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Solving a Puzzle Cube

Corbin R. Sheridan

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Solving a Puzzle Cube

Corbin R. Sheridan

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Table of contents Preface………………………………………………………………………………….5 Chapter 1: The Cube………………………………………………………………7 Chapter 2: The First Layer…………………………………………………….10 Chapter 3: The Second Layer………………………………………………..13 Chapter 4: The Third Layer……………………………………………………16

Orienting Last Layer Edges (Edge OLL)………………………………16 Permuting Last Layer Edges (Edge PLL)………………………..…..17 Permuting Last Layer Corners (Corner PLL)…………….…..…….18 Orienting Last Layer Corners (Corner OLL)………………………..20

Chapter 5: Patterns……………………………………………………………...22

Checkers………………………………………………………………………….22 Centers…………………………………………………………………………….22 Negatives………………………………………………………………………...23

Glossary………………………………………………………………………………..25

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Preface

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Preface This is a guide that will teach you how to solve a standard three by three puzzle cube. What is a standard three by three puzzle cube? Just look at the infamous Rubik’s cube and you have your answer. A three by three puzzle cube (referred to as just a “cube” or “puzzle” for the purposes of this document) is a cubic puzzle with smaller cubes called cubelets cut into it. The cube’s faces rotate around a central core inside the puzzle that allows other faces to be turned after the initial face, making it an easy puzzle to scramble and quite the devil to solve. Why do I want to solve this puzzle, you ask? Why take on such a notorious task as the Rubik’s cube? All you need to get from this book is the simple feeling of enlightenment as to the secrets of this famous mind-blower. Simply put, it is a quirky pastime and nothing more. The purpose of solving a puzzle like this is fun. No more, no less. This book will teach you how to solve a cube such as this in three simplified steps. I encourage you to be patient, though, as it is a process that does take some practice in a few parts. Some of the puzzle can be taught and memorized, yes, but there are parts that just take patience and spatial reasoning. Once you’ve learned how to solve the cube, this will also teach you a few fun patterns that you can use to amuse yourself or impress your friends (or enemies, as the case may be. Who doesn’t want to “one-up” the ol’ arch nemesis with a show of superior intellect?).

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Chapter 1: The Cube

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Chapter 1: The Cube The first thing to know about a cube is its’ dimensions. The cube is divided into a series of cubelets that cover the surface. The dimensions of the entire cube are three cubelets by three cubelets by three cubelets. The second item of note is the orientation of all the colors on the cube. Each face has four edge pieces (pieces with only two colors on them) and four corner pieces (pieces with three colors on them). The faces of the cube are defined by the center pieces, as they are affixed to the core. This means that you are moving the edges and corners around the center but the center piece on any given face never actually moves. Following this path of logic, it only makes sense that if two color centers are opposites, they will remain opposites forever. The cube should be solved accordingly, putting orange stickered pieces around the orange center, green stickered pieces around the green center, and any piece with both orange and green stickers between the two centers. This remains constant throughout the entire solving process. The color orientation of the puzzles colors are as follows:

White is always opposite yellow (or in some cases, yellow/white is opposite black)

Red is always opposite orange

Blue is always opposite green

This system ensures that no single piece will ever have two colors that are opposite, e.g. no piece has both a blue sticker and a green sticker. Now that we’ve established how the cube looks, let’s talk about how it works. The cube is cut into three vertical rows called slices and then cut into three horizontal rows called layers, making a three by three grid over each face of the cube. Slices and layers are defined by how you hold the cube, meaning that if you rotate the entire cube ninety degrees to the left, the previous layers are now slices and the previous slices are now layers. This system of cuts allows each face to be moved independently.

Figure 1.1: This is a three by three puzzle cube with layers defined by shade and slices defined by the arrows.

