2. Basis & Structure Surface – 6 of them, 6 colors Center Cube – 6 of them (fixed) Edge Cube –12 of them Corner Cube –8 of them Center Edge Corner

3. Graph for Edges Bijection A new game C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C3 C4 C1 C2 b B2 B3 B4A3 A2

4. Operation C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 B3 C2 b C1 A1 C3 B1 B2 A3 A2 B4 A4 C4 B3 C2 b C1 A1 C3 B1 B2 A3 A2 B4 A4 C4 B3 C2 b C1A1 C3 B1 B2 A3 A2 B4 A4

5. Graph for Corners Bijection Corners numbered in order Another new game 8 5 7 (inside) 2 14 3 6 1 7 8 6 5 2 4 3

6. Operation 1 7 8 6 5 2 4 3 4 7 8 6 5 1 3 2 3 7 4 6 8 1 5 2 3 1 4 7 8 2 5 6

7. Edge State in right position, can be error A state parameter needed Consider certain color & its opposite color –B VS G –W VS Y –O VS R R B Y W G O

8. Case 1 Both with same or opposite color e.g. AA BB or A’A BB’ or A’A BB –PS: A completed Rubik cube’s edges all have a parameter 0 Red 0 A A B B 0 A’ A B B Red 0 A’ A B B’

9. Case 2 Both not with same or opposite color e.g. AB AB or A’B AB’ or A’B AB Red 1 A B A B 1 A’ B A B Red 1 A’ B A B’

10. Case 3 & 4 Irrelevant color involved e.g. CA BB  state 1 (Case 3) e.g. CB AB  state 2 (Case 4) Red 1 C / C’ A B B Red 0 C / C’ B A B

11. Corner State Top/Bottom face  state 0 (when completed, final state) Clockwise face  state 1 (when completed, final state) The face left  state 2 (when completed, final state) State parameter = current state face’s parameter (current state) Top State Surface Bottom State Surface (inside) State face 0 1 2

12. Operation (Considering state) Case 1: Vertical State unchanged 1 7 8 6 5 2 4 3

13. Case 2: Horizontal State changing as left graph –Simply +1 or -1 1 7 8 6 5 2 4 3 + + - - + - - +

14. Case 3: w.r.t. axis State changing as left graph –Simply +1 or -1 1 7 8 6 5 2 4 3 - - + + - + + -

15. 1st Layer Observation + Operation Locus Method Avoidance Method

16. Locus Method Possible position after one operation b, c, d, e, f and a: relative locus c b a e f d

17. Steps & Example Target & destination square  relative locus Destination  Public locus Target replaces the destination one If needed, target  somewhere irrelevant to the Destination before any operation 1 2 3 Public Locus

18. Avoidance Method Sometimes, some other squares’ position may be affect when moving the target square to its destination position Like the last step in Locus Method But this time, we need to deal with 2 blocks

19. Steps & Example target block  somewhere irrelevant to the Destination before any operation 1 2 3

20. 2nd Layer Try & error  some method some method  derived method

21. Method A C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 B3 C2 b C1 A1 C3 B1 B2 A3 A2 B4 A4 C4 B3 C2 b C1 A1 C3 B1 B2 A3 A2 B4 A4

22. C4 B3 C2 b C1 A1 C3 B1 B2 A3 A2 B4 A4 C4 B3 C2 C1 A1 C3 B1 B2 A3 A2 B4 A4 C4 B3 C2 b C1 A1 C3 B1 B2 A3 A2 B4 A4

23. C4 B3 C2 b C1 A1 C3 B1 B2 A3 A2 B4 A4 C4 B3 C2 b C1 A1 C3 B1 B2 A3 A2 B4 A4

24. State changes StepB1B2B4C1A2A3 1++ 2+ 3++ 4+ 5++ 6+ 7+++ Total000000

25. Method B C1 A1 B1 B2 A3 A2 B4 A4 change all to and all to change the 5th and 7th steps from right- switching to left-switching method A  method B Always check the states

26. 3rd Layer Last layer More cubes’ position & state cannot be changed

27. Method C C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4

28. C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4

29. State changes StepA1A2A3A4B1B2B3B4C1C2C3C4 1++++ 2++++ 3++++ 4++++ 5++++ 6++++ Total000000000101

30. Observation No change in both position and state of the bottom and 2nd layers. But an obvious change in the top layer. Let C1, 2, 3 and 4 be A, B, C and D, --- state parameter followed D-0 A-0 B-0 C-0 D-0 C-1 A-1 B-0

31. Method D C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4

32. C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4

33. C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4

34. State changes StepA1A2A3A4B1B2B3B4C1C2C3C4 1++++ 2++++ 3++++ 4++++ 5++++ 6++++ 7++ 8++++ Total000000000110

35. Observation No change in both position and state of the bottom and 2nd layers. But obvious change in the top layer Let C1, 2, 3 and 4 be A B, C and D, --- state parameter followed D-0 A-0 B-0 C-0 D-0 A-0 C-1 B-1

