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Basis & Structure

Basis & Structure. Center. Surface 6 of them, 6 colors Center Cube 6 of them (fixed) Edge Cube 12 of them Corner Cube 8 of them. Corner. Edge. A1. C1. B1. B2. A2. A4. C2 b. C4. B4. B3. C3. A3. Graph for Edges. Bijection A new game. C4. C3. C1. C2 b. B2. B3. A2.

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Basis & Structure

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

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

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

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

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

  6. O B Y W G R 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

  7. B’ B B B B B 0 0 0 A A Red Red A Red A Red A’ Red Red A’ 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

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

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

  10. 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) State face 0 Top State Surface 1 Bottom StateSurface (inside) 2

  11. 8 1 7 2 3 6 4 5 Operation (Considering state) • Case 1: Vertical • State unchanged

  12. 8 - 1 + 7 + - 2 + 3 - 6 4 - 5 + • Case 2: Horizontal • State changing as left graph • Simply +1 or -1

  13. 8 + 1 - 7 - + 2 - 3 + 6 4 + 5 - • Case 3: w.r.t. axis • State changing as left graph • Simply +1 or -1

  14. 1st Layer • Observation + Operation • Locus Method • Avoidance Method

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

  16. 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 Public Locus 2 3 1

  17. 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

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

  19. 2nd Layer • Try & error  some method • some method  derived method

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

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

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

  23. State changes

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

  25. 3rd Layer • Last layer • More cubes’ position & state cannot be changed

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

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

  28. State changes

  29. D-0 D-0 A-0 C-1 C-0 B-0 B-0 A-1 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 + +

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

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

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

  33. State changes

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

  35. D-0 D-0 D-0 D-0 D-0 A-0 A-0 C-1 A-0 A-0 B-0 B-0 B-0 C-0 C-0 C-0 B-0 C-0 A-1 B-0 Method E • Derived by method C & D use in a different direction method C + + + + method E

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

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

  38. Demonstration (cont.) • A completedRubik Cube’s every edge cube’s state parameter is 0 • After everyoperation, 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

  39. 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.

  40. 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

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

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

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

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

  45. 3 8 3 8 3 3 8 8 8 7 1 3 3 4 4 7 1 1 7 7 8 8 7 7 8 7 8 1 2 2 2 4 7 7 2 2 6 3 2 2 6 1 2 2 1 6 5 6 5 6 6 6 5 5 1 3 6 6 4 1 3 4 1 4 5 5 4 4 5 5 5 4

  46. Observation • Corner 1, 3 and 4 change their position • State parameter +1 • Other 22 cubes remain unchanged

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

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

  49. 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

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