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# Sean Gallagher - Sr. Seminar Paper 40 Pages

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Senior Paper about the Rubik\'s Cube and It\'s Group Theory Applications.

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### Sean Gallagher - Sr. Seminar Paper 40 Pages

1. 1. Sean Gallagher<br />Professor Ray Mest<br />Sr. Seminar, MAT 492<br />May 2011<br />The Rubik’s Cube and its Group Theory Applications<br />When the Rubik’s Cube was invented in 1974, it took the world by surprise. When the idea was first thought up by Ernő Rubik, he may not have known how big the puzzle was eventually going to be. The Rubik’s Cube has sold over 350 million copies, making it the best selling puzzle of all time; it is considered by many to be the best selling toy in history. A few years after its release in the market, nearly every child in every household owned the puzzle. It has even made its way into the Oxford English Dictionary. <br />Introduction<br />The Rubik’s Cube is unlike any other puzzle. It has many distinct and unique features that allow it to stand out among many other puzzles. A few key features that Ernő Rubik pointed out himself are:<br /><ul><li>The cube (all of its pieces and all of its parts) stays together when being solved. Many other puzzles that require moving parts have separating pieces.
2. 2. Several pieces move at once, in contrast to other puzzles that may only move one piece at a time.
3. 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. 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. 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. 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. 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. 8. The complexity of the cube. For such a simple looking puzzle, the complexity of the cube is remarkable.
9. 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.
11. 11. (a * b) * c = a * (b * c). Associativity of *
12. 12. There is an element e in G such that for all x ∈ G,
13. 13. e * x = x * e = x. Identity element e for *
14. 14. Corresponding to each a ∈ G, there is an element a’ in G such that
15. 15. a * a’ = a’ * a = e. Inverse a’ of a (Fraleigh 37-38)</li></ul>Note: A binary operation * is a mapping from S x S into S (Fraleigh 20). (ex. +, /, …) <br />Now let’s prove that any sequence of moves performed on the Rubik’s Cube is a group.<br />Proof: To show this, we must show it satisfies the definition of a group as well as all three axioms. If our group is G,* on the Rubik’s Cube, then G and * must first be defined. The set G on the Rubik’s Cube is the set of all possible moves or sequence of moves. The binary operation * is defined as M1 * M2 where M1 is a move performed, followed by the execution of M2, where M2 is another move (“Group Theory” 11). It also must be shown that the set of moves is closed under the binary operation. This is quite easy to see. If M1 is a move, and M2 is also a move, then obviously M1 * M2 is a move. Now that the set and binary operation are defined, let’s prove the three axioms:<br /><ul><li>Associativity. It is important to note that a move can be defined by the change in the configuration of the cube that it causes. So if we perform a move M1 on a cubelet, the move puts the cubelet into another position M1(cubelet), that is the end result of the move performed on the given cubelet. If a move M2 is then performed after M1, it puts the cubelet into the position M2(M1(cubelet)), which equals (M1 * M2)(cubelet) (“Group Theory” 11).
16. 16. For associativity, we want to show that [(M1 * M2) * M3](cubelet) = [M1 * (M2 * M3)](cubelet).
17. 17. From what we proved above, [(M1 * M2) * M3](cubelet) = M3[(M1 * M2)(cubelet)] = M3[M2(M1(cubelet))].
18. 18. Similarly, [M1 * (M2 * M3)](cubelet) = (M2 * M3)[ M1(cubelet)] = M3[M2(M1(cubelet))].
19. 19. Therefore, [(M1 * M2) * M3](cubelet) = [M1 * (M2 * M3)](cubelet), and G,* is associative (“Group Theory” 11).
