Solving the Rubik s Cube

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Rosalind Richards
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2 Solving the Rubik s Cube Introduction Hungarian sculptor and professor of architecture Ernö Rubik invented the Rubik s Cube in When solved, each side of the Rubik s Cube is a different colour. Each face can be rotated, allowing the smaller cubes to be rearranged in many different ways. The challenge is to change a mixed state to the original state, in which each face of the cube consists of nine squares of a single colour. There are 12 edge pieces which show two coloured sides each, and 8 corner pieces which show three colours. Each piece shows a unique colour combination, but not all combinations are realized (For example, there is no edge piece showing both white and yellow, if white and yellow are on opposite sides of the solved cube). A Rubik s Cube has 43, 252, 003, 274, 489, 856, 000 different positions, about 43 quintillion. Despite the vast number of positions, all cubes can be solved in 29 moves or fewer. 1

3 2 Bringing Order to the Cube Without careful thought trying to solve the Rubik s cube can be a difficult, if not, impossible. By the use of just a few simple steps, however, it is possible to solve the Rubik s cube from any position. The method presented is very simple, and isn t very difficult to improve. With practice this method can solve most cubes in about three or four minutes. There are people who know about one-hundred different moves and a variety of different hand techniques that allow them to solve most cubes in about twenty seconds. It may seem a little odd, but the first two stages of the solution are the most difficult; there are a large number of different initial arrangements. Once the first two step are completed, the number of different positions decreases. Before you start select one colour to be your top colour. This is the colour of the face we solve first, in all the diagrams below this will be yellow. You may find it easier to follow the diagrams below if you select yellow as your top colour. If you do this you will be able to see the correspondence between your cube and the diagrams below. If you select yellow as your top colour, then white must be your bottom colour. Below there are various sequences of moves. We illustrate each sequence by cubes showing in dark grey the section of the cube you should move and a black arrow showing the direction you need to move it. Don t Forget: In every diagram an arrow means turn the dark grey part of the cube a quarter turn in the direction indicated. In some diagrams you will see 2 TURNS this means two quarter turns (the direction doesn t matter). To illustrate the cube we use a diagram like that one below

4 3 This diagram shows all six faces of the cube: three are shown as they are on the cube, whilst the other three are shown as projections around the cube. At some points we won t need to worry about what colours certain cubes are and so we will just colour them light grey. Step 1 Forming the Cross The first stage of the solution is to form a cross like in the diagram below. In this diagram, and all the subsequent ones, only the cubes we are interested in are coloured, the remainder are shaded grey. There are two main steps to reach this point, the first makes the cross in the first layer, the second extends the cross to the centre squares of the middle layer. Forming the top cross. Firstly move your cube so that the centre yellow square is at the top of your cube.

5 4 Once you have done this, the next step is to form a cross on the top. This step is very easy, but you may find the move in Figure 1 useful. It moves an edge from the bottom of the cube to the top. Don t forget the final step of this move, it ensures that you don t affect the other parts of the cube. Don t Forget: The diagram illustrates which parts of the cube you move in dark grey. Only those cubes that are important are coloured. The arrows show the direction you need to rotate the grey slice of the cube. Each arrow is a 90 degree turn in the direction shown. It is possible that you will find that you have an edge in the correct place, but flipped the wrong way. If this occurs then you can perform the edge flip in Figure 2 to correct the situation. This will flip the edge without affecting the other edges. (a) (b) (c) Figure 1: The edge swap move. The final turn here is important even though it doesn t move any of the yellow cubes. Don t forget it.

6 5 (a) (b) (c) (d) (e) Finish Figure 2: The edge flip move. Extending the cross. Once you have a cross on the top, you need to move the top edges around so that you get an extended cross: To complete the top layer two or more top cubes may need to be swapped around. There are two different types of move. The first of these, shown in Figure 3, swap neighbouring cubes. The second move is used to swap two cubes on the opposite sides of the top layer. This move is illustrated in Figure 4. In this figure each shaded part of the cube indicates two quarter turns.

7 6 (a) (b) (c) (d) (e) (f) Finish Figure 3: The first edge swap move. The notation 2 TURNS indicates two quarter turns of the shaded part of the cube. (a) (b) (c) (d) (e) (f) Finish Figure 4: The second edge swap move. Each twist of the cube is a half turn.

8 7 Don t Forget: The 2 TURNS in Figure 3 represents a half turn of the shaded part of the cube. Whilst in Figure 4 each twist of the cube is a half turn of the shaded part of the cube. This completes the first step. Once you have done this, have a practice to make sure that you are comfortable forming the cross from any starting position. Step 2 Finishing the Top Layer The second step completes the top layer by inserting the top corners. Before you begin check your cube looks something like the one below, remember we have selected yellow as our top colour. Our aim is to complete the top layer, so the cube looks like: To do this we only need one set of moves. Step A. Find a corner piece in the bottom layer that should be in the top layer. Turn the bottom layer so that that piece is direct below its home in the top layer. Hold the cube as illustrated in the diagram.

9 8 The corner piece doesn t have to be exactly the same way around as illustrated in the diagram, you only need the bottom piece below its home in the top. To put it another way, the corner piece illustrated in the diagram only needs to be the corner piece with yellow, blue and red faces. Step B. To move the piece from the bottom to the top you just repeat the set of moves in Figure 5 until the piece is in the top layer and the correct way around. You may find that after doing the move a few times the piece is in the top layer, but the colours are the wrong way around, just repeat the move until the piece in in its home and the colours are the correct way around. Don t Forget: You need to continue to apply the sequence of moves in Figure 5 until you have the piece the correct way around in the top layer. If you don t do this, then you won t complete the top layer correctly. Step C. The only time when you may encounter problems is when a piece is already in the top layer, but isn t the correct way around, or is not in the correct place. If the cube is already in the top layer and in the correct position but not the correct way around, then just apply the moves (a) (b) (c) (d) Figure 5: Putting corner piece into place.

10 in Figure 5 until it is. If the cube is in the top layer but in the wrong position, then use the moves in Figure 5 to move it to the bottom layer, then start from Step A above. 9 Step D. cube. Repeat Steps A C until you have completed the top layer of the Step 3 Completing the Middle Layer The next step is to complete the middle layer of the cube. To do this we need only put in the middle layer s edges, since we already have the middle cubes of the middle layer in place. By the end of this step your cube should look like: There are two different situations that can occur. The set of moves are basically the same, except in the second case you do them twice. Firstly you need to turn the bottom layer of the cube so that you are in either situation 1 or situation 2 as described below. If it is impossible to do this; that is, because there are middle layer cubes already in the middle layer but in the wrong place, or the wrong way around. Then perform the moves in Situation 1 to move these cubes from the middle layer to the bottom layer. From here you can now continue. Situation 1. If you are presented with the following arrangement of cubes (of course their colours can differ)

11 10 Then we want to move the cubes so that we end up with Performing the sequence of moves in Figure 6 will give us the desired result without destroying the parts of the cube we have previously assembled. This covers the first and the most straight forward situation. The second situation is almost identical. In this case, the cube we wish to move is in a slightly differ- Situation 2. ent place.

12 11 (a) (b) (c) (d) (e) (f) (g) (h) Figure 6: The moves for situation 1 of step 3. Our goal is the same though; we want to move the cube so that we end up with

13 12 To do this we perform a very similar sequence of moves to situation 1. Using the notation from Figure 6 we perform the following sequence of moves: a,b,c,d,e,f,g,h, b,c,d,e,f,g,h That is, we perform the sequence exactly as it is in Figure 6 once, then we perform it a second time without doing the initial move a. Step 4 Moving the Bottom Edges The fourth step puts the bottom edges into the correct place, but it won t put them in the correct way around, unless you are lucky. Moving the bottom edges so they are the correct way around is done later. At this stage our cube should look like: Step A. The first thing to do is turn the cube upside down, so what is the bottom now becomes the top. Since we made yellow our top colour, white should now be on the top of the cube. What you need to do is turn the top layer of the cube (that is, the new top layer with a white cube in the centre) so that as many edge cubes are in their home positions as possible. It doesn t matter if they are the wrong way around; we will deal with that later. Step B. Suppose you have the situation illustrated below.

14 13 (a) (b) (c) (d) (e) (f) (g) Figure 7: The moves for step 4. In this case the white-red and white-green cubes are in their home positions, but they are the wrong way around, whilst the white-orange cube and blue-orange cube are in the wrong places and need swapping. We want to swap the white-orange and white-blue cubes around so they are in the correct place, they don t have to be the correct way around though; if they are, then you are lucky. The sequence in Figure 7 will swap the white-orange and white-blue cubes around, so that you end up with the following arrangement. Of course, the situation you will be presented with may have different colours. Don t Forget: To place the edges into the correct places may require several applications of this move, but eventually you will end up with the edges in the correct place.

