Comments on the Rubik s Cube Puzzle

Transcription

1 Comments on the Rubik s Cube Puzzle Dan Frohardt February 15, 2013 Introduction The Rubik s Cube is a mechanical puzzle that became a very popular toy in the early 1980s, acquiring fad status. Unlike some other fads, however, the Rubik s Cube has sufficient intrinsic interest to insure that it will continue to have a following among the mathematically-oriented segment of the population. Rubik s cube is named after its inventor, a Hungarian architect, who apparently found the design as a solution to the structural problem of constructing a device that could be twisted without falling apart. I first encountered the puzzle at a group theory conference in California in In those pre-internet days the cube was not available in the United States, but several European attendees had brought along magic cubes and they were popular toys during the four week long meeting. One of the British attendees agreed to provide cubes to us interested Americans once he returned home. We paid him $10 per cube, which just covered his expenses. The two cubes I bought from him were among my most expensive. I acquired most of the cubes in my collection for 29 cents apiece during a close-out sale in the mid-80s, after the fad had crested. 1 Initial observations There are nine small squares on each face of the cube. In its original, pristine state, the nine squares on each of the faces all have the same color. It is easy to change that by twisting one of the faces. After a few such random twists, the cube will be sufficiently randomized that it is a very difficult puzzle to return it to its original state. I heard of an anecdote during the early 1980s, during the late stages of the Cold War. When visiting Russia, a traveler s suitcase was detained for several days before being returned. The only observed change in the contents was the condition of the Rubik s cube. Examining the puzzle, we note that there are 3 types of squares: centers, edges, and corners. Each side has a single center, four edge squares, and four corner squares. Each edge square is permanently attached to another edge square, which is necessarily of a different color. Each corner square is part of a block that shows 1

2 two other corner squares, the three squares having three different colors. The orientation of the three colors on a corner piece is fixed. To return a cube to its original condition, we have to arrange for each of the 8 corner pieces and each of the 12 edge pieces so that all of the colors match the color of the center square on the appropriate side. It should not be difficult to see that the relative positions of the six center squares remain fixed. No matter how we twist the pieces, we can always move the cube as a unit so that the top face is white and the face closest to us is yellow, at least when the cube starts with a standard coloring in which the white face and yellow face are adjacent. For that reason, I will ignore the centers for the most if not all of the rest of this talk. What distinguishes the Rubik s Cube puzzle from many of its predecessors is its complexity. It is a reasonable and non-trivial exercise to fix one face of the cube. To solve the cube, however, it is necessary to fix all of the faces at once. The straightforward approach is to fix the cube bit by bit, but this requires maneuvers that leave intact the parts that are in place while moving around those that aren t. This is very difficult to do without a systematic approach. The natural mathematical tool for attacking the cube systematically is group theory, and I will indicate where some of the basic concepts of group theory come into play as we solve the cube. 2 Notation and Terminology David Singmaster introduced useful standard notation in his 1981 book, Notes on Rubik s magic cube. Let R, L, F, B, U, and D stand for Right, Left, Front, Back, Up (top), and Down (bottom). The cube has 6 faces, each of which can be turned. We use the capital letters above to denote rotations by 90 clockwise (when viewed looking at the face). Let R, L, F, B, U, and D stand for roatations counterclockwise, and R 2, L 2, and so forth, stand for 180 rotations. Other basic turns are Beckon, denoted K which rotates the slice between R and L 90 so that the top comes to the front, Away, the inverse of K, which is denoted K, and Slice, denoted S which rotates the slice between U and D by 90, moving the front to the right. 3 The Outline of a solution I have found it convenient to think of my basic solution as consisting of the following steps, starting from Step 0: a random cube. 1. Bottom [Blue] edges completed located and oriented completely 2. Step 1 + Bottom corners completed 3. Step 2 + Top corners in correct locations 4. Step 3 + Top corners completed 2

3 5. Step 4 + Middle layer completed 6. Step 5 + Top edges located 7. Step 6 + Top edges oriented and thus completed I will discuss the individual steps later, as time allows. 4 How many arrangements are possible? Imagine removing the edge and corner pieces and reassembling the cube. Ignoring the orientations, we could place the 8 corners in 8! = distinct ways and the 12 edges in 12! = distinct ways. However, there would be 3 ways to orient each of the corners and 2 ways to orient each of the edges. So the total number of ways to reassemble the cube would be 8! 12! This is an upper bound for the number of arrangements possible. It turns out that we can only achieve only 1/12 of these through legitimate twists. Each basic move produces no net flipping of the edges and no net rotation of the corners. This accounts for a factor of 1/6. The remaining factor of 1/2 comes from the fact that the basic moves all induce even permutations of the set of 20 edges and corners. The previous sentence should make sense if and only if you have had a course in group theory. So the total number of distinct arrangements that can be reached using legal moves is 8! 12! /12. This is a very large number, approximately 100 times the age of the universe in seconds. So random fiddling around is hopeless. Only a systematic approach can work. We will not be concerned here with finding the shortest possible solution. Our goal is to find a reasonably workable way to straighten out the cube. We will sacrifice some efficiency to simplify the conceptual scheme. 5 Philosophy The subgroup lattice The set of all possible combinations of basic twists has a natural algebraic structure. We can agree that any combination that transforms a standard cube to a standard cube for example, rotating one of the faces through four successive quarter turns should be treated as not having done anything at all, then there is a multiplication defined on the set of all moves that satisfies the following basic properties. 1. The operation is associative. 2. There is an identity element. 3. Every element has an inverse. 3

