube so that the upper face is the bottom and vice versa (it’s more convenient). To begin with, we put all the corner cubes in place. To do this, use K8which moves 3 corner cubes in a cycle. Again, it is argued that this is sufficient for assembly. Further, after the corner cubes fall into place, they must be correctly oriented. A simple combination is used: RF 'R' FRF 'R' F. It twists the corner cube, which is indicated as CC in the figure on the left. You need to use it this way: execute the algorithm until the cube is oriented correctly (in this case, the yellow face up, the combination must be done once if the yellow color is in F and two times if in R), then rotate U so that it is in its place The next incorrectly oriented cube turned out to be and again use the combination until the cube stands up as it should. As a result, all corner cubes are oriented correctly. Note that in addition to turning the corner cube, the combination also spoils the other, correctly standing cubes. Do not worry - the cube is so arranged that you have to repeat the combination a multiple of three times, and after every third use of the combination everything returns to its place. So everything will turn out!

Total

Until now, the actions I have described are either almost obvious combinations, or combinations similar to those used in the assembly of a classic 3x3x3 cube. But then everything will be different. Although there are only two stages of assembly and several new combinations, they are more complicated than they were before, but after fully familiarizing yourself with this scheme, you can collect any cube of dimension <= 5.
To be continued ...