36. Method E Derived by method C & D D-0 A-0 B-0 C-0 D-0 C-1 A-1 B-0 D-0 A-0 C-0 B-0 D-0 A-0 B-0 C-0 D-0 A-0 C-0 B-0

37. Demonstration Arbitrary start Deal with the top layer Put C1 to the correct position but not consider its state parameter C1 C4 C3 C2 C1 Ck Cq Cp Unknown state parameter All state parameters are 0

38. Demonstration (cont.) Case 1: C1 = 0  completed  Jump to next step Case 2: C1 = 1  use method D C1= 1 Ck Cq Cp C1= 0 Ck Cq CpC1= 0 Ck Cq Cp

39. Demonstration (cont.) A completed Rubik Cube’s every edge cube’s state parameter is 0 After every operation, the total parameter changes is +4 i.e. the total parameter must be an even number. A1~4, B1~4,C1 all equal to 0 C2, C3 and C4: can’t be one ‘1’ or three ‘1’ among them

40. Demonstration (cont.) Case 1: no ‘1’ among C2, C3 and C4 –i.e. C1=0, C2=0, C3=0, C4=0  use method E Case 2: 2 ‘1’ among C2, C3, C4. –If these 2 ‘1’ are adjacent  use method D –change both of them to ‘0’  case 1 –If these 2 ‘1’ are not adjacent  use method C –change both of them to ‘0’  case 1.

41. Method F Corners of the top layer left after the previous steps More and more complicated TASK: deal with the 4 corners CAUTION: cannot affect the other 22 cubes A new method needed

42. C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 C1 A1 B4 B1 B2 B3 A2 A3 A4

43. C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4

44. C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4

45. All edge cubes’ parameter equal to 0 No changes occurred Table of state parameter omitted Consider corners

46. 1 7 8 6 5 2 4 3 1 7 8 5 4 2 3 6 3 7 8 5 4 1 6 2 1 8 3 5 4 7 6 2 7 8 3 5 4 2 1 6 7 8 3 6 5 2 4 1 4 8 3 6 5 7 1 2 3 7 8 6 5 4 1 2 4 7 8 6 5 2 3 1

47. Observation Corner 1, 3 and 4 change their position State parameter +1 Other 22 cubes remain unchanged Steps12345678 1+-+- 2 3-++- 4 5-+-+ 6 7-+-+ 8 Total-2=+10+1 0000

48. Method G Derived by Method F (Opposite process ) C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 C4 C3 C2 b C1 A1 B4 B1 B2 B3 A2 A3 A4 1 7 8 6 5 2 4 3 3 7 8 6 5 2 1 4 1.3 and 4 corners’ parameters minus 1

49. Last 4 Corners Divide into 2 parts –1. Change all the parameter to 0 –2. Switch them to the correct position

50. Part 1 Sum of all parameters must be a multiple of 3 –Case 1: 0, 0, 0, 0 –Case 2: 0, 1, 1, 1 –Case 3: 0, 0, 1, 2 –Case 4: 0, 2, 2, 2 –Case 5: 1, 1, 2, 2

51. Part 1 (cont.) Case 1  Done. Case 2: 0,1,1,1 –Use method G: 0, 1-1, 1-1, 1-1; Result: 0, 0, 0, 0 Case 3: 0,0,1,2 –Use method F: 0+1, 0+1, 1, 2+1; Result: 1, 1, 1, 0  case 2 case 4: 0,2,2,2 –Use method F: 0, 2+1, 2+1, 2+1; Result: 0, 0, 0, 0 case 5: 1,1,2,2 –Use method G: 1-1, 1-1, 2-1, 2; Result: 0, 0, 1, 2  case 3

52. Part 2 –Put corner cubes in order –Introduce method H –For interchanging 2 corner cubes which are beside each other

53. Method H 1 5 8 2 3 4 6 7 1 7 8 6 5 2 4 3 1 7 8 4 3 2 6 5 Move twice 1 5 8 7 2 4 3 6 8 5 2 7 3 4 1 6 8 5 2 3 1 4 6 7

54. 8 7 2 4 1 3 6 5 8 7 2 5 4 3 1 6 1 7 8 5 2 3 4 6 1 7 8 6 5 3 2 4 2 7 8 6 5 1 4 3

55. Observation Step12345678 1+--++--+ 2+- -+ 3-+-+ 4--++ 5-++- 6+- -+ 7+-+- 8++-- 9+--+ 10 Total00+3=0-3=00000

56. Part 2 (cont.) With only method H –locate the corners to correct position –in right state Combine all the methods  Deal with the whole Rubik Cube

57. Proof by State Graphs 1. Interchanging 6 centers or 4 centers is possible, but not 2 centers 2. Impossible to make odd number edge cubes’ 2 sides interchanged 3. Impossible to make only 1 corner cubes’ 3 sides interchange (Trivial)