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

Graph for Edges

• Bijection

• A new game

C4

C3

C2

b

C1

A1

B4

B1 B2

B3

A2

A3

A4

C3C4 C1

C2

bB2

B3

B4 A3

A2

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

C1 A1

C3

B1 B2

A3

A2

B4

A4

Graph for Corners

• Bijection

• Corners numbered in order

• Another new game8 5

7 (inside)

2

1 4

3

61

7

8

6

5

2

4

3

Operation1

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

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

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

Red

0

A

A

B

B

Red

Red

0

A’

A

B

B

Red

Red

0

A’

A

B

B’

Case 2

• Both not with same or opposite color

• e.g. AB AB or A’B AB’ or A’B AB

Red

Red

1

A

B

A

B

Red

Red

1

A’

B

A

B

Red

Red

1

A’

B

A

B’

Case 3 & 4

• Irrelevant color involved

• e.g. CA BB state 1 (Case 3)

• e.g. CB AB state 2 (Case 4)

Red

Red

1

C / C’

A

B

B

Red

Red

0

C / C’

B

A

B

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 (curren

t state)

Top State Surface

Bottom StateSurface(inside)

State face 0

1

2

Operation (Considering state)

• Case 1: Vertical

• State unchanged1

7

8

6

5

2

4

3

• Case 2: Horizontal

• State changing as left graph– Simply +1 or -1 1

7

8

6

5

2

4

3

+

+

-

-

+

-

-

+

• Case 3: w.r.t. axis

• State changing as left graph– Simply +1 or -1 1

7

8

6

5

2

4

3

-

-

+

+

-

+

+

-

1st Layer

• Observation + Operation

• Locus Method

• Avoidance Method

Locus Method

• Possible position after one operation

• b, c, d, e, f and a: relative locus

c

b

a

ef

d

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

23

Public Locus

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

Steps & Example

• target block somewhere irrelevant to the Destination before any operation

1

23

2nd Layer

• Try & error some method

• some method derived method

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

C4

B3

C2

bC1

A1

C3

B1 B2

A3

A2

B4

A4

C4

B3

C2

C1

A1

C3

B1 B2

A3

A2

B4

A4

C4B3

C2

b

C1

A1

C3

B1

B2

A3

A2B4

A4

C4

B3

C2

b

C1

A1

C3

B1

B2

A3

A2B4

A4C4B3

C2

b

C1

A1

C3

B1 B2

A3

A2

B4

A4

State changes

Step B1 B2 B4 C1 A2 A3

1 + +

2 +

3 + +

4 +

5 + +

6 +

7 + + +

Total 0 0 0 0 0 0

Method B

C1

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

3rd Layer

• Last layer

• More cubes’ position & state cannot be changed

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

A4C4

C3 C2

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

C4

C3

C2

C1

A1

B4

B1 B2

B3

A2

A3

A4

C4

C3C2

C1

A1

B4

B1 B2

B3

A2

A3

A4

State changes

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

1 + + + +

2 + + + +

3 + + + +

4 + + + +

5 + + + +

6 + + + +

Total

0 0 0 0 0 0 0 0 0 1 0 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 D-0

A-0

B-0

C-0

D-0

C-1

A-1

B-0

Method D

C4

C3

C2

b

C1

A1

B4

B1

B2

B3 A2

A3

A4

C4

C3

C2

C1

A1

B4

B1

B2B3

A2

A3

A4

C4

C3

C2

bC1

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

B2B3

A2

A3

A4

C4

C3

C2

C1

A1

B4

B1

B2B3

A2

A3

A4

C4

C3C2

C1

A1

B4

B1

B2B3

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

State changes

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

1 + + + +

2 + + + +

3 + + + +

4 + + + +

5 + + + +

6 + + + +

7 ++ ++ ++ ++

8 + + + +

Total 0 0 0 0 0 0 0 0 0 1 1 0

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 followedD-0

A-0

B-0

C-0

D-0

A-0

C-1

B-1

Method E

• Derived by method C & DD-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

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

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

Cp C1=0

Ck

CqCp

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

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.

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

C4

C3C2

C1

A1

B4

B1

B2 B3

A2

A3

A4C4

C3

C2

C1

A1

B4

B1 B2

B3

A2

A3

A4

C4

C3 C2

C1

A1

B4 B1

B2

B3

A2

A3

A4C4

C3

C2

C1

A1

B4

B1

B2

B3

A2

A3

A4

C4

C3C2

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

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

• All edge cubes’ parameter equal to 0

• No changes occurred

• Table of state parameter omitted

• Consider corners

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

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

Steps 1 2 3 4 5 6 7 8

1 + - + -

2

3 - + + -

4

5 - + - +

6

7 - + - +

8

Total -2=+1 0 +1 +1 0 0 0 0

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

41.3 and 4 corners’ parameters minus 1

Last 4 Corners

• Divide into 2 parts– 1. Change all the parameter to 0– 2. Switch them to the correct position

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

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

Part 2

– Put corner cubes in order– Introduce method H – For interchanging 2 corner cubes which are

beside each other

Method H1

5

8

2

3

4

6

7

1

7

8

6

5

2

4

3

1

7

8

4

3

2

6

5Move twice

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

7Move twice

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

ObservationStep 1 2 3 4 5 6 7 8

1 +- -+ +- -+

2 +- +- -+ -+

3 - + - +

4 - - + +

5 - + + -

6 +- +- -+ -+

7 + - + -

8 + + - -

9 + - - +

10

Total 0 0 +3=0 -3=0 0 0 0 0

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

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)