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

Back in 2008, a Rubik's Cube of non-standard sizes fell into my hands. How to collect such a miracle, I then had no idea. At first, my friends and I collected it partially, without having any idea about the assembly algorithm, but then I still wanted to learn how to assemble it completely. Through Google, I found some semblance of an assembly algorithm, but unfortunately it was incomplete and sinned by inaccuracies. For some time, after analyzing the classic 3x3 die cube algorithm and I realized the complete cube assembly algorithm not only 5x5x5 but 4x4x4 (although I didn’t have such a cube, I wrote a program to simulate such a cube in 3D and checked the algorithm). Anyone who would like to learn how to collect such a cube - welcome under cat.

I. Basic concepts

The 5x5x5 cube consists of several types of small cubes that cannot be swapped with each other due to the internal structure of the cube:

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

Each of these categories of small cubes when turning turns only into itself, so the whole cube is arranged. For example, central cubes always remain central and, moreover, their relative position relative to each other always remains the same. Those. if in your cube the yellow and white centers are on opposite sides, then whatever one may say, 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 go to corner ones only. In addition, in the cube of any size, the corner assembly algorithms are the same. I would also like to say a few words about the side intermediate dice. Take a look at the edge of a large cube - there are two such cubes on it, they have the same colors on the edges. But the cubes themselves are not the same. The fact is that inside they have attachments that go deep into the cube, and therefore these cubes have a mirror symmetry (you can also say that they differ approximately as the right and left triples of vectors). Thus, at the assembly stage of this category of cubes, it will be strictly determined which of them will be in which place (in more detail - in several sections below). We now turn to the actual assembly of the cube.

Ii. Upper bound

Choose one of the colors on the cube that you like the most, let it be green, for example. With it we will start the assembly. Find the face where the center is green and place the cube in space so that this face looks up. Further, for convenience of assembly, we will use the legend.

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

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

So, you hold the cube and the green center is in the U. First, we will assemble the central cross. Find the desired cubes on the cube (hereinafter, we will call the whole construction a cube, and its components, which we can see on each face, are 25 pieces - cubes, although if we disassemble the cube, then they will look like cubes only remotely). They are located around the centers F, B, L, R, and D. The first 4 cases are symmetric. Let the cube be in F. Then turn 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 a part of the cross has already been assembled and the cube is in place of another of the same green cube, turn F in front of this so that this does not happen. If the cube is in D, turn D so that it stands exactly below that place in U where it should be. Next, rotate the corresponding L2, R2, F2, or B2 180 degrees to move it to the top.

Central square

Now we will collect the central square in the upper face - we will add also the central angular ones. Again we get 2 fundamentally different cases - a cube in the side or in the lower face. For the side face, use the K1 combination (we will use the following notation in the combination description - specifying a face / layer means turning clockwise if you look at it “from the front.” If there is an apostrophe after the face name, then you should turn it counterclockwise). Also use a combination symmetrical to this one and rotate in front of the combination of U and F so that the necessary cubes are in the places shown in the figure. Looking ahead, I will say that we will use this technique in the future - before the combination turn some faces so so that this combination moves the cubes we need. Of course, after the combination, you need to reverse everything back in place, although at this stage of the assembly it is not even necessary. You should not worry about this, such “pre-training” will include in us a maximum rotation of two faces. Now it remains to consider the case when the cube is in the bottom face. Place the cube in D exactly under that place in U where it should appear and use the K2 combination. You don’t need to memorize it - just run the algorithm a couple of times and you’ll understand what it does. In general, if you twist the cube and figure out how it works, then you will assemble the upper bound without my instructions, even by a faster method - what and where it goes when turning becomes obvious. Nevertheless, I will give here a strict algorithm for how to assemble this cube from the beginning to the end, so let's go further!

Lateral middle cubes

We assembled the central square 3x3 in U. Put in place 4 side medium cubes. This is done in the same way as in the original 3x3x3 cube. Notice that we collect the top face with a correctly assembled top bar (as in the picture at the beginning of this large section), so be careful. Let the center be red in front of us. Then find the green-red side middle cube. If it is located in the central horizontal layer or the upper edge, move it with the necessary rotation to the lower edge (if it is of course already worth it, you don’t need to do anything :))) If you touched one of the “ready” cubes, and this could happen only if the cube was in the middle horizontal layer, then rotate D by 90 degrees in any direction and return the “spoiled” edge to its place. Now the desired die is exactly in D and in U nothing has gone wrong. Next, turn D so that this the cube was at F. If its red color is at F, then simply turn F by 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 color is green) must match the color of the corresponding. center. What should happen can be seen in the figure on the right.