4 This makes the set G of moves a group under this operation, and there is a vast theory that applies. We consider the following subsets of G. E 1 E 2 all moves that return all of the corners to their original locations all moves that return all of the edges to their original locations all moves that return all of the corners to their original locations with the original orientations all moves that return all of the edges to their original locations with the original orientations Each of these sets is closed under the operation which means that they are subgroups of G. If you know some group theory you should be interested to know that these are in fact normal subgroups. Without getting technical, this means that they are nice to work with, and I will indicate why this is so later in the talk. Conceptually, as we proceed to solve the cube, the moves required to complete the task will lie in smaller and smaller subgroups, so we move down the following diagram. G E 2 = 1 E 1 E 1 E 2 E 1 E 2 4

5 The trick is to find enough combinations lying within each of the smaller pieces to accomplish the task. The nicest combinations are those that do something, but very little. 6 Finding nice moves Idea 1 Repeat what you just did. Despite the cliché about the definition of insanity, repeating the same thing over and over can actually be part of an effective strategy. By a basic theorem in finite group theory, if you scramble the cube with a series of twists you can unscramble it by repeating that precise series of twists enough times. 1 Idea 2 Take advantage of the failure of the commutative law. The associative law is a more fundamental algebraic principle than the commutative law because function composition is automatically associative, but not necessarily commutative. More subtly, while ab need not equal ba, it may be the case that ab is fairly close to ba. This means that the product aba 1 b 1 = (ab) (ba) 1 may be close to the identity 2. That s exactly the sort of combination we want! Idea 3 Exploit the normality of the subgroups C i and E j. Each of the sets,, E 1, E 2 is the kernel of a homomorphism defined on the group of all combinations of twists. As you should learn in your first course in abstract algebra, this means that they are normal subgroups so that you will automatically stay in the appropriate subgroup if you take something there, pre-multiply it by an arbitrary element x and then post-multiply it by the inverse of x. 3 Using these ideas as a guide, experimenting with combinations yields some convenient moves. Notes: g 1 and g 2 have no effect on the middle and bottom layers. They can be used to set the locations of the four top corners in Step 3. g 3 only twists two of the corners, leaving all of the small cubes in their original locations. It and minor variations make Step 4 one of the easiest steps. g 4 and g 5 leave all corners alone and simply move three of the edges. g 4 moves the three edges next to a given corner while g 5 moves three of the four edges on a given face. I find it easier to remember and execute g 5. Steps 5 and 6 are actually fairly similar. g 6 and variations are convenient for Step 7. 1 Unfortunately, the number of repeats required may be much larger than the number of times that you can reliably reproduce the exact series of twists. 2 In technical terms, an expression of the form aba 1 b 1 is a commutator. 3 Getting technical again, an element of the form xgx 1 is called a conjugate of g. A subgroup is normal exactly when it contains all of the conjugates of all of its elements. 5

6 Table 1: Key words name description effect g 1 F RU R URUR F U 2 (ulf) (bul.ubr) + (uf.lu) + (ub) + g 2 F RU R U 2 RUR F (urf.flu)(ubr.ulb) (uf.lu)(ub.ur) g 3 (RF R F ) 2 U (F RF R ) 2 U (urf) + (ulf) g 4 RBLF UF L B R U (ur.rf.uf) g 5 KUK U 2 KUK (ur.fu.ul) g 6 (SR) 4 U (SR) 4 U (ur) + (uf) + Because all of the moves g 3, g 4, g 5, and g 6 are very clean and nice, once one has completed Step 3 it is possible to fix complete the remaining steps by doing the edges first. Comments on the forms of g 1, g 2, g 3, g 4, g 5, and g 6 : 1. g 1 = aba 1 b 2 and g 2 = ab 2 a 1 where a = F RU R and b = U 2. g 3 = aba 1 b 1 where a = (cdc 1 d 1 ) 2, b = R, c = R, d = F. 3. g 4 = aba 1 b 1 where a = RBLF and b = U. 4. g 5 = aba where a = KUK and b = U 2. The more natural KbK 1 b 1 has order 3, but the three edges do not lie on the same face or next to the same corner. The other natural choices, aba 1 b 1, aua 1 U 1, and KUK 1 U 1 have order 5 and thus don t work as well. 5. g 6 = aba 1 b 1 where a = (SR) 4 and b = U. 6

7 Schematic diagram of solution G g 1, g 2 g 3 g 4, g 5 E 1 g 6 1 returns all corners to their original locations leaves all corners with their original orientations in their original locations E 1 leaves all corners with their original orientations in their original locations and also returns all edges to their original locations 7