12. Rubik’s Magic Cube
Robert Snapp
[email protected]
Department of Computer Science
University of Vermont
Robert R. Snapp © 2012
12. Rubik’s Magic Cube
CS 32, Fall 2012
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Rubik’s Magic Cube
Ernö Rubik invented this celebrated puzzle in 1974.
When completed, each of the six faces displays a
common color, usually white, yellow, red, orange, blue
and green.
Questions:
1
How many different ways can six
colors be assigned to the six faces?
2
How are the colors of each pair of
opposite faces related at right?
Rubik’s standard color
arrangement.
The cube actually consists of 26 visible cubies, consisting of
6 single faced, centers, which are stationary.
12 double faced, edges.
8 triple faced, corners.
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David Singmaster’s Notation
David Singmaster1 published one of the first analyses of the Magic Cube. He
introduced the following notation:
U
U , for the Upper face,
B
F , for the Front face,
D , for the Down face,
L
R
B , for the Back face,
L, for the Left face, and
F
D
R, for the Right face.
Note that the Magic Cube can be oriented 24 ways within this coordinate system:
the upper face can be chosen 6 different ways.
for each upper face, the front face can be chosen 4 different ways.
6 4 D 24.
1. David Singmaster, Notes on Rubik’s Magic Cube, Enslow, Hillside, NJ, 1981.
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Singmaster’s Operations: U
Once the cube has been positioned, we define a
set of rotation operations that maintain the
orientation of the center cubies.
U
For example, U denotes a quarter turn of the
Upper face in the clockwise direction.
U2
U 2 denotes a half turn of the Upper face. (N.B.,
U 2 D U U .)
U 0 denotes a quarter turn of the Upper face in
the counter-clockwise direction. (N.B.,
U 0 D U 3 .)
Robert R. Snapp © 2012
12. Rubik’s Magic Cube
U0
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Singmaster’s Operations: F
F denotes a quarter turn of the Front face in the
F
clockwise direction.
F 2 denotes a half turn of the Front face. (N.B.,
F 2 D FF .)
F 0 denotes a quarter turn of the Front face in the
counter-clockwise direction. (N.B., F 0 D F 3 .)
Robert R. Snapp © 2012
12. Rubik’s Magic Cube
F2
F0
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Singmaster’s Operations: D
D denotes a quarter turn of the Down face in the
D
clockwise direction.
D 2 denotes a half turn of the Down face. (N.B.,
D 2 D DD .)
D2
D 0 denotes a quarter turn of the Down face in
the counter-clockwise direction. (N.B.,
D 0 D D 3 .)
Robert R. Snapp © 2012
12. Rubik’s Magic Cube
D0
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Singmaster’s Operations: B
B denotes a quarter turn of the Back face in the
B
clockwise direction.
B 2 denotes a half turn of the Back face. (N.B.,
B 2 D BB .)
B 0 denotes a quarter turn of the Back face in the
counter-clockwise direction. (N.B., B 0 D B 3 .)
Robert R. Snapp © 2012
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B2
B0
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Singmaster’s Operations: L
L denotes a quarter turn of the Left face in the
L
clockwise direction.
L2 denotes a half turn of the Left face. (N.B.,
L2 D LL.)
L0 denotes a quarter turn of the Left face in the
counter-clockwise direction. (N.B., L0 D L3 .)
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L2
L0
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Singmaster’s Operations: R
R denotes a quarter turn of the Right face in the
R
clockwise direction.
R2 denotes a half turn of the Right face. (N.B.,
R2 D RR.)
R0 denotes a quarter turn of the Right face in the
counter-clockwise direction. (N.B., R0 D R3 .)
Robert R. Snapp © 2012
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R2
R0
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Restore the Cube: Outline
Part I: Restore the upper face.
1. Restore the upper edges.
2. Restore the upper corners.
Part II: Restore the middle layer.
3. Turn the entire cube upside
down.
4. Restore the middle edges.
Part III: Restore the final face.
5.
6.
7.
8.
Robert R. Snapp © 2012
Invert the upper edges.
Reposition the upper edges.
Reposition the upper corners.
Twist the upper corners.
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Part I: Step 1 — Restore the Upper Cross
1a Select a color for the upper face (e.g, green), and
an adjacent color for the front face (e.g., white).
1b Identify the cubie that belongs in the upper-front
(uf ) edge, e.g., the green-white edge. It should
be easy to bring this cubie to the correct
location.