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Each face on a cube has a name that is used when referring to that face in algorithms, which we will cover momentarily. Just like the location of slices and layers remains the same when the cube is rotated, a faces’ name changes when it is rotated. Tilt a cube ninety degrees to the left and your top face has become your new left face, as your right face has become your new top face and so on. The names and abbreviations of the faces are as follows:

Up (U)

Down (D)

Left (L)

Right (R)

Back (B)

Front (F)

Figure 1.2: This three by three has had its respective faces labeled according to the way the cube is positioned. The names and letters of a cubes faces are part of a system called cube notation. Cube notation is important in understanding the letters written out in a series of steps with the purpose of simplifying a process, called an algorithm. When reading a puzzle cube algorithm, each letter is an instruction to make a move. Each space is used to separate two given moves that are made with those instructions. Basically, if you see a letter of cube notation, turn that particular face clockwise. If the letter has an apostrophe, like U’, turn that particular face counter clockwise. If a letter has a number right after it without any spaces, turn that face twice (it doesn’t matter which direction because turning a face twice in any direction just ends in an effective 180 degree turn).

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Chapter 2: The First Layer

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Chapter 2: The First Layer Now that you understand cube notation, orientation, and algorithms, you’re ready to begin solving. First, practice moving pieces around the cube. Move any edge piece to a specific location and don’t think about any of the other pieces. You’ll want to use multiple faces to accomplish the job, and figure 2.1, shown below, might just help you.

Figure 2.1: To move an edge piece into its correct position, first find one center to which it belongs (a center of one of the colors on the piece). Twist the face that holds that edge until the two colors are matched. Now that they are matched, rotate the face on which they are matched until the other sticker on the edge piece matches its center. Once you think you’ve got the hang of it, move the edge pieces into a formation called the cross. This is done by picking one face to begin completing and aligning all of its respective edges so that each edge piece matches the center you chose and its respective second center. The face that you chose will be referred to as the first layer. (the finished product should look like figure 2.2). Figure 2.2: The cross formation on the puzzle. The red center is matched up to each of its respective edges, which are matched up to each of their respective second centers.

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Once all of the edges have been put into place it’s time to plug in the corner pieces. Make sure that the first layer is currently your top layer (U). The first step to plugging in corners is to find a corner that belongs in your first layer (a corner with one sticker that matches the color of your first layer). If the corner you’ve selected is in the bottom layer, skip the next paragraph. If not, read on. If the corner you’ve selected is in the top layer (your first layer), set the cube so that it’s in the top right position on the front face. Proceed to do the following algorithm until the corner is in the bottom right position on the front face: R’ D’ R D When the corner is in the bottom layer, your next step is to rotate the down face until that piece is directly below the position in which it needs to be, as shown by figure 2.3 (top). Once it’s in position, rotate the entire cube until that piece is in the bottom right position of the front face and use the algorithm R’ D’ R D until the cube is in the correct position and oriented the correct way (meaning that the colors of all three stickers on the corner match the stickers on adjacent cubelets, as shown in the bottom of figure 2.3). Figure 2.3: (Above) A corner piece is directly below its proper location. (Below) By use of the R’ D’ R D algorithm, the corner piece was moved from below the correct position to the correct position, and is oriented the correct way.

Repeat this process until all corners belonging to the first layer have been plugged in correctly. Once completed, congratulate yourself- you’ve just solved the first layer, one face of the puzzle. The finished product of the first layer should look like figure 2.4. Figure 2.4: Puzzle cube with a completed first layer.