Side intermediate

Now we put in the right place 8 side intermediate cubes. For cubes located on the side edges, use a combination of K3 and K4 , as well as symmetrical to them on the right side. Before the combinations, of course, rotate D2 and U2 so that the combination performs the correct movement. If the cube is in D, then a simple combination return 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 hits L, and finally, return L to its place. Now use the algorithm described by several lines above. Of course, place the cubes so that the side color coincides with the side color of the adjacent side middle cube, because we are also collecting the top strip! Here, by the way, you may notice that visually identical (for example, green-red) cubes are actually different - one of them will rise to the left of the corresponding side 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 angular cubes. They are also assembled as in a 3x3x3 cube. We collect in turn. Choose a corner cube with one green edge that is not worth it. If it is in U, then move it to D by rotating the side face containing it by 90 degrees, then turn D so that this cube no longer lies in the turned side face and return this facet to its place. Even if this cube is in its place, but only oriented incorrectly, still perform these actions. Now everything depends on where the green face "looks". She can look either sideways or down. If down, then turn D so that it is under that corner cube in U, where there is still no desired cube, i.e. the one that will be there in the assembled cube. Next, rotate the side face containing our cube so that its green side looks sideways and is in D, then by turning D remove it from the rotated side face and return this facet to its place. Now our cube is looking sideways and is at the bottom. 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 be obtained (see the picture on the left): the upper edge, upper strip, and the color of the strip on each face coincide with the color of the center of this face.

Iii. All lateral central and angular cubes.

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

Side 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 combination K6 , which moves the cube from D to the middle horizontal layer (see figure). There will sometimes have to use a symmetrical combination (depending on the orientation of the colors of the cube being moved). If you want to move the cube from the middle horizontal layer to another place in the same layer, then you must first move it to the bottom layer (simply replacing any other cube from D with 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 - the color of the corresponding side face. Then correctly orient the cubes of the lower face.
Point one. Using the K7 algorithm, place all the side middle cubes in D in the correct positions. The algorithm moves in a circle 3 of the 4 necessary cubes. First, make sure that exactly one die in D stands in its position, and then use this combination to replace all the others. It is alleged that this is enough for assembly :)
Point two. Orient the cubes correctly. It just so happened that only 0, 2 or 4 dice can be incorrectly oriented (this is the device of the cube). If 0, that it is not necessary to do anything for obvious reasons, if 4, then you will have to build an algorithm for two twice.
Algorithm for 2. Turn D so that the cube to be oriented correctly is in F. Then do a simple operation 4 times: turn F in a clockwise direction, turn the central horizontal layer in a clockwise direction as viewed from the U side. Now turn D so that the second the cube that needs to be flipped turned out to be in F and again 4 times do the procedure described above. Then simply return D to its original position. In the end, we get what is in the picture on the right.

Angle cubes in the bottom

At the time of the assembly of these cubes, we turn the cube so that the top face is lower and vice versa (this is more convenient). To begin, put all the corner cubes in their places. To do this, use K8 , which moves through the cycle 3 corner cubes. Again, it is argued that this is enough for assembly. Next, after the corner cubes have fallen into place, they must be correctly oriented. A simple combination is used: RF 'R' FRF 'R' F. It twists the corner die, which is designated as CC in the figure to the left. It should be applied as follows: execute the algorithm until the cube is oriented correctly (in this case, the yellow face goes up, the combination must be executed once, if the color is yellow in F and twice in R), then turn U so that the next incorrectly oriented cube turned out to be and again apply the combination until the cube rises as it should. As a result, all the corner cubes are oriented correctly. Note that in addition to turning the corner cube, the combination also spoils the other, already correctly standing cubes. Do not worry - the cube is so arranged that you have to repeat the combination multiple of three times, and after every third use of the combination everything returns to its place. So everything will work out!

Total

So far, the actions I described were either almost obvious combinations, or combinations similar to those used in the 3x3x3 classic cube assembly. But then everything will be different. Although there are only two stages of assembly and a few new combinations, they are more complicated than they were before, but having read the entire scheme, you can collect any cube of dimension <= 5.
To be continued ...