Can all the disturbed Rubik's Cube be restored with a formula?

My friend, I am playing the Rubik's Cube now. I tell you that all the disturbed Rubik's Cube can be restored, as long as you master the method and formula! Stratification is very popular now! Although the Rubik's Cube has only 26 small squares, it changes a lot. The total number of changes in the Rubik's cube is

Or around 4.3? 10 19。 If you can turn the Rubik's Cube three times in one second, you will need to turn it for 454.2 billion years, which is about 30 times the current estimated age of the universe.

The reason for the total number of changes in the third-order Rubik's cube is this: after the six central squares are oriented, a coordinate system is formed. In this coordinate system, all 8 character blocks are arranged in 8! And each character block has three orientations, so it is 8! * 38, 12 prism blocks are all arranged, each with two orientations, that is, 12! *2 12, so multiplication is a numerator. The meaning of 3*2*2 on the denominator is to keep other color blocks still, and it is not allowed to change the direction of a character block (corresponding to 3), change the direction of a prism color block (corresponding to 2) or exchange the positions of a pair of prism color blocks or a pair of character blocks (corresponding to another 2) separately. As for why, I suggest you think about it first. I wrote some preliminary works. You can go here and have a look.

It can be seen that it is not easy to change back to the same color in a short time with so many changes. However, the fastest person in the world can restore a Rubik's cube in 7.08 seconds (the record was set at the Czech Open on July 12, 2008), and the record holder is Erik Akkersdijk from the Netherlands.

Why are those people so fast? Because he can remember many Rubik's cube algorithms, or there are also top players in the world called Rubik's cube formulas. It is said that he can remember more than 600 algorithms. The introductory Rubik's Cube solution we introduce here involves few algorithms and is very simple. As long as you learn, everyone can easily learn to play the Rubik's Cube.

Before we begin, let's look at the basic structure of the Rubik's cube. The relative positions of the six central squares of the Rubik's Cube are fixed. You'll know this when you dismantle the Rubik's cube. I'm sure that when you start wriggling in the following way, it's easy to forget what color you started with, so it's messy. So be sure to set a direction you like at the beginning. Here I choose blue as the top, green as the bottom, red as the front, orange as the back, white as the left and yellow as the right. Of course, your sticker may be different from mine. There should be 5*3 stickers on six sides of the Rubik's Cube! =30 kinds of stickers, why? Because if you specify the blue face as the top face, then the bottom face should have five options, and the remaining four faces form a ring, except for rotational symmetry, there are three! This is a paste method, right:) I chose the following picture.

(Step 1) Make a cross on the first side to form the following appearance: (You can also watch the video explanation of step 1)

Note, be sure to look at the cross as shown above, and the edge and center of each side are the same color. There are many ways to do this step, so I suggest you play it freely. If it is really difficult, I provide a definite solution here, that is, replace the blue prism block in the middle layer with the bottom surface, then color the side surface and turn it up. Here's an example, you have to understand that for position B on the left, only the following three steps are needed.

The animation will play automatically, or you can watch it step by step by using the and on the right side of the playback bar. The leftmost one returns to its original state. F

D

R2 180

In the initial state, turn the blue-yellow block to the bottom, rotate the bottom and turn up the yellow side.

Did I make myself clear? The D position is exactly the same as the B position, that is, the blue prism block in the middle layer is replaced by the bottom surface, and then the side surface is colored and turned up. For the A and C positions, you can rotate the surface so that it becomes the B or D position. If the blue surface of a prism is already on the top or bottom surface, I believe everyone will have a solution. What you have to remember is that if you encounter difficulties, you can change the blue color to the bottom surface and rotate it at will.

Sometimes, you will encounter the situation shown in the left picture. When the blue-yellow block turns to the bottom, it affects the aligned red face. At this time, you need to restore the position of the red face to the previous step. See the figure below for specific operation.

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The initial state flips the blue-yellow block to the bottom, but

This will affect the rotation of the bottom surface of the good red surface and the recovery of the red surface of the good yellow surface.

R2 180

According to the above, you should be able to make a cross by repeating four sides.

