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Sean Gallagher - Sr. Seminar Paper 40 Pages
1.
2. Several pieces move at once, in contrast to other puzzles that may only move one piece at a time.
3. Most pieces of the cube have what is called “orientation.” This means that not only does each piece have a correct “positioning,” but each piece has a correct orientation as well. In other words, a single piece could be placed in the correct spot but could be flipped (colorwise) the wrong way. Rubik says that the only other puzzles that have this quality are assembly puzzles, which are very, very different types of puzzles as compared to the Rubik’s Cube (viii).
4. The three-dimensionality of the cube is a unique characteristic trait. Three-dimensional moving-piece puzzles are very rare. In Rubik’s eyes, this is a very important feature (viii).
5. The cubicality of the cube. Simply put, the cube is a very satisfying shape to handle. It is the most basic three-dimensional shape. On a cube it is easy to make specified turns because everything is symmetrical and everything lines up nicely (Rubik viii)
6. The colors of the cube. It has great aesthetic appeal; some other puzzles lose their appeal. Rubik put much thought into the colors of his puzzle. At first, he wished to make opposite sides of the cube complementary colors. Later, he realized that he wanted a white side to “brighten” the effect of the cube. So what he ended up doing was separating colors on opposite sides by a factor of yellow. For example: yellow-white, red-orange, and blue-green (Rubik viii).
7. The mechanism of the cube. This may be the most remarkable aspect of the puzzle. When Ernő Rubik first proposed the idea of the Rubik’s Cube, people laughed at him and said that the puzzle was impossible to physically make. He ended up developing an amazing core mechanism that fit together with each individual piece and allowed the puzzle to exist.
8. The complexity of the cube. For such a simple looking puzzle, the complexity of the cube is remarkable.
9. The mathematics of the cube. That is what this paper focused on. The Rubik’s Cube is a great example of permutation groups and group theory.
19. Therefore, [(M1 * M2) * M3](cubelet) = [M1 * (M2 * M3)](cubelet), and G,* is associative (“Group Theory” 11).
20. Identity. Let e be the “do nothing move.” The “do nothing move” is defined as the move where you do nothing to the cube. So the move M1 * e = e * M1 means to perform the move M1 followed by the “do nothing move,” or do nothing to the cube (vice versa for the other way around). This is obviously the same as performing just the M1 move. Therefore, there is an identity element for all sequences of moves in the cube (“Group Theory” 11).
21.
22.
23. Now we pick up 2 in the first permutation and continue in the above manner. 2 goes to 4, then 4 goes to 3. So 2 goes to 3. In the first permutation 3 goes to 5, and there is no 5 in the second permutation. Then 5 goes to 3, then 3 goes to 4. So 5 goes to 4. Now this cycle is closed with 4 elements in it (“Mathematics of the Rubik’s Cube” 6).
24. Now we write both cycles together to give us the ending permutation of:
25.
26. support(P) ∩ support(M) = Ø. This means that moves P and M affect completely different cubelets (“Mathematics of the Rubik’s Cube” 13).If moves P and M have affected cubelets in common, then the commutator is not the identity. This is when we measure the relative commutativity by applying the commutator and noting the number of affected cubelets. Predictions of relative commutativity can be made by looking at the number of affected cubelets that the two moves have in common. Many useful algorithms attempt to minimize the number of changed cubelets in common (“Mathematics of the Rubik’s Cube” 13). <br />Let’s look at a very practical use of conjugates and commutators. There are many different methods for solving the Rubik’s Cube, the layer method being one of them. The bottom layer is solved first, then the middle layer, then the top layer. After solving the first two layers, the top layer may be disarranged in many different forms. One form may include a linear, horizontal line of the designated top layer color. In other words, some of the top layer edge pieces may be flipped incorrectly. It would look something like this:<br />196278543878500<br />If we want to solve the cube, we would want to correctly flip these edge pieces, while simultaneously leaving the bottom two layers intact. We can now use a commutator and conjugate to solve this. Let’s use the commutator RUR’U’ (“Mathematics of the Rubik’s Cube” 15). After performing this move, seven cubelets are affected and two of them are not in the top layer. A conjugate can be used to fix this problem, making it so that the only cubelets that are affected are in the top layer. So if we perform F before RUR’U’, then perform F’ afterwards, we have the complete conjugate of RUR’U’ by F (“Mathematics of the Rubik’s Cube” 15). This results in the complete algorithm of FRUR’U’F’. Executing an F turn would result in this:<br />184150021399500<br />Then executing the commutator RUR’U’ would result in this:<br />172212017208500<br />Then executing the F’ turn to complete the conjugate would result in this:<br />181038518415000<br />As we can see, the top layer edge cubelets are now correctly flipped. However, this does not mean that all of the edge cubelets are in the correct cubicle. That involves a completely different algorithm. So anytime that an algorithm is found that rotates three pieces, flips pieces, etc., it is possible to apply the support to desired pieces by conjugating the algorithm with the appropriate face turn (“Mathematics of the Rubik’s Cube” 16). <br />Cube Solving<br />Now it’s time to put everything together. We have discussed notations, groups, subgroups, permutations, parity, conjugates, and commutators. All of these things are used in solving the Rubik’s Cube. What fun is a paper on the Rubik’s Cube and Group Theory without a demonstration of a solving technique? <br />There are many different techniques that can be used to solve the Rubik’s Cube. Some are faster than others and require fewer moves. Others may take longer and require more moves, but are easier to execute. The level of difficulty often depends on the complexity of the algorithms used. Every method is just a combination of many different algorithms. Some algorithms may be upwards of 20 or more face turns. Others can be as simple as 3 face turns. <br />Experienced cubers have their own arsenal of algorithms memorized. This is how a cuber is able to solve the Rubik’s Cube so quickly. They glance at the cube’s configuration, recognize the positioning, and apply an appropriate algorithm to arrange specified cubelets. The speed at which a cuber recognizes positioning and applies the algorithm, determines how quickly the cube is restored to its solved state. <br />One method, and probably one of the quickest, is called the Petrus Method. In this technique, a single corner cubelet is focused on. From there, a small 2x2 square is formed around that corner piece. The correctly oriented 2x2 square is extended to a 2x2x3 rectangular shape. This leaves most of the cube solved except for two adjacent faces. These two faces are then rotated to correctly arrange the remaining cubelets on the cube. <br />Another method is sometimes called the Cross Method. This is where an “X” is correctly formed on all six sides of the cube. This is done using simple algorithms. Then, using what is called a “key” cubicle, the remaining edge pieces are inserted into their correct cubicles. Finally, any middle layer cubelets that are flipped incorrectly are corrected. <br />One of the most trivial methods is called the Layer Method. This is the method that will be explained and analyzed in this paper. For this method, first the bottom layer is correctly solved using mainly recognition. Then the middle layer edge pieces are solved for. Then the top layer is correctly solved. As one can see, this method solves from the bottom layer up. <br />The Layer Method<br />The first thing we want to do is pick a side to begin. For simplicity, we will start with green first. Locate the green center facelet and place all four of the green edge facelets around the center. This will create a cross patter on the green side. There aren’t really any algorithms 1857375117157500to do this, it is just simply recognition and a little bit of practice. It will look like this:<br />1971675148844000Search for a green corner cubelet in the bottom layer (green being the top layer). Note the other two colors that are on that same corner cubelet. Twist the bottom layer so that the corner cubelet is between the two faces of the same color:<br />Notice how the green-yellow-red corner cubelet is in the bottom layer and between the red and yellow faces. Now here is when our first algorithm occurs. Rotate the entire cube so that the specified corner cubelet is in the bottom right (we would be looking at the yellow face in this example) and execute the following algorithm:<br />R’D’RD<br />Notice how this algorithm is a commutator with moves R and D. It is a very simple commutator because it is still in the very early stages of solving the cube. If we recall, a commutator can give us insight about the relative commutativity of two moves. Since most of the cube is still disoriented at this point, it does not matter that this commutator has many support pieces in common. This algorithm is executed as many times as it takes until the corner piece is correctly placed and oriented in the top layer. <br />2238375173863000This same procedure is executed for all four green corner cubelets. If there is not a green corner piece in the bottom layer, the above algorithm can be used to remove a green corner piece from the top layer and put it in the bottom. The result will be the entire first layer solved:<br />Now the cube is flipped over for the rest of the solving (green on bottom, blue on top). The middle layer is next. Locate an edge piece in the top layer that does not contain blue. This is because we want to complete the middle layer, which consists of red, yellow, orange, and white. This edge cubelet will contain two colors and one of the facelet colors will be on the top face. Match the other color with the matching center facelet color by twisting the top layer. Depending on the two colors, this edge cubelet will either have to go left, or right. There are two 1866900139065000different algorithms for each separate case:<br />186690062166500 URU’R’U’F’UF<br /> U’F’UFURU’R’ <br />Both of these algorithms should be performed while looking at the red face in this example. Now let’s take a closer look at them. Both algorithms are a pair of two different commutators. It is one commutator followed immediately by another commutator. Notice how these algorithms are definitely more complex than the previous one. That is because we now have an entire green face that cannot be messed up permanently. These commutators allow for the green face to be temporarily messed up but then fully restored at the end of the move. Also notice how both algorithms only utilize the