20. 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. 21. Inverse. For any sequence of moves M1 you perform, you can exactly reverse the moves. Let’s call this reverse of moves M1’. So for any given configuration state that the cube is in, if you perform M1 and then perform M1’, you are right back to the configuration state where you started. This would be the same as performing the “do nothing move.” So we can see that M1 * M1’ = e. Therefore, there is an inverse for all sequences of moves in the cube (“Group Theory” 11). </li></ul>Therefore, any sequence of moves performed on the Rubik’s Cube, G,*, is a group. ∎<br />Definition: The order of an element g in a subgroup is defined as m, such that gᵐ = e, the identity (“Mathematics of the Rubik’s Cube” 9). <br />The order of an element can also determine the size of the subgroup that element generates. We can use these ideas of order and generators to help us understand move sequences of the Rubik’s Cube. For example: what is the order of a certain move? In other words, how many times must that move be executed in order for the cube to be returned to its original state, or the identity? Notice how if you perform the move FF twice to a solved cube, the cube returns back to its solved form. FF becomes a generator of order 2. <br />Subgroups<br />Now that we know that any sequence of moves is a group, let’s further discuss some subgroups of the cube. Since any sequence of moves can be of any length, the number of different sequences is limitless. This is why discussing specific and limited subgroups is a practical thought. <br />The F Subgroup<br />This subgroup is very simple and consists of all the possible configurations of the cube that can be obtained from twisting only the F face. This subgroup contains only four elements: the identity, the result from performing the move F, the result from FF, the result from F’. Performing the move F four times brings the cube back to its original state (“Cube Groups”). <br />F = e, F, FF, F'<br />1104900152019000This is an example of a Cayley graph for the subgroup generated by F. A Cayley graph allows us to gain insight into the structure of a subgroup (“Mathematics of the Rubik’s Cube” 12). Each g ϵ G, the subgroup, is a vertex:<br />Suppose that we wanted to draw the Cayley graph for the subgroup generated by U. It would look exactly the same as the Cayley graph for F. If two groups have the same Cayley graph, they essentially have the same structure and are called isomorphic (“Mathematics of the Rubik’s Cube” 13). In the Rubik’s Cube, two isomorphic groups will have the same order and same effect on the cube. For example, executing the algorithm FFRR is the same as turning the entire cube to the right (so now you’re looking at the previous L face) and executing RRBB. <br />The Slice Squared Subgroup<br />This subgroup consists of all the possible configurations of the cube that can be obtained from the following: RRLL, UUDD, FFBB. The subgroup gets its name from the fact that performing either one of these moves is equivalent to twisting a center “slice” 180°. RRLL can also be written as R2L2, hence the “squared” part. Let’s let X = R2L2, Y = U2D2, and Z = F2B2. Notice that this subgroup is abelian. Performing XY gives the exact same result as performing YX. Also notice that performing XYZ results in a “checkerboard” pattern for the cube. Since the slice squared subgroup is abelian and each element is its own inverse, it can be represented by the Cayley Table below:<br /> EXYZXYXZYZXYZEEXYZXYXZYZXYZXXEXYXZYZXYZYZYYXYEYZXXYZZXZZZXZYZEXYZXYXYXYXYYXXYZEYZXZZXZXZZXYZXYZEXYYYZYZXYZZYXZXYEXXYZXYZYZXZXYZYXE<br />Cayley Table of the SLICE SQUARED Subgroup<br />The Slice Subgroup<br />This subgroup is larger than any of the subgroups mentioned so far, but it is easy enough to understand without too much difficulty. It consists of all the moves: RL’, UD’, and FB’. It is called the “Slice Subgroup” because doing any of the above moves has the exact same result as simply moving a middle “slice” (“Cube Groups”). This subgroup contains much symmetry on each side, no matter what slice moves are performed. On each face of the cube, all four corner piece facelets will always be the same color. Also, opposing edge piece facelets will be the same color. The pattern would look similar to the following:<br />19145254000500<br />2171700102870000One unique pattern that can be obtained from the slice subgroup moves LR’BF’UD’LR’ is often called “dots.” It is called this because every face of the cube has a different color center facelet like so:<br />The Slice Subgroup contains the Slice Squared group as a subgroup. As shown earlier in the table, the Slice Squared subgroup has order 8. There is a theorem called Lagrange’s Theorem. Let’s define Lagrange’s Theorem:<br />Let H be a subgroup of a finite group G. Then the order of H is a divisor of the order of G (Fraleigh 100).