Here, I suggest a clearer and faster method for beginners. After the blue edge becomes the bottom surface, there is no need to turn it up in a hurry. You can turn it into the left picture. Note that the four sides of the bottom surface can be in any order, which will reduce your difficulty. Here are two examples to broaden your mind.

Finally, turn them over one by one. For example:

Now I want to talk about marking. You may have noticed some strange letters in the last table. Their meaning is simple.

F = front, front, B = back, back, R = front, right.

L = left, left, U = top, top, D = bottom, bottom.

Take the marks in the above table as an example. F stands for clockwise rotation of 90, F' stands for counterclockwise rotation of 90, and R2 stands for right rotation180. It's that simple. You got it?

In addition, if your Rubik's cube is a Rubik's cube with numbers, or a Rubik's cube with figures and patterns, then your six-sided central block has orientation problems. In this step, you can refer to this page to align the side center block.

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(Step 2) Align the first side and add a T-shape to the four sides to form: (You can also watch the video explanation in Step 2)

To do this step well, you only need to learn one trick. That blue character block, after turning around, has six positions.

For position a, only the following three steps are required. For positions C, D, E and F, please go to the back first and see how to become A or B. ..

Note that the color of the small corner we return must correspond well. For example, the small corners of blue, red and yellow here must go up to the red and yellow corners, and you can't make mistakes, otherwise the T word will not come out. Maybe some friends already think I look like the Tang Priest, but some friends do have this problem, so I'll talk about it in detail. A position conversion method is as follows: Rubik's cube

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In the initial state, this step is very clever, and at the same time, it achieves two purposes: one is to make the target corner position at the top level "meet" with the blue, red and yellow corner blocks at the bottom, and the other is to make the blue, red and yellow corner blocks ready for their proper positions. Turn the blue, red and Huang San corners to the target position. Turn back to the top floor.

For position b, it is exactly the same, that is, the last three steps are made into a mirror image of the diagonal line at the top, a square.

As shown on the left, we will encounter the mirror image of the top diagonal for countless times in the future, so please pay attention to the meaning of this mirror image.

The diagonal mirror image of F D F' above is R' D' R, please see,

The animation is set not to play automatically.

Please press the play button to start.

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The graph of position B is a diagonal mirror image of position A, so his algorithm is that FDF looks at the mirror diagonally. The diagonal mirror image of f is R', which you should understand very well. Please refer to the picture of the mirror above. The mirror image of d is the mirror image of D' F' and r.

For positions C, D, E and F, you can turn them to position A or B by rotating the sides and bottom. Here are a few examples: (Don't learn these algorithms by rote, you can find them by yourself for a few laps.)

For f, for c, for d and e.

F D2 F D

F D F D

R&D.

F D F '

Repeat four corners and you will get it.

So we have laid the foundation, let's make it simple.

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(Step 3) Put the prism block on the second floor and make it look like it (you can also watch the video explanation in Step 3).

From the previous step to this step, you must have seen that in this step, we have to deal with four prism blocks in the middle layer: red, white, red, yellow, orange, white and orange. This time, you turn the Rubik's Cube upside down, with the blue side facing down and the green side facing up, and then see which of the four prismatic blocks is on the top (the green center). Pick any one and we'll start with it. Take red and white as an example. Usually, you will encounter two situations.

We want to insert the red and white color block at the top into the position indicated by the red arrow in the above picture. You should put the corner between red and white in the nearest position (right front), and then one or two situations are very easy to identify, and the rows of the same color are in front and right respectively.

For the first case, the following algorithm is used,

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The whole algorithm is URU'R'+U'F'UF, is it regular? Let's see, it seems that the second half of U'F'UF is just a diagonal mirror image of the first half of URU'R', right? We see that the first four steps of URU'R' are used to combine the two small blocks we want in the left picture. The fifth picture marked with red in the above table shows the combined two color blocks, and the last four steps of U'F'UF are used to fill the two small blocks we have combined in the correct position. Rubik's Cube

So, if you meet,

What should I do? Maybe many friends see that our algorithm should be a diagonal mirror image of the above algorithm, because all we have to do is look at the diagonal mirror image, because the first half and the second half are diagonal mirror images of each other, so the algorithm is to turn the first half and the second half upside down and turn them into U'F'UF+URU'R', the Rubik's Cube.

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In some cases, the prism you need is not at the top, but in the wrong position or direction on the second floor. What should I do? First of all, the prism block at the top you have to do will become top by itself. If they are still disobedient in the end, as shown in the left picture, we will use the above algorithm to get an irrelevant prism block in place, and the red and white prism block we want will naturally become the top, which is a bit troublesome, but we can only endure our elementary solution, and maybe it will inspire everyone to learn advanced solutions:)

Besides, you can also try this algorithm, F' U2 L' U L U2 F, which is similar to the above algorithm. Please note that the starting states are not the same.

And his diagonal mirror image: R U2 B U' B' U2 R'

Repeat the other three sides, and the third step is completed. We have learned two algorithms so far. Come on.

In the following steps, I will provide you with two paths. They are very similar, but they are only one step apart. The first path is this page. Its advantage is that the shape is easy to recognize, which is very convenient for you to improve the speed, but the algorithm in the sixth step is a bit difficult to remember. Please click here to get the second path. Its advantage is that the algorithm is easy to remember, but it takes a little time to identify the shape in the fifth step. Personally, I like the first way.

(Step 4) Draw a cross on the top of the new Rubik's Cube (you can also watch the video explanation in Step 4).

Become:

Next, we will learn a new algorithm. The algorithm will switch the top level in the following four situations. After the four prism blocks on the top surface are rotated, only these four situations may occur.

1 2 3 4

Probability 1/8 probability 1/4 probability 1/2 probability 1/8

In this step, we treat all the character blocks as gray and only look at the prism blocks. For example, if you look at the left picture, it is the third situation above, and if you look at the right picture, it is the second situation above. (Reprinted from Rubik's Cube Station)

Before applying the algorithm, you should determine the direction of your Rubik's cube by referring to the green appearance at the top of the picture above. We just need to go through it, and we don't need the color on the side of the cross to match the two layers below. Please note that this algorithm will switch between these four situations in turn from left to right, that is, if you encounter a "point" (that is, the first picture above), you have to apply this algorithm three times (you have to position yourself according to the above picture before each time), and if you encounter a "one", you have to apply the algorithm twice.

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In addition, you can also try it at the time of "centralized".

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Through observation, we can see that this algorithm is the inverse of the above algorithm. He can make a good cross directly from the word "one"

When learning an algorithm at first, it is always easy to make mistakes. It is impossible to remember clearly at first, so it is easy to screw up and often depressed. I suggest that you should find some hints on the pattern of this figure, especially some relatively "whole" figures. I usually remember that the biggest hint of an algorithm is the bottom blue color block, because they are the most complete graphics, how they are separated and reassembled, and then how they are reassembled. Imagine a process or a "story" so that the algorithm is easy to remember, including the following algorithm.

Another small idea: here is a problem. Why do you say that there are only four situations, so it won't happen?

The answer is no, if your Rubik's Cube really has the above situation, then the most likely reason is that someone has dismantled your Rubik's Cube and installed it incorrectly. If he randomly assembles and there is no color at all, then he has a probability of112 installation error, and the solution is to turn an edge at will.

They don't appear because you can't keep other prism blocks still, but you can only flip a single prism block, which is mentioned when we first calculated the total variation of the Rubik's cube. Why? If you are interested, you can look at this page.

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(5) rotate that four corners of the top surface of the Rubik's cube and color the top surface to form the Rubik's cube.

(You can also watch the video explanation in step 5. )

There are only eight possible situations in the four corners at the top of the Rubik's Cube. The first one is already aligned, and the other seven are as follows. Rubik's Cube

Probability 4/27 4/27 4/27 2/27 4/27

If your patterns are outside these 8 kinds, then you definitely need to tear down the Rubik's Cube and adjust the orientation of a corner block. To know how the above probability is calculated, you can look at this page first. If not, you can also send a letter to ask me below.

I'm going to preach again If you are in a hurry to see the algorithm, you can skip it first.

First of all, we observe two cases: 1 2. In both cases, the three dislocated blocks are in the same position in the sense of rotation, right? This is one of their characteristics. Green has three directions at each corner. If we mark four corners, we might as well call it the case of 1 (from the upper right corner113), the case of 2 (2223), the 3 in brackets indicates that the green block is already on it, and the 3 (1233) is. 6 situation (12 12), 7 situation (2 1 12), have you found any rules? The sum of the numbers in brackets must be an integer multiple of 3! Why does it have to be an integer multiple of 3? I suggest you look at the total variables of the original Rubik's cube and prove that the sum of the angles of the corner blocks should be an integer multiple of 360 degrees. This restriction determines that we can only have eight situations.

This problem is not particularly simple. First, let's ask a question. If the four corner positions of the top surface of the Rubik's cube can be marked as 1, 2, 3 as above, they can be marked as (1113) (1233) and so on. The same situation can be removed after rotation.

If the same rotation is not removed, then the four angles can be distinguished. The answer is simply 3 4 = 8 1? But if the same rotation is removed, it will be more complicated, for example,113,311.

This is a classic problem of combinatorial mathematics, called the necklace problem (that is, wearing a necklace with several colors of beads), or called Polya theorem. You can go here if you are interested, but I still suggest you think about it yourself first. The formula here will make you dizzy at once, if you are unfamiliar with the concept of "Euler number" or have never heard of it.

In this case, we don't need the formula in mathworld, just enumeration:) But if we think about it again, the answer is 24. I looked it up and gave the same answer as the horror formula. Among these 24 kinds, the remainder of the "sum" of 4 numbers divided by 3 should be evenly distributed to 0, 1 2, which I haven't proved yet, hehe, so there should be 8 divisible by 3. That's right+these 7 kinds:

Now let's talk about the algorithm. For the first case, we should put the corner whose top surface is already green at the top (that is, the "left back corner"). At this time, you don't have to care which of the two floors faces forward, Rubik's cube.

Apply the following algorithm. The function of this algorithm is to keep the orientation of the upper corner (left back corner) unchanged, and at the same time turn the green of the lower three corners upward.

Algorithm 1:

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For the second case, we should find that it is a diagonal mirror image of the first case, right? So use the above algorithm to do diagonal mirror image to get the algorithm of the second case: Rubik's cube.

Algorithm 2:

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For case 3-7:

You have a two-thirds chance of meeting them. In fact, you can try the algorithm of 1 2 several times to see what will happen and find the law slowly. 3-7 cases are all solved by the combination of 1 2 algorithm, and there are at most 2 algorithms. Rubik's Cube

If you can't wait or feel that your method is not fast enough, you can see my method in References 3-7 here.

For the case of 3-7, I will introduce two methods to you. The first one has a deduction process. Master the second trick and you will know why you did it. The second method will obviously be faster. In practice, I recommend the second method.

Method 1: First, you need to put any square whose top surface is not green on the top (left back corner), and then you have the remaining three corners, right? You have to choose one of the first two algorithms, so that only one of the three angles turns to it. Then you will get one of the cases of 1 2. Why don't you turn it over? I don't have to tell you.

For example, for the case of 5, let's take a small piece with the top not green (the second picture), and then we have to turn only one green, right? Then you must use the algorithm 1 (why use the algorithm 1? Look at the graph of the algorithm 1 above and see how it turns green. You will understand. ), when used up, it will become the third picture on the left. Turn around and you will know that this is the case of 2.

One more, for the case of 6, the top is not green, and the remaining three are turned over. Which one should we turn? The right rear and left front positions are the same. To turn them over, you must use the algorithm 1, but you must turn them up at the same time. No, we just want one of the three corners to be turned to the top. You have to turn the right one, then you have to use algorithm 2, and then we will find that it becomes 2. Simple. Have fun.

Don't understand? Ok, we also have dummies.

Method two, you see, this is the algorithm 1, right?

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All you have to do is put your Rubik's cube in the lower position. Then apply the 1 algorithm, and it will become the case of 1 or the case of 2.

3 4 5 6 7

This stupid picture is actually very regular. You see, if you have two greens that are not on the top, then the top corner (left back corner) of the green is facing back. If you have four greens that are not on it, then the top corner (left back corner) of the green faces to the left. Then use the algorithm 1, it's as simple as that:) In fact, as long as you remember the sentence "4 left and then 2" 3-7, you will know it. What if I forget this sentence? Nothing, just use the algorithm 1 to turn it over several times, and finally it comes out.

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(Step 6) Adjust the order of the four corners to form: (You can also watch the video explanation in Step 6)

This step is simple. First turn the top surface to see if it is aligned (probability 1/6). If not, first find an edge with two corners of the same color (2/3 probability), like the following example. There is also a probability of 1/6 that you can't find such an edge, so we'll talk about it later. Rubik's Cube

We don't have to care about the color of the edge between the two corners and the color of the two layers below. Now put this side on the back, taking the last one in the above table as an example, it becomes,

Front and back

The follow algorithm is then applied,

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If after applying the algorithm, the four corners have not matched the colors of the lower two layers, turn around. The basic function of this step is to exchange the positions of the first two corners. Or do this counterclockwise rotation, and the two effects are equivalent after the top surface rotates. Rubik's Cube

In the second case, two edges with the same corner color cannot be found.

If you can't find such an edge with two corners of the same color, you can definitely find it by applying the above algorithm with your eyes closed. If you are interested in why, you can look at this page. In this case, you actually need to exchange two corners on a diagonal line.

In addition, you can also try the left and right mirror images of this algorithm, using the same method and the same effect.

B2·B2·R2

Ok, the sixth step is completed, just the last step! Let me tell you a good news first. In the last step, we don't need to learn new algorithms. We can do it with algorithm 1 and algorithm 2 in the fifth step:)

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(Step 7) Arrange the last prism blocks in sequence (you can also watch the video explanation of Step 7).

Last step, are you excited? Finally, victory is ahead. The last step is simple, you just need to look at the direction of the Rubik's cube. Rubik's Cube

In the first case, you want the three sides to rotate counterclockwise. (probability 1/3)

First of all, you have to put the color-matched edge in front.

Then use the fifth step algorithm 1+U2+ algorithm 2+U2. namely

R ' U ' R U ' R ' U ' 2 R

U2 F U F' U F U2 F '

U2

Step 5 Algorithm 1

Step 5 Algorithm 2

You can go up and look at the picture of step five. You can also have a look.

U2 between the algorithm 1 and 2 can also be replaced by rotating your whole Rubik's Cube 180, so the final U2 is not needed.

That's it,

In the second case, you want the three sides to rotate clockwise. (probability 1/3)

Then put the color matching side on the right. At this time, you will find that the second graph is a diagonal mirror image of the first graph.

So it is the fifth step: algorithm 2+U2+ algorithm 1+U2.

F U F' U F U2 F '

U2 R ' U ' R U ' R ' U ' 2 R

U2

Step 5 Algorithm 2

Step 5 Algorithm 1

Or, remove U2 in the middle and rotate the whole Rubik's Cube 180.

How to swing clockwise and then counterclockwise, which algorithm does not need to be memorized, is also difficult to memorize. You just need to pay attention to: 1. The direction from the second step and the fourth step to the top surface is the direction of the three arrows on the top surface; 2. The orientation of the Rubik's Cube will allow you to "cut" to the right side in the first step, and you will know which algorithm to use first and how to put the Rubik's Cube.

Probability112 probability 1/6

If you encounter such a situation that needs to be changed. It's actually quite simple. You can apply the above clockwise or counterclockwise algorithm at will, and then it becomes clockwise or counterclockwise.

Although we don't need to remember the new algorithm, the method described above is very long and needs 16 steps. You can remember another algorithm if you like. Rubik's Cube

R2 R2 R2 R2 R2 R2

These two algorithms are mirror images of the top horizontal centerline. This is much easier.

If you want to make your Rubik's Cube cooler, you can change your Rubik's Cube a little.

Rs2 Fs2 Us2 or (Rs Fs)3 Rs Us Fs Rs' Us2 (Fa Ra)3.

Among them, marks such as Fs, Fs', Ra, Ra' indicate that you move back and forth or left and right like a clip, s = slice, a = reverse slice, for example,

Mark Fs Fs' Ra Ra'

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He is equivalent to F B' F' B R L R' L'