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Chapter 3: The Second Layer

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Chapter 3: The Second Layer Now that you’ve finished the first layer, flip the cube over so that your previous up face is now down and your previous down face is now up. The first layer should be your bottom layer. The second layer is by far the shortest layer. There are two easy algorithms used in this part and only four pieces to move around. The only pieces you need to worry about in the second layer are those that do not have any colors of the first layer or the third layer (the color of your current top center piece). The first step to solving the second layer is to find a piece that belongs there (as defined by the previous paragraph). The piece that you have chosen needs to be in or be moved to the top layer for the moment. If the piece you chose is already in the top layer, skip the next paragraph. Once again, if it is not, read on. If the piece you chose is currently in the second layer, rotate the entire cube until that piece is in the middle right position of the front face. Once it’s in the right position, use the algorithm U R U’ R’ F R’ F’ R . This should put that particular piece in your top layer and on your back face. Once the piece you chose is in the top layer, rotate the up face until that piece matches one of its respective centers. When it does, your next step is to determine where it needs to go. There are only two options, provided that it is a correct second layer piece that’s matched up with a correct center. The piece either needs to be down and to the left of its current position, or down and to the right. Set the piece so that it’s in the top middle position of your front face and perform one of the following algorithms, depending on where the piece needs to go:

Down and to the left (figure 3.1): U’ L’ U L F’ L F L’

Down and to the right (figure 3.2): U R U’ R’ F R’ F’ R Figure 3.1: A piece in the top layer needs to be moved down and to the left into the second layer. Use the algorithm U’ L’ U L F’L F L’

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Figure 3.2: A piece in the top layer needs to be moved down and to the right into the second layer. Use the algorithm U R U’ R’ F R’ F’ R

Repeat this process with every cubelet that belongs in the second layer. Once you have done this to all four cubelets, you have completed the second layer. You’re almost done, there’s just one layer left. Your finished product for the second layer should look like figure 3.3.

Figure 3.3: (Left) Completed second layer from left view. (Right) Completed second layer from right view.

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Chapter 4: The Third Layer

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Chapter 4: The Third Layer The third layer is often considered the most difficult part of the puzzle. In it, you will use four different algorithms and you will need to be able to intuitively find where to use them. However, if you completed the first and second layer, this should be a breeze.

Orienting Last Layer Edges (Edge OLL) If you completed the second layer, your third layer edges could possibly be oriented correctly already. If this is the case, your third layer’s edges should all be correctly matched to the third layer center (not necessarily to their respective second centers, though). In the event that your third layer edges are already oriented correctly, skip to the section of chapter four entitled “Permuting Last Layer Edges (Edge PLL)”. The first part of the third layer is a process called Edge OLL. OLL stands for “orienting last layer”, remember that orienting a piece means moving it around so that it’s stickers face the right way and match their centers. In this case, we’re just trying to make the stickers on the edge pieces match the center of the third layer, not any of the respective second centers. There are four different patterns that are displayed when you look at the third layer from above. These patterns are called the dot, the “L”, the bar, and the cross. On each pattern, use the algorithm F R U R’ U’ F’ as instructed by the description of figure 4.1 to get the next pattern until you’ve gotten the cross. Figure 4.1 has all of these images in the order in which they occur. Figure 4.1: A top view of the most basic pattern in edge OLL, the dot (top left). Use the algorithm above to progress to an upside down “L” pattern (small square, top middle), remembering that the bottom edge of the picture is your front face. Turn the upside down “L” around so that it looks like the “L” (top right) and use the algorithm again to progress to the bar(bottom right). Use the algorithm one more time to progress to the cross (bottom left) and you’ve finished edge OLL.

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Permuting Last Layer Edges (Edge PLL) If you completed the second layer and edge OLL, your third layer edges could possibly be in the correct position already. If this is the case, your third layer’s edges should all be correctly matched to the third layer center and to their respective second centers. In the event that your third layer edges are already oriented correctly and matched with all the correct centers, skip to the section of chapter four entitled “Permuting Last Layer Corners (Corner PLL)” (you may have to rotate the top layer a few times before it becomes clear that the edges are permuted correctly). Permutation is moving a piece to change its physical location on the puzzle, as opposed to orientation which is moving a piece around in its current location to change which way it faces. To permute the edges in the third layer and maintain their orientation, first rotate the third layer until two of the edge pieces are permuted correctly. This will be apparent in one of two ways. The first way in which the edges can be permuted is displayed in figure 4.2. Figure 4.2: In this permutation, there are two correctly permuted edges that are positioned across from each other with the center in between them (white arrows in the picture indicate which edges match both the third layer center and their respective second centers). Remember, you may have to rotate the top layer a few times before you realize that they are permuted correctly.