<br />Therefore, the order of the Slice Squared subgroup (8) must divide the order of the Slice subgroup evenly. Suppose someone claimed that the order of the Slice subgroup was 108. This person would be incorrect because 8 does not divide evenly into 108. <br />The (F²R²) Subgroup<br />This is a cyclic subgroup that consists of the moves FFRR repeatedly. If we perform FFRR 6 times in a row, we will have the cube back in its starting configuration. This shows that the subgroup is cyclic with order 6 and is generated by <FFRR>. The subgroup is also abelian of order 6. <br />This subgroup has many practical uses and processes, many of which can be used when physically solving the cube. Performing FFRR 3 times swaps exactly four cubelets: the uf, df, ur, and dr cubelets. So when solving the cube, these cubelets can easily be swapped by twisting only two faces. Let’s define a few terms before proceeding.<br />Definition: Let H be a subgroup of a group G. The subset aH = {ah | h ϵ H} of G is the left coset of H containing a, while Ha is the right coset of H containing a (Fraleigh 97). <br />Definition: gxg⁻¹ is defined as a conjugation of x by g (Fraleigh 141).<br />We are able to create cosets of the (F²R²) subgroup now. If we perform any move before performing FFRR, then the configuration that remains is not an element from the (F²R²) subgroup. Rather, it is an element from one of its left cosets. If we perform a move after performing FFRR, then the configuration that remains is also not an element from the subgroup. Similarly, it is an element from one of its right cosets. These ideas can be elaborated on and can create very important and useful techniques for cubing.<br />We have defined that multiplying a subgroup on the left by an element creates a left coset. Similarly, multiplying a subgroup on the right by an element creates a right coset. Also, multiplying something on the left and right by an element and its inverse, respectively, creates a conjugate. Let’s look at some properties of conjugates:<br />(a * b * a⁻¹) * (a * c * a⁻¹) = a * b * (a⁻¹ * a) * c * a⁻¹ = a * (b * c) * a⁻¹<br />This property shows that conjugates by a are closed under the group operation (“Cube Groups”).<br />(a * e * a⁻¹) = e<br />This property shows that the identity e is a conjugate by a (“Cube Groups”).<br />(a * b * a⁻¹)⁻¹ = (a * b⁻¹ * a⁻¹)<br />This property shows that the inverse of a conjugate by a is also a conjugate by a (“Cube Groups”). Along with associativity, this shows that conjugates by a form a subgroup. <br />Here’s where the usefulness comes in. As stated earlier, performing FFRR three times swaps two pairs of opposing edge pieces. Performing other moves three times that are similar to FFRR will also swap opposing pairs of edge pieces. Now let’s apply this idea of conjugates to the (F²R²) subgroup. Suppose that we do a D move, then (FFRR)³, then D’. This could also be written as D(FFRR)³D⁻¹, which is exactly what a conjugate form looks like. It would be stated as “a conjugate of (F²R²)³ by D.” When this sequence of moves is performed, it still swaps four edge pieces, but they are not two pairs of opposing ones. The move swaps four completely different and more random edge pieces. A conjugate is used to create a useful process from an already existing sequence of moves (“Cube Groups”). This is a very common practice for speed cubers and cubologists. Using conjugates can help create an organized plan when solving the Rubik’s Cube.<br />The (FRBL) Subgroup<br />This is a cyclic subgroup of repetitions of the process FRBL (“Cube Groups”). Since this group is cyclic, performing FRBL over and over again will eventually bring the cube back to its original configuration. Let’s discuss the order of this subgroup.<br />If we pick up the cube and start performing the move FRBL repetitiously, we see that it rotates and switches the orientation of corner and edge cubelets. Through experimentation, we notice that after 5 full repetitions of FRBL, the top four corner cubelets are restored to their original cubicle and configuration. 