15 14 Step 5 Moving the Bottom Corners This step must be performed exactly as stated otherwise you will have to go back to Step 1. In this step we move the bottom corners into the correct places, later we will put them the correct way around. Don t Forget: To complete this step you need to perform at least two swaps. Sometimes you need to apply four swaps. You should never have to perform an odd number of swaps, if you do, then someone has moved the stickers on your cube! At this stage the cube is positioned so that the white face is now at the top. The previous step put the edges of the new top layer (the one with the white cube in the centre) in the correct places. Suppose we have the situation illustrated below In this case, one cube is in the correct position and the correct way around. There are three cubes that aren t in the correct place. In this case we need to perform the following swaps: 1. Swap the orange-blue-white cube, with the red-green-white cube. 2. Swap the orange-green-white cube with the newly moved, orangeblue-white cube. To perform this sequence we use the following steps: Step A. Position the cube so that the first two cubes we wish to swap are position as illustrated in the diagram below:

16 15 Step B. To interchange the positions of the two cubes illustrated in Step A, perform the set of moves in Figure 8. When you have finished this sequence DO NOT MOVE THE CUBE. If you do you will have to start all over again! (You will notice that the two layers below the top one are all messed up, this will be sorted out in a second.) Step C. Now turn the top layer of the cube (remember white is on the top now) so that the second two cubes you want to swap are in the same positions as we illustrated above. Step D. Then perform the moves in Figure 8 again. When you complete the set of moves for the second time the lower two layers will be back to their starting positions. Don t Forget: Remember you must always perform this operation an even number of times to return the bottom layers to their original formation. When you have completed your swaps, you may need to turn the top layer so that the cubes are in their correct places. Step 6 Flipping the Bottom Edges This stage must be performed exactly as stated otherwise you will have to go back to Step 1. At this point every cube is in the correct position, but may not be the correct way around. The goal of this step is to flip the edges so that they are the correct way around. This move is like the previous one in that you must do it twice to return all the pieces to their original positions.

17 16 (a) (b) (c) (d) (e) (f) Figure 8: The moves for step 5. In these figures 2 TURNS indicates you apply two quarter turns to the shaded part. To perform an edge flip, position the cube so that it is as illustrated in the diagram below. Step A. Perform the moves in Figure 9. The orange-white edge will be flipped so it is the correct way around. However, when you have finished the moves, DO NOT MOVE THE CUBE. Step B. To flip the blue-white cube, turn ONLY THE TOP LAYER so that blue-white cube is in the same position as the orange-white cube was before we performed the moves in Step A. Now follows the moves in Figure 9.

18 17 (a) (b) (c) (d) (e) (f) (g) (h) Figure 9: The moves for step 6. Note that in the odd numbered moves it is the middle slice of the cube that is begin rotated. Don t Forget: The orange-white and blue-white flips are only used to illustrate the moves you could have to perform. You may need to flip different coloured edges. When you have completed your flips, you may need to turn the top layer so that the cubes you have flipped are in the correct position. Your cube should look like the one below, and you are nearly done. Step 7 Flipping the Bottom Corners This stage must be performed exactly as stated otherwise you will have to go back to Step 1. Now we come to the final stage, flipping the cor-

19 18 ners. In this case there are two different situations. Don t Forget: Once you begin you should only move the top layer between each different flip (it is possible to get away without doing this, but it is better not to try). If you find that you need to perform one Move 1 and one Move 2, then make sure that after you have perform Move 1 you only move the top layer to bring the corner you wish to flip into position. The same holds true if you need to do more Move 1 s and so on. Situation 1. Suppose you have an edge in the follow position: Then the sequence of moves in Figure 10 will adjust the white-red-blue cube so that it is the correct way around. However, this sequence of moves will, on a temporary basis, destroy the bottom two layers. For those of you that are interested, if you do the moves in Figure 10 three times the bottom two layers of the cube return to their original state, this is what makes the corner flips work, and also why it is important to only move the top layer between moves. Situation 2. If you find that you have an edge in the following position

20 19 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) Figure 10: The moves for step 7 situation 1.

20 then perform the moves in Figure 11 to flip the corner so it is the correct way around. If you do Move 2 three times the bottom two layers of the cube return to their original state.

21 20 then perform the moves in Figure 11 to flip the corner so it is the correct way around. If you do Move 2 three times the bottom two layers of the cube return to their original state. If you do Move 1 then Move 2 you also return the bottom layers to their original state (or you can do Move 2 then Move 1 ). Like the previous steps, when you have completed your flips, you may need to turn the top layer so that the cubes are home. When you have perform these moves on the all the corners of the cube, you should have returned the cube to its original state. Congratulations, you have just solved the Rubik s Cube. Now, you can try and commit this to memory and impress your friends.

22 21 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) Figure 11: The moves for step 7 situation 2.

Further Mathematics Support Programme

Further Mathematics Support Programme Stage 1 making a cross Solving the Rubik s cube The first stage is to make a cross so that all the edges line up over the correct centre pieces in the middle layer. Figure 1 Find a white edge piece (in