Extended Essay Mathematics: Finding the total number of legal permutations of the Rubik s Cube

Transcription

1 Extended Essay Mathematics: Finding the total number of legal permutations of the Rubik s Cube Rafid Hoda Term: May 2010 Session number: Trondheim Katedralskole Word Count: 3635

2 Abstract In this essay we will be finding the total number of legal permutations of the Rubik s Cube. Appropriate notation is introduced first, enabling us to handle the cube mathematically. The cube is then converted to a mathematical cube called the cube group {G, }. Subsequently, we show that the generators of {G, } are: By looking at permutations, we show that every state of the cube is actually a permutation. Permutations of the cube can be written in cycle notation. The permutations in cycle notation can be decomposed into 2-cycles or pair exchanges, which we will show. By looking at the number of decomposed 2-cycles we know if a permutation is even or odd, and we thereby show that the cube group only contains even permutations, thus odd permutations are not achievable on the cube. We will also explicitly consider some illegal cases to help us answer our question. We will then consider twists and flips of the little cubies. We will see that the sum of all edge flips must be 0 or a multiple of 2. Similary the sum of all corner twists must be 0 or a multiple of 3. We will explicilty look at some illegal cases. Finally we will use the illegal cases of the cube to compute the legal permutations of the cube. This is done by: 12!8! = 12!8! = Which leads to the final result that the total number legal permutations of the cube is exactly which can be approximated to Word count: 241 2

3 Acknowledgement I am grateful to my supervisor and teacher Hella Sakshaug, who provided me with constructive feedback throughout the process of this essay. 3

4 Table of contents Abstract 2 Acknowledgement 3 1. Introduction 5 2. Notation 6 3. Groups Turning the cube into a group Testing the axioms Generators Generators of the cube group Permutations Cycles Permutation cycles of the cube Decomposition of cycles Even permutations Illegal cases Flips and twists of edges and corners Flips of edges Twists of corners Finding the total number of permutations Positioning of the corners and edges Flipped edges and twisted corners Calculating the total number of legal permutations of the cube Conclusion 30 Bibliography and resources 31 Appendix 32 Summary of terminology and notation 32 4

5 1. INTRODUCTION Ernö Rubik invented the Rubik s Cube to improve his student s three-dimensional skills. The Rubik s Cube quickly became a worldwide phenomenon and was picked up by people of all ages. 1 A typical Rubik s cube looks as follows: Figure 1: Left and right view of the cube. The cube consists of 6 faces, each colored a unique color. Traditionally, these colors are white, red, blue, orange, yellow and green. We can turn each of the six faces a multiple of 90 in either the clockwise or counterclockwise direction. By turning the faces of the cube, we can mix up the colors. The purpose of the toy is to mix up the colors and then solve it by making each face a solid color again. It is apparent that by turning the sides of the cube we can get many different states of the cube, but exactly how many? We will later see that these states are actually permutations of the cube and so our research question becomes: Finding the total number of legal permutations of the Rubik s Cube. The question is worthy of investigation because we get an idea of how complex a toy like the Rubik s Cube can be. 1 Alexander H. Frey, Jr. and David Singmaster, Handbook of Cubik Math (New Jersey: Enslow Publishers, 1982) page.iv, Preface.

6 2. NOTATION It is crucial to introduce appropriate notation, as it can get confusing otherwise. We will use David Singmaster s notation, described in the Handbook of Cubik Math. 2 Naturally, the cube consists of 6 sides, of faces, where each face has a unique color: Figure 2: Two figures of the whole cube in dice view and how the sides come together to form the cube. The cube is built up of 3 different types of smaller cubes, or cubies: Center cubies: Edge cubies: Corner cubies: Figure 3: Center cubie Figure 4: Edge cubie Figure 5: Corner cubie Edge and corner cubies will sometimes be called edges and corners throughout this essay. There are in total 6 centers, 12 edges and 8 corners. Singmaster has chosen the following six names for the six faces: Front, Back, Right, Left, Up, and Down. 3 2 Singmaster, pages Singmaster, page 10. 6

7 The following table highlights the different faces: Names: Figures: Names: Figures: Front = F Up = U Right = R Down = D Left = L Back = B Table 1 If we turn the up face, 90 in the clockwise direction, that will be a move, denoted by U. To undo the move we just turn the up face 90 in the counterclockwise direction, which will be U -1. Note that clockwise or counterclockwise refers to as we look at that face. In general we have: X is the move to turn the X face 90 in the clockwise direction. We turn the X face 90 in the counterclockwise direction to undo the move, and return to our initial state. This move is denoted by X -1 and is the inverse of X. 4 The result of doing X and then X -1 is the identity denoted by e. The identity will be explained later in detail. We can keep turning any face however many times we like. So; X, X 2, X 3, X 4 are all moves which we can apply to the cube. The exponent acts as; X 2 = XX, X 3 = XXX = X -1 etc. Note, that X 4 would be to turn the X face 360 in the clockwise direction, which just leads us back to the initial state. 4 Singmaster, page 18. 7

8 The following basic facts can be derived: X, X 2 = XX, X 3 = XXX = X -1, X 4 = X 2 X 2 = XXXX = e, X 5 = X, etc We can turn two or more distinct faces to execute a more complex move. The moves are read from left to right. Examples: Move: UF Explanation: Turn the Up face 90 clockwise, then turn the Front face 90 clockwise. UF 2 R Turn the Up face 90 clockwise, turn the Front fact 180 clockwise/counterclockwise, then turn the Right face 90 clockwise D -1 BL 2 Turn the Down face 90 counterclockwise, the Back face 90 clockwise, the Left face 180 clockwise/counterclockwise. Table 2 Below are examples of three moves executed on the cube: U: UR: URF: Figure 6: U Figure 7: UR Figure 8: URF A summary of the notation is included in the appendix. 8

9 3. GROUPS 5 Let us understand what groups are, before turning the cube into a group. Definition: A group is a mathematical structure made up of a set G of elements together with an operation. There are certain rules that a group must satisfy in order to be a group, these are: (1) Closure: For all a, b G a b G. Hence the group G is closed under. Example; = 3 where 1,2 and 3 Z (2) Associativity: The following holds; (a b) c = a (b c). Example: (1 + 2) + 3 = 1 + (2 + 3) (3) Identity: Every group has an identity element e such that; a e = e a = a. Example: = 0 +1= 1 (4) Inverses: Every element a in a group must have an inverse a -1 such that; a a 1 = a 1 a = e. Example 1 +(-1) = (-1) + 1 = 0. 0 is the inverse of itself. The four rules stated are called the group axioms. The examples under the four examples show that the integers under addition form a group. 5 This section is based on: Mark Cartwright. Groups (Hampshire: THE MACMILLIAN PRESS LTD, 1993), pages

10 3.1 Turning the cube into a group Let us now turn the cube into a group. We need to define the elements and the group operation of our group, and check if the axioms hold. Let us denote our cube group as: {G, } where G is the set containing the elements, and is the operation. We will henceforth use {G, } or the cube group when referring to our group. We define the different elements in the set G to be moves. Remember from earlier that a move is just a single or a combination of two or more single turns of faces. So the following moves are examples of elements contained in the set G {R, F, D, B, L, U, R 2, F 2, D 2, R -1, FR, FRU, RUR -1 } are all elements of G Above are just some of the examples. It is immediately apparent that there are a lot of elements in G. Now we need to define our group operation. The natural way to define the operation would be to use it to combine two element(moves): A B = AB where A, B G (A and B are just arbitrary moves.) A B is doing move A followed by move B. And therefore is the same as followed by. Now that we have our set and group operation, let us check if they together form a group 10

11 3.2 Testing the axioms (1) Closure: The closure law is satisfied under {G, }. Suppose A and B are arbitrary moves, then A B = C where A, B G. As A and B are both moves in G then C must also be in G as C is a move composed by A and B. (2) Associativity: Assume A, B and C are arbitrary moves, then: (A B) C = AB C = ABC A (B C) = A BC = ABC ABC = ABC Hence, associativity holds under {G, }. As long as one executes multiple moves in the same order, it does not matter if we combine moves. (3) Identity: The identity will be the move that does nothing. We can call this identity e, then for any arbitrary move A, we have: A e = e A = A (4) Inverses: Every element or move has its inverse such that A A 1 = A -1 A = e. So if we apply a move to the cube and then apply the inverse of the move, we are back to where we started. Let us show that Q -1 is the inverse of Q, where Q = ABC and Q -1 = C -1 B -1 A -1 : Q Q 1 = ABC C 1 B 1 A 1 Q Q 1 = AB e B 1 A 1 Q Q 1 = A e e A 1 Q Q 1 = e Hence Q -1 is the inverse of Q. 11

12 4. GENERATORS A generator of a group is the set of elements that with their inverses can be combined by the group operation(repetitions allowed) to produce all the other group elements 6. Hence we can produce all elements of a group by just one or a few elements. Let us consider the group of all integers under addition, i.e. { Z, + }. What we notice is that by using the numbers 1 and -1(inverse of 1) we can produce all of the numbers of { Z, + }. Examples: 0 = 1 + (-1); 1 = 1; 2 = 1 + 1; 3 = ; etc - 1 = -1; -2= (-1)+( -1) 3 = (-1)+(-1)+(-1); etc (Note that subtraction is not an operation here, the negative numbers are just inverses.) The notation we use to denote that the element 1 generates the group of integers is: { Z, + } = Generators of the cube group The elements that generate the cube group are: This is very easy to see, because the moves; U, F, R, L, B and D are the fundamental moves that when combined can generate all moves. Examples: UF 2 R: UFFR R 3 F 2 U -1 : RRRFFUUU D -1 RL 2 : DDDRLL etc. 6 Edited by David Nelson, The Penguin Dictionary of Mathematics(London: Penguin Books, 2003) pages

13 5. PERMUTATIONS 7 Permutations are very important for this essay, as we will later see that all the elements in { G, } are really permutations. A permutation is a rearrangement of objects 8. When we turn the faces of cube, we are changing the position, or rearranging the edge and corner cubies of the cube. Let us consider a simple case, before we proceed to the cube. If we consider the set {1, 2, 3}, there are 6 possible rearrangement of the objects: 1. (1, 2, 3) 2. (1, 3, 2) 3. (2, 1, 3) 4. (2, 3, 1) 5. (3, 1, 2) 6. (3, 2, 1) And thus there are 6 possible ways to rearrange the objects of {1, 2, 3}. Let us look at one of the permutations from an intuitive point. Consider 3 boxes, numbered 1, 2 and 3 and placed in increasing order from left to right. Each box has an object inside it; box 1 has object 1, box 2 has object 2 and box 3 has object 3. What is in a box at a given time can be called its content. 7 This section is based on: Tom Davis, Permutation Groups, 2. April 2003 < 7. November 2009, pages Davis, page 1. 13

14 Below follows an illustration of the situation: Figure 9: The brown figures represent boxes, while the yellow figures represent objects. The boxes do not move. We can however move the objects around, but there can only be one object in a box at a given time. Now let us perform a permutation that we listed previously, (1, 3, 2): Figure 10: Objects 2 and 3 are swapped. What Figure 12 is showing is that object 2 and 3 have been swapped. 14

15 6. CYCLES The permutation on the previous page can be written as (23) in cycle notation. This indicates that the contents of box 2 moves to box 3, and the contents of box 3 moves to box 2. In other words, it represents a cycle. 1 is not included in (23), because nothing happens to the contents of box 1. Let us consider another permutation listed in cycle notation: (14)(23) Contents of 1 moves to 4, contents of 4 moves to 1. Contents of 2 moves to 3, contents of 3 moves to 2. Observe that since it s a cycle: (14)(23) = (41)(32) The above is correct because the numbers inside the cycle are just shifted, they still have the same order. Of course we are not restricted to having 2 objects inside the parenthesis: (234)(576) (1432)(56) (12)(367) are all examples of permutations. The number of numbers we have within a parenthesis tells us what type of a permutation cycle it is. So for instance: (12) is a 2-cycle, while (243) is a 3-cycle, (1768) is a 4-cycle etc. 15

16 7. PERMUTATION CYCLES OF THE CUBE Let us get specific to the cube. We need to find an effective way to name the movable cubies of the cube, so we can work with them. Fortunately, a neat way of naming the individual cubies is used and described in the Handbook of Cubik Math 9, which we will use. If we take the cube and label every facelet of each side with the first letter of that side in lowercase, this is how the cube will look like: Figure 11 By labeling the cube this way, we can easily refer to the individual cubies. Remember that when we first introduced permutations, we had boxes with objects in them. The boxes acted as containers for the movable objects. We will be using the same principle as with the boxes. 9 Singmaster, pages Figure 12 16

17 What is highlighted in the Figure 14 is the cubicle: urf. We could also call it rfu or fur, just as long as we read the letters clockwise. A cubicle acts as a box and does not move, what moves is the cubie in a cubicle. We do not need to worry about naming the cubies, because they are moving. Below follows a flat view of the Up side, but all the corner and edge pieces are labeled in their full form: We turn the Up side 90 clockwise: Figure 13 Figure 14 17

18 The arrows represent where each edge and corner piece is moved. This is what happens: Edges: uf ul ul ub ub ur ur uf Corners: urf ulf ulf ulb ulb urb urb urf uf ul means that what ever cubie is in uf is moved to ul. We write what happens in cycle notation: (uf, ul, ub, ur)(urf, ulf, ulb, urb) 7.1 Decomposition of cycles 2-cycles are interesting, because we can take any cycle larger than 2 and break it into a sequence of 2-cycles 10. This is fairly easy to show, because a 2-cycle is just a pair exchange. So a 2-cycle or pair exchange is when we swap any two cubies. Consider the cube permutation: (uf, ul, ub, ur)(urf, ulf, ulb, urb) 10 Singmaster, page

19 This is how the pair exchanges(edges and corners) are happening: Edges: Corners: Figure 15 We can see from the figures above that what we actually are doing is swapping two cubies at a time, and hence we can decompose any permutation of the cube into these 2-cycles or pair exchanges: (uf, ul, ub, ur)(urf, ulf, ulb, urb) Can be decomposed into: (uf, ul)(uf, ub)(uf, ur)(urf, ulf)(urf, ulb)(urf, urb) In general: (a, b, c): contents of a move to b, contents of b move to c, contents of c move back to a. (a, b)(a, c): contents of a move to b, contents of b move to a. contents of a(which was originally in b) move to c, contents of c move to a. Hence: (a, b, c) = (a, b)(a, c) 19

20 7.2 Even permutations We say that a permutation is even if it can be decomposed into an even number of 2- cycles(pair exchanges) 11. Conversely we have that an odd permutation is one which can be decomposed into an odd number of 2-cycles. Let us again look at the permutation where we turned the up face 90 clockwise: (uf, ul, ub, ur)(urf, ulf, ulb, urb) We can decompose it into six 2-cycles: (uf, ul)(uf, ub)(uf, ur)(urf, ulf)(urf, ulb)(urf, urb) Since six is an even number, the permutation is an even permutation. Hence, U is an even permutation. If we repeat this process by turning the right, left, front, back and down side, we will get the exact same result, namely even permutations. Hence R, L, F, B, D are even permutations. From page 12, we know that the generators of the cube group are: This means that the cube group exists only of even permutations, because U, F, R, L, B and D all produce even permutations. And all permutations are just combinations of the six fundamental permutations. It is similar to saying that however many even numbers we add up, the sum will always be an even number. This leads to the conclusion that every single permutation of the cube is even. Conversely, it means that odd permutations are not valid permutations on the cube. 11 Davis, page 8. 20

21 This fact will allow us to show that certain states of the cube cannot be achieved. It is important to notice that we have only dealt with the positioning of the cubies. 7.3 Illegal cases We can prove that certain permutations are illegal, meaning not achievable by turning the faces. For now we will only consider pair-exchanges(swapped edges or corners). We will look at twists and flips of corners and edges later. Consider the following state of the cube: Figure 16 We cannot see the whole cube, but the blue-white and red-white edges are swapped. This means that it is a 2-cycle. The permutation will be (uf, ur). Since it is a single 2-cycle, it is an odd permutation, and thus an illegal permutation. We can take another example where we have 3 2-cycles: (uf, ur)(uf, ul)(uf, ub) This is also an illegal permutation as we have 3 pair-exchanges, and thus an odd permutation. We will not go through all cases, because that will take too much space, but this section is important for our final calculation, and we will be using what we have shown here. 21

22 8. Flips and twists We have not yet considered that edges can be flipped and corners can be twisted. A flipped edge cubie will have the faceletes swapped. Unfortunately, we cannot use the same method as we used for the positioning of the cubies. This is because the notation for writing permutations would have to be enhanced to include twists and flips. Instead we will use another method described in the Handbook of Cubic Math 12, where we assign number values to different flips and twists of edges and corners. 8.1 Flips of edges We will denote the original position of the edge as 0(no change), while when it is it flipped it will be +1. This is shown in the diagram below: Figure 17: The edge cubie on the left is place right, while the one on the right is flipped. 12 Singmaster, pages

23 Implemented on the cube, we have: Figure 18 As we can see, the figure to the left is untouched and hence the sum of the flips is 0. The figure to the right however has one edge flipped, and hence the sum of flips is +1. If we flip the flipped edge we get +1+1= 2 and hence we are back to where we started. This means that the second state shown in Figure 19 is not achievable(hence its illegal), because it flips a single edge while leaving everything untouched. This means that the sum of flips being 2 is the same as 0. In fact, any multiple of 2 will give a sum of 0. This means that we are working with modulus 2, ie. multiples of 2. So we have: 0 = 0 (mod 2) = 2 = 0 (mod 2) = 3 = 1 (mod 2) This means that one single edge cubie cannot be flipped, while leaving everything else fixed. Two edges can however be flipped, because = 2 = 0 (mod 2). Three edges cannot be flipped, four edges can be flipped, etc. This fact will be used later on when we calculate the number of total permutations of the cube. 23

24 8.2 Twists of corners We will handle the twists of corners similar to how we handled the flips of edges. A corner cubie can be twisted in three different ways: Figure 19 As illustrated in the figure above; the original state of the corner cubie will be 0, a twist of 120 in the clockwise direction is +1, while a twist 120 in the counterclockwise direction is -1. Notice that here we work with a modulus of 3, since =3 = 0(mod 3). The following is obvious: +1 = 1 (mod 3) +1+1 = 2 = -1 = 2 (mod 3) = 3 = 0 (mod 3) So if we twist three corner cubies 120 in the clockwise direction, that is the same as doing nothing. So as with the edge, we cannot have a single corner twisted, while everything else is fixed, because this gives a sum of +1 or -1. We can however have two corners flipped, where one is +1 and the other is -1, because they cancel each other. We cannot have two corners twisted with +1, because that will give a sum of 2. What we have shown in 8.1 and 8.2 will be used for our final result. 24

25 9. Finding the total number of permutations We can now use everything we know to show exactly how many permutations of the cube are achievable. 9.1 Positioning of corners and edges Let us first look at the positions corner and edge cubies can have. We have 12 cubicles to place the 12 edge cubies in: Figure 20: The edge cubicles are highlighted. The 12 edge cubies can be placed in the 12 cubicles in 12! different ways. The first edge cubie can be placed in 12 cubicles, the second in 11 cubicles, the third in 10, etc. The 11 th edge cubie can be placed in 2 ways. But this is really not true. 25

26 The figure below illustrates where the 11 th edge cubie can be placed: Figure 21 Notice how only one of the positioning will lead to a valid permutation. If the edge piece(in Figure 24) is placed in the red face cubicle, then the last edge cubie must be placed in the last cubicle, which will lead to an illegal permutation, because we will have two edges swapped. Let us address with this problem after we have considered the corner cubies. We have 8 corner cubies: Figure 22: The corner cubicles are highlighted. Similar to the edge cubies, we can place the 8 corner cubies in the 8 corner cubicles in 8! different ways. 26

27 We see that we meet the same problem as with the edge cubies. The 7 th corner cubie can be placed in 2 places: Figure 23 In 1 out of 2 cases the 7 th corner cubie will be placed wrong, which will make the last corner cubie being misplaced, as with the edges. We cannot have a state of the cube where two edges or corners are swapped. Below is a list of the states we have: (1)Two edge cubies swapped. (2)Two corner cubies swapped (3)Two edges swapped and two corners swapped. (4)All edges and corners in their right place. 27

28 Below follows figures for each point: Figure 24 Number 4 is obviously a legal state, as nothing is done. However, number 3 is also a valid permutation, because it consists of two pair exchanges or 2-cycles, i.e. it is an even permutation. So half of the permutations are valid. Hence, the number of permutations(not counting twisted corners and flipped edges) are: 12!8! Flipped edges and twisted corners The edge cubies can be placed in an edge cubicle in 2 ways, either as they are or flipped. Since there are in total 12 edge cubies, there will be: = 2 12 Ways to place the 12 edge cubies in the 12 edge cubicles. However, the 12 th and last edge cubie really has no choice, the 11 previous determine its flip. If the sum of the 11 first flips 28

29 is even, then the 12 flip must also be even. Conversely, if the sum of the 11 first flips is odd, then the 12 flip must also be odd. This is because the final sum of all the flips can only be an even number for the state to be considered legal. This is how the 12 th edge cubie loses its choice, and hence the number of ways the 12 edges can be flipped is: Similarly, the 8 corners can be twisted in 3 8 ways. Again the 8 th edge cubie s twist is determined by the previous 7 corner twists. If the sum of the first 7 twists is not a multiple of 3, the 8 th corner(with +1, -1 or 0) must make the final sum a multiple of 3. This is because the sum must always be a multiple of 3. The 8 th corner loses its choice and hence the number of ways the 8 corners can be flipped is: And that is it! That covers everything. 9.3 Calculating the total number of legal permutations of the cube Since we are calculating the total number of legal permutations of the cube we need to multiply the results in 9.1 and 9.2: 12!8! = 12!8! = So the total number of legal permutations of the Rubik s cube is , which can be approximated to

30 10. Conclusion In this essay we wanted to find the total number of legal permutations of the Rubik s Cube. To do this, we first converted the cube into a group { G, }, which had the generators: We then looked at cube permutations. Since we could decompose any cycle into 2-cycles, we showed that only even permutations are valid with the cube. With this we showed some illegal cases of positioning cubies. Since we could not use the same method as positioning for considering flips and twists of the cubies, we introduced another method. We found that legal flips of edges had to sum up to 0, or a multiple of 2. Twists of corners had to sum up to 0, or a multiple of 3. We looked at general illegal cases. We finally used everything we had shown to show how many legal permutations we could get of the cube. The exact value was calculated to be: , which is approximately As the result of this essay demonstrates, the number of legal permutations of the Rubik s Cube is a relatively large number, which in turn shows the complexity of the toy. 30

31 Bibliography and recourses Books: Alexander H. Frey, Jr. and David Singmaster, Handbook of Cubik Math (New Jersey: Enslow Publishers, 1982) Mark Cartwright, Groups (Hampshire: THE MACMILLIAN PRESS LTD, 1993) Edited by David Nelson, The Penguin Dictionary of Mathematics (London: The Penguin Group, Third Edition, 2003) Internet: Tom Davis, Permutation Groups, 2. April 2003, 7. November 2009, < Applications used: Microsoft Word 2008 Equation Editor Macromedia Flash 8 (Used to make illustrations) Big number scientific calculator (Used for final calculation)

32 Appendix The following table is taken directly from page 4 from Handbook of Cubik Math: SUMMARY OF TERMINOLOGY AND NOTATION: Terminology Definition or Abbreviation Cubies The small cube pieces which make up the whole cube. Cubicles The spaces occupied by cubies Facelets The faces of a cubie Types of Cubies: Corner, Edge, and Center A corner cubie has three facelets. An edge cubie has two facelets. A center cubie has one facelet. Positional Names for Cube Faces Up Down Right Left Front Back Table(A) 1 32