basis & structure
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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)