Once your third layer is in the same position as figure 4.2 (one correct edge in the back and the other in the front), use the algorithm R U R’ U R U2 R’ to get the permutation displayed in figure 4.3 (one correct edge in front and the other to the right). Figure 4.3: When you have the new permutation, rotate the entire cube ninety degrees counter clockwise (once to the left) so that the correctly permuted edges are now on the right face and the back face, as shown to the left.

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Once your third layer is in the same position as figure 4.3, or if you had this permutation after edge OLL (remember to make sure that the matched edges are on the right and back faces), use the algorithm R U R’ U R U2 R’ U and you will have successfully permuted the edges of your third layer correctly, as displayed in figure 4.4.

Figure 4.4: (Left) Completed first and second layers with correctly permuted and oriented third layer edges, front view. (Right) completed first and second layers with correctly permuted and oriented third layer edges, back view.

Permuting Last Layer Corners (Corner PLL) If you completed the second layer and/or edge PLL, your corners could possibly be in the correct position already. If this is the case, your third layers corners should all be in the correct position, as indicated in figure 4.5 (though not necessarily oriented correctly, as in figure 4.5). If this is the case, skip to the section of chapter four entitled “The Final Step: Orienting Last Layer Corners (Corner OLL)”. Figure 4.5: The corners in the third layer are permuted correctly, but not oriented correctly. They’re in the right spot, but not turned the right way.

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In this section, we’re doing the same thing to the third layer corners as we did to the third layer edges in the previous section- moving them around until they’re in the correct spot (though not necessarily oriented correctly). The first step is to find a corner that is already in such a position. During this process the third layer edges have to stay matched, so such a corner cannot be found by rotating the third layer. When you find a correctly permuted corner, rotate the entire cube until that corner is in the bottom right position when the cube is looked at from above and skip the next paragraph. If you don’t have any correctly permuted corners, read on. If you don’t have any correctly permuted corners, simply make sure the third layer is your top layer and use the algorithm U R U’ L’ U R’ U’ L. After this, you should have exactly one correctly permuted corner. Rotate the entire cube until the corner is in the bottom right position when the cube is looked at from above. With the correctly permuted corner in the bottom right position when looked at from above (top right position when looked at from the front), use the algorithm U R U’ L’ U R’ U’ L. This will cycle the other three corners counterclockwise between themselves, as shown in figure 4.6. You may only have to do this one time, but you will only have to do it twice at the most. After each use of the algorithm, the correctly permuted corner should be in the same spot, ready to use the algorithm again if necessary.

Figure 4.6: Set the correctly permuted corner to the bottom right position, when looked at from above, and use the algorithm above to cycle the other three corners counterclockwise. Once you’ve permuted the corners in the third layer, the puzzle should resemble the image in figure 4.5 on the previous page (the corners could be oriented in any way, figure 4.5 is just one example of how the puzzle should look after corner PLL).

With Corner PLL done, there’s only one last thing to do: orient the corners to match the faces to which they belong.

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The Last Step: Orienting Last Layer Corners (Corner OLL) If you have finished any one of the previous sections and the cube is already solved, congratulations, you just solved a Rubik’s Cube for the first time. If your corners still need to be oriented correctly, we’re almost done so keep reading. This part is fairly short, there are only a maximum of five small steps, the first of which being that you tilt the cube to the left so that your third layer becomes your new left face, as displayed in figure 4.7.

Figure 4.7: The cube is tilted to the left so that the third layer is now contained in the left face. Proceed with caution on the next part, as it can be easy to mess up. Make sure the corner in the top left position of your front face is disoriented (not oriented correctly). If it is oriented correctly, rotate the left face (your third layer) until the top left position on your front face contains a disoriented corner. The rest of the puzzle will be partially scrambled when you do this, so don’t turn any of the other faces, just follow the instructions until the puzzle is solved. Use the algorithm R U’ R’ U over and over until the piece is correctly oriented. Once the piece is correctly oriented, rotate the left face until the top left corner of your front face is another disoriented corner and repeat the algorithm until the piece is oriented. Repeat this process of rotating the left face and using the algorithm as necessary until all corners in the third layer are oriented correctly. At this point, rotate the third layer until all the colors match. The puzzle is now solved.

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Chapter 5: Patterns

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Chapter 5: Patterns Now that you’ve solved the Rubik’s Cube, you may want something more from your puzzle. If that is the case, this section will teach you three different patterns that you can make on your puzzle to entertain yourself. These patterns are the checkers pattern, the centers pattern, and the negatives patter.

Checkers The checkers pattern is by far the easiest pattern to make, yet definitely one of the most fun. To do the checkers pattern, turn each face and it’s opposite twice. The algorithm for the checkers pattern is U2 D2 B2 F2 R2 L2. After doing the algorithm, the puzzle should now look like figure 5.1. Figure 5.1: Use the algorithm above to make the checker pattern all the way around the cube.

Centers The centers pattern is one of the more difficult patterns to pull off. To make the centers pattern, move the middle slice of the puzzle up, rotate (don’t tilt) the puzzle ninety degrees to the right, and move the middle slice up again. Repeat this process until your cube looks like figure 5.2.

Figure 5.2: After following the process described above, the puzzle should look like the image to the left.

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Negatives The negatives pattern makes three patterns on the cube that reflect over two faces each. The patterns are made with two colors each and the colors are inverted between the two faces involved in the individual pattern. The algorithm for the negatives pattern is R U2 R’ L’ U2 L. When you’ve completed the algorithm, the cube should look like figure 5.3 Figure 5.3: The negatives pattern includes a pattern reflected across the edge between the front and top faces, a pattern reflected across the back and down faces, and mirrored between the left and right faces.

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Glossary

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Glossary A Algorithm: A series of short steps intended to simplify a larger process.

B Back: The face of the cube facing away from the person holding it.

C Center: The piece on a face of the cube that only has one sticker on it and is surrounded by all other pieces on that face. Core: The small, three axis piece inside a puzzle cube to which the center pieces are affixed. This device allows the cubelets to rotate around it, making the puzzle cube possible. Corner: The piece of a puzzle on which there are three stickers of different colors. This piece is located between two edge pieces on any given face. Cube: A three dimensional, six sided figure on which each face is a square.

Cubelet: Any of the smaller, relatively cubical pieces that rotates around the central core of a Rubik’s type puzzle cube.

D Disorientation: 1.The condition of a cubelet when it isn’t matched to at least one of its respective centers. 2.to change a cubelets position so that it is no longer oriented correctly. Down: The face of the puzzle that’s on the bottom of the cube.

E Edge: The piece of a puzzle on which there are two stickers of different colors. Fits directly between two center pieces.

F First Layer: The first horizontal row of cubelets that you assemble when solving a puzzle cube. Front: The face of the cube facing towards the person holding it.

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L Layer: A horizontal row of cubelets on a puzzle cube. Left: The face of the cube facing to the left of the person holding it.

O Orientation: 1. The way a cubelet is turned in one location. 2. To change a cubelets position in one location so that it matches at least one of its respective centers.

P Pattern: A specific arrangement of cubelets to coordinate the colors on them. Permutation: 1. Any sequence of the location of various cubelets. 2. Moving one or more cubelets around each other to change their position.

R Right: The face of the cube facing to the right of the person holding it. Rotate: To twist one of the slices or layers of the puzzle.

S Second Layer: the second horizontal row of cubelets that you assemble when solving a puzzle cube. Slice: A vertical column of cubelets on a puzzle cube. Sticker: The colored adhesive on each cubelet or center piece.

T Third Layer: The third horizontal row of cubelets that you assemble when solving a puzzle cube.

U Up: The face of the puzzle that’s on the top of the cube.

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