U, F, and R faces. This is because these are the only faces that need to be addressed considering that the edge piece is in the U layer, and it is placed between the F and R faces. <br />2133600134683500This same procedure is applied to all four edge pieces that do not contain blue. Just like the previous algorithm, an edge piece can be taken from the middle layer by simply performing one of the above formulas. This results in the first two layers being successfully solved:<br />Now comes the final and most challenging layer. This layer is obviously the most challenging because it must be solved without messing up the bottom two layers. First, we want to obtain a blue cross on top. If a blue cross already exists, then this step is over. If not, the top layer will be in 1 of 3 configurations. One of the configurations is shown above, with a blue “dot” in the center. The other two are as follows:<br />326707572390004476757747000<br />Twist the top layer so that it is positioned like one of examples above (looking at the red face in this example) and execute:<br />FRUR’U’F’<br />2171700252984000This algorithm was briefly discussed earlier in this paper. If we analyze it once more we can see that it is both a commutator and a conjugate. The algorithm is a conjugate of the commutator RUR’U’ by F. The commutator allows to swap common support cubelets, while the conjugate by F allows to focus the swaps on designated cubelets. This is a great example of how commutators and conjugates are used to swap and flip certain pieces of the cube. Applying this algorithm a certain amount of times will yield this, the blue cross:<br />2181225166687500Now that all of the blue edge cubelets are correctly oriented, we want to place them in their correct cubicles. While looking at red, twist the top layer until the red-blue cubelet matches with the red face. Now check to the right to see if the yellow-blue cubelet matches with the yellow face. If not, then perform the following algorithm:<br />RUR’URUUR’<br />2228850242570000Execute this algorithm until the yellow matches. Then check left to see if the white-blue cubelet matches with white. If not, look at the white face and execute the above algorithm. This algorithm is a conjugate of UR’URUU by R. This is another great example of how a conjugate is used. The conjugate allows for three of the top layer edge cubelets to rotate in a clockwise direction. This is how to successfully place all of the top layer edge cubelets in their correct cubicles:<br />2162175203835000 Finally, we want to correctly place the top four corner cubelets in their designated cubicles and then correctly orient them. The first step is to locate a corner piece that is in the correct cubicle (not necessarily flipped correctly though). Rotate the entire cube so that this piece is in the top right and execute the following algorithm. If there are no corner cubelets in the correct cubicle, then execute this algorithm while looking at any side:<br />URU’L’UR’U’L<br />If we split this algorithm into two parts, it’s easier to understand. The first part is URU’L’. The second part contains an inverse to each face turn in the first part. There’s a U and a U’. There’s an R and an R’. There’s a U’ and a U. There’s an L’ and an L. The second part in total is UR’U’L. This is a very unique algorithm as it rotates three of the top layer corner cubelets in a counterclockwise direction. It must be executed as many times as need until all four top layer corner cubelets are in their correct cubicles:<br />2190750000<br />2190750351726500Lastly, it’s time to correctly orient all four top layer corner pieces. This is very easy. Begin by looking at the red face and noting the corner cubelet in the top right cubicle. If it is correctly oriented, then turn the U layer until a piece that is not correctly oriented slides into that cubicle. Anytime that a corner piece in that exact cubicle is incorrectly flipped, simply execute the very first algorithm (R’D’RD) as many times as need until that specified corner cubelet is in the correct orientation. Follow this pattern (while continuing to look at the red face) until all four corner cubelets are oriented correctly and the cube is restored to its original state:<br />One may ask, “How come the entire cube stays intact when this simple algorithm is performed?” The answer is quite simple: parity. The cube’s parity cannot be altered, so each cubelet has the same respect relative to each other. For example: if you take a solved cube and execute the inverse of R’D’RD three times, while following the pattern above, the cube will be completely solved except for three flipped U layer corner cubelets. When a cube is solved, it will always reach a point that can be reached from a solved cube. This is actually how some algorithms are created; sequences of moves are performed backwards until something recognizable appears.<br />We have not just developed a method for solving the cube, but we have done more. We have analyzed each and every move and attempted to explain why each algorithm works, using group theory applications. <br />The Rubik’s Cube is one of the most amazing puzzles in the world. To any kid, it brings hours of play time. To any adult, it brings hours of confusion. To any mathematician, it brings hours of conversation. Most people do not realize the mathematical nature that the Rubik’s Cube holds. From its symmetry to its group theory applications, it’s bewildering to even the most brilliant of minds. It’s almost unreal how such a simple looking puzzle has so much complexity within it. <br />