1c If this colors of the uf edge need to be flipped,
then apply the sequence
F 0 UL0 U 0 :
1d Rotate the entire cube one-quarter turn, and
repeat the above until all four upper edges are in
place. You should see a green cross.
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Part I: Step 2 — Restore the Upper Corners
2a For each corner cubie in the Down layer that
belongs in the Upper layer:
i Rotate the Down layer (using the D operation) until
this cubie is directly below its desired postion.
Rotate the entire cube so that the desired position
is under your right thumb (upper-right-front
position).
urf
drf
ii Apply the operation R0 D 0 RD one, three, or five
times, until this corner cubie is in the correct
position, with the correct orientation. (This will not
destroy the cross, obtained in Step 1.)
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Part I: Step 2 — Restore the Upper Corners (cont.)
2b For each Upper layer corner cubie that is
incorrectly placed, or incorrectly rotated,
i Rotate the entire cube until the misplaced cubie is
under your right thumb.
ii Place the cubie in the Down layer using R0 D 0 RD:
iii Then apply step 2a (above) to move this cubie in
the correct position.
2c Apply the above steps until the entire upper layer
is complete.
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Part I: Step 2 — Restore the Upper Corners (cont.)
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Part II: Step 3 — Turn the Cube Upside Down
Turn the entire cube upside down, so that the completed green layer is the bottom (or down) layer. The
new upper layer should have a blue center.
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Part II: Step 4 — Restore the Middle Layer
The key operation is RU 0 R0 FR0 F 0 RU 0 which swaps and inverts ul and fr.
4a Rotate the entire cube until a front-right (fr) edge
is incorrect, or flipped. (Assume the right edge of
the white face is incorrect.)
4b Locate the correct edge (e.g., the red-white
edge).
Case A: If the correct edge is in the middle
layer:
i Rotate the entire cube so that
the correct edge is a front-right
(fr) edge. (Note, the red-white
edge is in the middle layer.)
ii Perform the sequence
RU 0 R0 FR0 F 0 RU 0 which will
place the correct edge in the
upper layer (at ul ).
iii Apply Case B.
Robert R. Snapp © 2012
12. Rubik’s Magic Cube
ul
fr
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Part II: Step 4 — Restore the Middle Layer (cont.)
Case B: If the correct edge is on the top layer:
i Ensure that the misplaced edge
is still the front-right (fr ) edge.
ii Rotate the upper layer (using U
operations) so that the correct
edge is an upper-left (ul ) edge.
iii Apply the operation
RU 0 R0 FR0 F 0 RU 0 .
iv If the correct edge needs to be
flipped, apply Case C.
Case C: If a middle edge is flipped in the
correct location:
i Apply the operation
RU 0 R0 FR0 F 0 RU 0 twice.
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Part II: Step 4 — Restore the Middle Layer (cont.)
The top row illustrates two successive occurrences of Case B. The left two diagrams show how
the red-yellow edge is moved into its correct position with RU 0 R0 FR0 F 0 RU 0 . The right two,
show how the orange-yellow edge is moved into its correct position by the same operation.
The bottom row illustrates an occurrence of Case B, that leads to a Case C. First the
orange-white edge is moved into its correct position, but with an incorrect orientation. Applying
RU 0 R0 FR0 F 0 RU 0 moves it back into the top layer, but flipped. A third application, brings
the orange-white edge into the correct position and orientation.
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Part III: Restoring the Upper Layer
Now that the bottom and middle layers are complete, every cubie in the upper layer
has a single blue face. In order to restore the upper face, one needs to
5. Flip the edge cubies so that the blue face of each
faces upwards.
6. Move the edge cubies to their final locations,
without destroying their orientation.
7. Move the corner cubies to their final locations.
8. Rotate the corner cubies (in place) so that the
blue face of each faces upwards.
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Part III: Step 5 — Flip the New Upper Edges
5. Orient the cube so that it matches one of the four orientations:
“Blue Dot”
“Blue Corner”
“Blue Line”
“Blue Cross”
a. If the ”Blue Cross” is displayed, move on to Step 6.
b. If the ”Blue Cross” is not displayed, apply the maneuver
FRUR0 U 0 F 0
and repeat Step 5 as many times as required.
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Part III: Step 6 — Restore the New Upper Edges
At this point of the solution, the bottom two layers should be solved, and a blue cross,
should appear on the top face. If you are very lucky, the red, white, yellow and orange
sides of the blue cross match all four of the corresponding center cubies. (Twist the
upper layer using a succession of U operations, to see if this occurs. If so procede to
Step 7.) If you are not so lucky, twist the upper layer until exactly one of the sides of
the blue cross matches its center cubie. Rotate the cube so that the matching side
cubie is in the front face. In the figures below the matching cubie happens to be red.
RWYO
ROWY
RYOW
Apply the sequence RUR0 URU 2 R0 until the sides of the four top edge cubies match.
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Part III: Step 7 — Place the Upper Corners
We shall now ensure that each upper corner is in the
correct position. (Don’t worry now about their orientations; those will be restored in Step 8.)
ulb urb
Compare the colors of each upper corner with those of
the adjacent centers. If all three match, even if the orientation is wrong, then this piece is in the correct position. In the diagram at right, the upper-left-front (ulf)
corner (red-white-blue) is in the correct position. The
upper-right-front (urf ) corner (yellow-orange-blue) is
not.
ulf
The key sequence of Step 7 is L0 URU 0 LUR0 U 0 , which rotates (or cycles) the upper
three corners (ulf, ulb, urb ), in a clockwise direction, while maintaining the positions
and orientation of the remaining 23 cubies.
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Step 7 — Place the Upper Corners (cont.)
7a. If no upper corners are in their correct positions, apply L0 URU 0 LUR0 U 0 (once
or twice) until one is. Then continue.
7b. If one corner is in its correct position, then rotate the entire cube so that the
correctly placed corner is near your right thumb, in the upper-right-front (urf )
position. Then apply L0 URU 0 LUR0 U 0 (once or twice) until all four upper corners
are correctly placed.
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Part III: Step 8 — Twist the Upper Corners
At this point every cube is in the correct position. However, two or more corners may
have an incorrect orientation.
The key sequence of Step 8 is R0 D 0 RD , which you already practiced in Step 2.
8a. Rotate the entire cube until an incorrectly
oriented (twisted) corner is located near your
right thumb. (It should be in the urf position.)
8b. Apply the sequence R0 D 0 RD (two or four times)
until this corner cube has the correct orientation.
Don’t worry about the middle and bottom layers:
they are temporarily messed up.
Robert R. Snapp © 2012
12. Rubik’s Magic Cube
urf
urf
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Part III: Step 8 — Twist the Upper Corners (cont.)
8c. Now rotate only the upper layer, by applying one
or more U operations, until the next twisted cube
is near your right thumb in the urf position.
8d. Repeat steps 2 and 3 until every corner is
correctly oriented.
8e. Finally, restore the cube using one or more U
operations.
urf
urf
urf
8f. Fix yourself an ice-cream cone.
urf
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Summary
Step
Operations
upper
(green)
cross
Use the six basic operations to move the desired edge immediately below its home, without moving the other upper edges. Then rotate that
face one-half turn.
Goal
To flip an inverted edge, apply F 0 UL0 U 0 .
upper
(green)
corners
Use R0 D 0 RD to swap (and twist) the urf and drf corners. After each
misplaced corner has been moved to the down (blue) layer, use the D
operator to move it immediately below its home. Then apply R0 D 0 RD
a sufficient number of times, so that it is correctly placed and correctly
oriented.
flip
entire
cube
Easy as pie! Turn the entire cube upside down so that the blue center
on top and the completed green face is the new down layer.
middle
edges
Use RU 0 R0 FR0 F 0 RU 0 to swap and flip the ul and fr edges, without
displacing the other cubies on the lower two layers.
Robert R. Snapp © 2012
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Summary (cont.)
Step
Operations
orient
upper
edges
If the blue facets on the upper face form a corner, rotate the cube so
that the corner is at ul, u, and ub. If the upper facets of the upper edges
form a blue line, rotate the cube so that the blue line runs from left to
right (ul, u, ur). Apply FRUR0 U 0 F 0 until a blue cross is displayed.
restore
upper
edges
Goal
Apply U until the the uf edge matches the color of the front face. Then
apply RUR0 URU 2 R0 until every upper edge matches the side faces.
place
upper
corners
If an upper corner is correctly placed, rotate the entire cube so that
this becomes the urf corner. Then apply L0 URU 0 LUR0 U 0 until each
corner is correctly placed.
twist
upper
corners
Apply U until urf is twisted. Then apply R0 D 0 RD until this urf is correct. Repeat until every corner is untwisted. Apply U to restore the
cube.
Robert R. Snapp © 2012
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urf
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How Many States are in the Cube?
Claim: A 3 3 3 Rubik’s cube can be placed in exactly
N D 43; 252; 003; 274; 489; 856; 000
different configurations, using a sequence of legal moves based on L, R, U , D , B
and F , more than the number of seconds in 10 billion centuries.
Counting this number is sort of like counting the number of anagrams that can be
formed from a given set of letters. We thus count permutations.
Recall that there are three kinds of cubies: 8 corners, 12 edges, and 6 centers. First
note that it is impossible to exchange a three-sided corner with a two-sided edge,
and likewise we can’t exchange a center with either a corner or edge.
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How Many States are in the Cube?
We will use the multiplication principle to count the number N of configurations that
can be obtained by a sequence of the operations, L, R, U , D , B and F .
Let,
N1 D number of configurations of the 6 centers
N2 D number of configurations of the 12 edges
N3 D number of configurations of the 8 corners
Then, our first estimate of N is
N D N1 N2 N3 :
What is the value of N1 ?
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Estimating N1
Since the locations of the centers are unchanged by each of the six basic operations,
they are also unchanged by any sequence of these operations. Thus,
N1 D 1:
Thus,
N D 1 N2 N3 :
What is the value of N2 ?
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Estimating N2
Since there are 12 locations (cubicles) for each edge, there are 12Š ways to order the
edges. In addition, each edge can be flipped in two different ways: e.g., the red-blue
edge can be red-side up, or blue-side up. This suggests that there are at most
N2 D 12Š 212 D 1; 961; 990; 553; 600
ways to arrange the 12 edges.
What can we say about N3 ?
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Estimating N3
Since there are 8 corner cubicles (locations for the corners), there are 8Š ways to
order the corners. In addition each corner can be twisted three different ways. This
suggests that, at most,
N3 D 8Š 38 D 264; 539; 520
ways to arrange the eight corners.
Does
N D 1 .12Š 212 / .8Š 38 /‹
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Counting the Configurations of Rubik’s Cube
This number,
1 .12Š 212 / .8Š 38 / D 519; 024; 039; 293; 878; 272; 000
actually represents (exactly) the number of different ways that Rubik’s cube can be
reassembled, assuming that the centers are not rearranged.
Anne Scott (cf., Berlekamp, Conway, Guy, 2004), showed that this value
overestimates the correct value of N by a factor of 12.
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Invariants
Consider a “puzzle” that concerns the value of a variable x . Initially, x D 0. Every
second a coin is tossed. If the coin lands heads then we add 4 to x . If the coin lands
tails, we subract 2. Here is a sample sequence.
time (s.)
coin toss
0
x
0
1
H
4
2
T
2
3
H
6
4
H
10
5
T
8
6
T
6
7
T
4
8
T
2
9
T
0
10
H
4
Question: Can x ever equal 1?
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Invariants
Correct! The answer is no. Since x begins as an even number, and every possible
operation (adding 4 or subtracting 2) preserves evenness, x will always be even.
In this context, evenness is said to be an invariant property, or an invariant (for short),
of x .
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Invariants and Loyd’s 14-15 Puzzle
Sam Loyd (1841–1911) created many popular puzzles, including the celebrated
14–15 puzzle, shown above. Can you interchange just tiles labeled 14 and 15, by
sliding tiles horizontally or vertically into the space? (Loyd offered a $1000 prize to
anyone who could.)
How many states are realizable?
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Invariants (cont.)
For the space to wind up in the lower-right corner, there must have been an even
number of vertical moves, and an even number of horizontal moves. Consequently,
only permutations that swap and even number of pieces are possible. For Loyd’s
puzzle, only half of the 16Š states are realizable.
Anne Scott used invariants to exactly count the number of possible states for Rubik’s
cube.
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Reexamining the allowed corner twists
Place a 0, 1, or a 2 on each corner face,
as shown at right. The initial sums are then
computed for each face, and recorded under
column I of the table. Sums are also computed following each legal quarter turn. Note
that ever entry is a multiple of 3. This latter
property is preserved for every sequence of
legal operations.
2
1
0
1
0
2
1
2 1
2
1 2
Consequently, only one-third of the total
number of corner twists 38 can be realized
using a sequence of legal operations.
Robert R. Snapp © 2012
1
21
2
1
0
0
0
However, if one were able to twist a single
corner, one-third of a turn, in either direction, the sums of the adjacent faces change
to numbers that are not multiples of 3.
0
0
2
0
Face Sums
Face
I
L
R
U
D
F
left
6
6
6
6
6
3
B
3
right
6
6
6
6
6
3
3
upper
0
3
3
0
0
3
3
down
0
3
3
0
0
3
3
front
6
3
3
6
6
6
6
back
6
3
3
6
6
6
6
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Reexamining the allowed edge flips
Place a 0 or 1 on each edge, and construct
a stationary blue window for each face, as
shown. The initial sum of the values that appear in the blue windows is computed under
column I in the table. It can be shown that
the window sum will always be a multiple of
2, and even number, after every sequence of
operations. (After F U , for example, it equals
6.)
0
1
0
0
Thus only one-half of the 212 edge states are
realizable.
Robert R. Snapp © 2012
1
1
1 0
1
1
0
1
1
1
0
0
0
0
However, flipping any single edge results in
an odd window sum. Consequently, it is not
possible to invert a single edge using a sequence or rotations.
1
1
0
0
1
0
Blue-Window Sums
sum
12. Rubik’s Magic Cube
I
L
R
U
D
F
B
12
8
8
8
8
8
8
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How many states are expressible by the cube?
The final reduction factor is obtained by observing that only one-half of the 12Š 8Š
permutations of the locations of the 12 edges and 8 corners are realizable. Each
sequence of operations always moves a multiple of 4 pieces. It is thus impossible to
interchange just two corners, or just two edges.
Thus,
ND
1 1 1
12Š 212 8Š 38
2 2 3
D 43; 252; 003; 274; 489; 856; 000:
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Some Symmetrical States
Let Fs D FB 0 denote a move called a front slice. Similarly,
let Rs D RL0 denote the right slice, and Us D UD 0 denote the upper slice.
“Dots”
“Chessboard”
“Cross”
0
Rm Fm0 Rm
Fm
Fs2 Rs2 Us2
R0 L2 Fs2 U 2 Rs2 Fs2 D 2 R0
0
The definitions of Rm , Rm
, Fm , and Fm0 appear below.
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Singmaster’s Operations: Rm
Start with yellow on top, blue in front, and red at
right. Rm denotes a quarter turn of the middle
layer (only) parallel to the direction of R. The
easiest way to complete this is to rotate both the
right face, and the middle layer behind the right
face, one quarter turn clockwise, followed by R0 .
Rm
2
Rm
2
Rm
denotes a half turn of the middle layer
behind the right face.
0
Rm
denotes a quarter turn of the middle layer,
behind the right face, in the counter-clockwise
0
3
direction, i.e., parallel to R0 . (N.B., Rm
D Rm
.)
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0
Rm
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Singmaster’s Operations: Fm
Fm denotes a quarter turn of the middle layer
(only) parallel to the direction of F . The easiest
Fm
way to complete this is to rotate both the front
face, and the middle layer behind the front face,
one quarter turn clockwise, followed by F 0 .
Fm2 denotes a half turn of the middle layer
Fm2
behind the front face.
Fm0 denotes a quarter turn of the middle layer,
behind the front face, in the counter-clockwise
direction, i.e., parallel to F 0 . (N.B., Fm0 D Fm3 .)
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Singmaster’s Operations: Um
Um denotes a quarter turn of the middle layer
(only) parallel to the direction of U . The easiest
Um
way to complete this is to rotate both the upper
face, and the middle layer behind the upper face,
one quarter turn clockwise, followed by U 0 .
Um2 denotes a half turn of the middle layer
Um2
behind the upper face.
Um0 denotes a quarter turn of the middle layer,
behind the upper face, in the counter-clockwise
0
direction, i.e., parallel to U 0 . (N.B., Um
D Um3 .)
Robert R. Snapp © 2012
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References
1
Christoph Bandelow, Inside Rubik’s Cube and Beyond, Birkhäuser, Boston, 1982.
2
Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Winning Ways For Your
Mathematical Plays, Second Edition, Vol. 4, A. K. Peters, Natick, MA, 2004.
3
John Ewing and Czes Kośniowski, Puzzle It Out: Cube Groups and Puzzles, Cambridge
University Press, Cambridge 1982.
4
Alexander H. Frey, Jr. and David Singmaster, Handbook of Cubik Math, Enslow, Hillside,
NJ, 1982.
5
Martin Gardner, ed., The Mathematical Puzzles of Sam Loyd, Dover, NY, 1959.
6
David Joyner, Adventures in Group Theory: Rubik’s Cube, Merlin’s Magic & Other
Mathematical Toys, Johns Hopkins University Press, Baltimore, 2002.
7
Ernö Rubik, Tamás Varga, Gerzson Kéri, Györgi Marx, and Tamás Vkerdy, Rubik’s Cubic
Compendium, Oxford University Press, Oxford, 1987.
8
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Robert R. Snapp © 2012
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