5 repetitions is equivalent to 20 total moves. We can also note that after 3 repetitions of FRBL, the four top layer edge cubelets become restored. This is equivalent to 12 total moves. So after 5, 10, 15 repetitions and after 3, 6, 9, 12, 15 repetitions, the corner cubelets and edge cubelets will be restored, respectively. Therefore, after 15 total repetitions, the entire top layer will be completely restored. In other words, 15 is the least common multiple of 5 and 3 (“Cube Groups”). 15 repetitions is equivalent to 60 moves.<br />We will continue to analyze the order of the (FRBL) subgroup with a layer by layer analysis. Let’s move on to the middle layer. Through experimentation, we notice that after 5 repetitions of FRBL, only one middle edge cubelet (the lf cubelet) returns to its original cubicle and orientation. The other three middle edge cubelets are restored after 7 repetitions. These are equivalent to 20 and 28 total moves, respectively. The least common multiple of 5 and 7 is 35. Therefore, after 35 repetitions (140 total moves) the entire middle layer will be restored (“Cube Groups”). <br />Finally, we move to the bottom layer. It is found that after 7 repetitions (28 total moves) the four bottom edge cubelets become restored. We can also notice that after 9 repetitions (36 total moves) the four bottom layer corner cubelets return to their original positioning (“Cube Groups”). The least common multiple of 7 and 9 is 63. So after 63 repetitions, or 252 total moves, the bottom layer will be fully restored. Since the six center facelets never change position, we do not have to bother with those.<br />Therefore, to find the total number of repetitions necessary to restore the entire cube, we must analyze the required repetitions of each layer together. We found that it takes 15 repetitions, 35 repetitions, and 63 repetitions to restore the top, bottom, and middle layer respectively. The least common multiple of 15, 35, and 63 is 315. Therefore, it will take 315 repetitions of FRBL (or 1260 total moves) to fully restore the cube back to its original configuration (“Cube Groups”). This means that the (FRBL) subgroup has order 315. <br />Permutations<br />When performing move sequences on the Rubik’s Cube, different cubelets are rearranged. These rearrangements can also be viewed as permutations of the cubelets. Thus, every move sequence can be written as a permutation (“Mathematics of the Rubik’s Cube” 6). Every algorithm that is used when solving the Rubik’s Cube is designed to rotate the configuration or flip the orientation of specified cubelets. For example, a “corner rotation” algorithm (a very common and easy one) rotates three top layer corner cubelets in a triangular pattern, leaving the fourth corner cubelet alone. Let’s look at a simple permutation like this one using numbers. We will write it in what is called cycle notation. <br />(1)(234)<br />This cycle notation generally represents the rotation of the corner pieces that is stated above. This type of notation allows us to “read” the permutation that is occurring using cycles. Here’s how to read this:<br /><ul><li>Since the 1 is by itself, the 1 stays in place and is not rearranged. The 1 in this case represents the corner cubelet that is not affected.
22. 22. The 2, 3, and 4 are cycled. It is read as “the 2 goes to 3, the 3 goes to 4, the 4 goes to 2.” Once we get back to 2, the cycle is closed and starts over again. The 2, 3, and 4 in this case represent the three corner cubelets that are being rotated, or cycled. </li></ul>Let’s analyze a situation where two sequences of moves are performed back to back. For simplicity, we will designate numbers to cubelets. Suppose you perform a move that results in the following permutation: (124)(35). Suppose that immediately after that sequence you perform a move that results in this permutation: (612)(34). Earlier in this paper, we explained that our binary operation, *, represents the execution of one move followed by the execution of another move. This would be written as (124)(35)*(612)(34). Using this, we can actually develop a cyclic notation that represents the end result after performing these two sequences back to back. Here’s how to develop this:<br /><ul><li>We start with 1 in the first permutation. 1 goes to 2. Now we move on the other side of the binary operation and pick it up from 2. We see that 2 goes to 6 in this permutation. We now see that 1 goes to 6. Continuing with 6, we see that 6 goes back to 1. Therefore, the cycle is closed and 1 and 6 form a single 2-cycle (“Mathematics of the Rubik’s Cube” 6). This looks like: (16)
23. 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. 24. Now we write both cycles together to give us the ending permutation of: