MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
from FS2009
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
L ECTURE 19
Part/ References
Topic/Sections
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
Lecture 19:
Rubik’s Cube: Beginnings
Contents
7
Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due
Multivariable GFs
19
IV.3, IV.4
Analysis
Analytic Methods andComplex
19.1
Rubik’s
Cube terminology
notation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
FS: Part B: IV, V, VI
Singularity Analysis
Appendix B4
IV.5 V.1
19.1.1
Move Notation:
. . . . . . . . . . .
Stanley 99: Ch. 6
Nov 2
Asymptotic methods
Handout #1
19.1.2
Position
and Piece Notation: . . . .
(self-study)
9
VI.1
26
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
. . Sophie
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
19.2
Moves . . . . . . .Introduction
. . . . to. Prob.
. . . . Mariolys
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 Impossible
A.3/ C
6
18 A Catalog
IX.1
Limit Laws and
19.3
of Useful Move Sequences
. Comb
. . . . Marni
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
9
10
11
12
Random Structures
19.3.1 Cornerand
Moves
. . .
Limit Laws
FS: Part C
23
IX.3
(rotating . . . .
19.3.2 Edge Moves
presentations)
25
IX.4
20
IX.2
Asst #2 Due
Discrete Limit Laws
Sophie
Combinatorial
.instances
. . . of. discrete
. . .
Mariolys
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuous Limit Laws
Marni
19.4 Strategy for Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
30
IX.5 The Layer Method . .
19.4.1
14
Dec19.4.2
10
Solving the
Presentations
Top
Layer
Quasi-Powers and
.Gaussian
. . . limit
. . laws
. .
. . Sophie
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
19.4.3 Solving the Middle Layer
. Asst
. .#3. Due
. .
. . . . . . . . . . . . . . . . . . . . . . . . . .
7
8
8
9
9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
19.4.4 Solving the Bottom Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
In this lecture, we summarize Rubik’s cube terminology and notation that we have been using so far, as well
as
introduce
Singmaster
notation
each
piece of, and each position on, the cube. There is no ”one size fits
Dr. Marni
MISHNA, Department
of Mathematics,
SIMONfor
FRASER
UNIVERSITY
Version of: 11-Dec-09
all”
notation when modelling Rubik’s cube, we’ll see that each notation has its benefits depending on what you
are trying to do with it.
19.1
Rubik’s Cube terminology and notation
The notation we use was first introduced by David Singmaster in the early 1980’s, and is the most popular
notation in use today.
19.1.1
Move Notation:
Fix an orientation of the cube in space. We may label the 6 sides as f , b, r, l, u, d for front, back, right, left, up,
and down.
Face moves:
A quarter twist of a face by 90 degrees in the clockwise direction (looking at the face straight on) is denoted by
the uppercase letter corresponding to the name of the face. For example, F denote the move which rotates the
front face by 90 degrees clockwise. See Table 1 for a complete description of cube moves and notation.
Jamie Mulholland, Spring 2011
Math 302
19-1
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
L ECTURE 19
Part/ References
Topic/Sections
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
from FS2009
Slice moves:
We1 also
the names of some slice
moves.
Septindicate
7 I.1, I.2, I.3
Symbolic
methods These are moves in which one of the three middle slices is
Combinatorial
rotated. For example, if
the slice between the l and r face is rotated upwards, that is, in the clockwise direction
Structures
2
14
I.4, I.5, I.6
Unlabelled structures
Part A.1,
A.2 then we denote this move by S` . We could also view this move from the left
when viewed from theFS:
right
face,
R
Comtet74
3
II.1, II.2, II.3
structures I
side
as21a counterclockwise
soLabelled
we could
denote it by S`−1
Handoutrotation,
#1
L . Similarly, we have slice moves for the slice
study)
4
28 to the
II.4, II.5,
II.6 d(self
II to the f and b face. These moves are denoted by:
parallel
u and
face,
and for theLabelled
slicestructures
parallel
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
S`RParameters
= S`−1
L ,
Asst #1−1
S`U =
S`DDue, S`F = S`−1
B .
Multivariable GFs
We can also square these moves. See Figure 1.
10
11
12
IV.5 V.1
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Complex Analysis
Singularity Analysis
Asymptotic methods
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
(a) IX.4
Slice move S`R .
13
30
IX.5
14
Dec 10
Sophie
(b)Laws
Slice Marni
move
Continuous Limit
S`U .
(c) Slice move S`F .
Quasi-Powers and
Sophie
FigureGaussian
1: Three
basic slice
moves of Rubik’s Cube.
limit laws
Presentations
Asst #3 Due
Whole cube moves:
The whole cube, as a single object, can be rotated in space. For example, we can rotate the cube about an axis
through the centres of the left and right faces. If the rotation is in the clockwise direction as viewed from the
right face then we denote the move by R. This could also be viewed as a counterclockwise rotation from the
left face perspective, so we could also denote it by L−1 .
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
Figure 2: Whole cube rotation R. Also denoted by L−1 .
19.1.2
Position and Piece Notation:
The 26 pieces of the cube, called cubies, split up into three distinct types: centre cubies (having only coloured
sticker), edge cubies (having two coloured stickers), corner cubies (having three coloured stickers).
We call the space which a cubie can occupy a cubicle, and we call the space a sticker can occupy a facet. We
can also describe a facet as the face of a cubicle. As the pieces move around, the cubies move from cubicle to
cubicle, and the stickers move from facet to facet. In the 15-puzzle, Oval Track, and Hungarian Rings puzzles,
we called the location a piece could occupy a position or spot, the term cubicle is customary to use when talking
about the Rubik’s cube.
To solve the puzzle each cubie must get restored to its original cubicle, we call this the cubies home location,
and each sticker must get returned to its original facet (i.e. the facets must also be correctly positioned), we
Jamie Mulholland, Spring 2011
Math 302
19-2
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
notation
L ECTURE 19
Sections
Part/ References
from FS2009
pictorial
Topic/Sections
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
description of basic move
(Singmaster)
from front)
1
Sept 7 I.1, I.2, I.3(viewCombinatorial
(clockwise/counterclockwise
refers to viewing the face straight-on)
Symbolic methods
2
F
, 14F −1
I.4, I.5, I.6
3
21
II.1, II.2, II.3
structures
FUnlabelled
= quarter
turn of front face in the clockwise direction.
−1
structuresturn
I
FLabelled
= quarter
of front face in the counterclockwise direction.
4
28 −1 II.4, II.5, II.6
5
Oct 5
B,
B
III.1, III.2
6 , 12R−1
R
IV.1, IV.2
7
19
IV.3, IV.4
8
L
26L−1
,
9
Nov 2
U
10
, U −1
11
D
, D−1
12
S`
R
IV.5 V.1
9
VI.1
12
A.3/ C
18
IX.1
20
IX.2
23
IX.3
, S`−1
R
25
IX.4
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Combinatorial
parameters
FS A.III
(self-study)
GFs
RMultivariable
= quarter
turn of right face in the clockwise direction.
−1
RComplex
= quarter
Analysis turn of right face in the counterclockwise direction.
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Analysis
LSingularity
= quarter
turn of left face in the clockwise direction.
−1
#2 Dueface in the counterclockwise direction.
LAsymptotic
= quarter
of left
methodsturnAsst
Sophie
U = quarter turn of up face in the clockwise direction.
−1
to Prob.turnMariolys
UIntroduction
= quarter
of up face in the counterclockwise direction.
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Limit Laws and Comb
Marni
Combinatorial
instances of discrete
Mariolys
D = quarter turn of down face in the clockwise direction.
Discrete Limit Laws
Sophie
D−1 = quarter turn of down face in the counterclockwise direction.
S`R = quarter turn of vertical slice in the clockwise direction.
−1
Continuous
Limit Laws Marni
S`
R = quarter turn of vertical slice in the counterclockwise direction.
Quasi-Powers and
13
IX.5
S`U 30, S`−1
U
14
Labelled structures II
B = quarter turn of back face in the clockwise direction.
−1
BCombinatorial
= quarter turnAsst
of #1
back
Due face in the counterclockwise direction.
Parameters
Sophie
Gaussian
limit lawsturn of horizontal slice in the clockwise direction.
S`
R = quarter
−1
of#3
horizontal
slice in the counterclockwise direction.
Presentations S`R = quarter turn
Asst
Due
Dec 10
F 2 , B 2 , R2 , L2 , U 2 , D2 denote the corresponding half-turn of the face.
Since a clockwise half-turn is equivalent to a counterclockwise half-turn then
F 2 = F −2 , B = B −2 , R2 = R−2 , L2 = L−2 , U 2 = U −2 , D = D−2
F , B, R, L, U , D
denote clockwise rotations of the whole cube
behind the indicated face.
Table 1: Summary of cube move notation
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
call this the cubies home orientation.1 See Figure 3 for an example of this distinction. Once all cubies are
in their home locations and home orientations the puzzle will be solved.
Figure 3: Cubie U F is in its home location, but not in its home orientation since it is flipped. Similarly for cubie U B.
We will describe a labeling of facets and cubicles below. It is important to keep in mind that facets and cubicles
don’t move, only the pieces (cubies and stickers) move. So when describing a labeling of the cubies and facets
it is best to think of this label as appearing on a fictitious layer of skin surrounding the puzzle. The pieces can
move around under the skin but the skin remains in place.
1 This
can also be called its home position.
Jamie Mulholland, Spring 2011
Math 302
19-3
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
Part/ References
from FS2009
Facet Notation
L ECTURE 19
Topic/Sections
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
1
Sept47 and
I.1, I.2,
Symbolic
methods
Figures
5 I.3
showsCombinatorial
a labeling of the
facets
of the cube. This labeling is due to mathematician, and puzzle
Structures
2
14
I.4, I.5, I.6 Singmaster. Our typical
Unlabelled
structuresuses numbers (see Lecture 1), but this labeling uses strings
enthusiast,
David
labeling
FS: Part A.1, A.2
Comtet74 to this labeling is that it allows us to easily determine where a facet position is
of 3symbols.
That
advantage
21
II.1,
II.2, II.3
Labelled structures I
Handout #1
located. For example, (self
thinking
back to our numerical labeling, if asked where facet position 41 is, you likely
study)
4
28
II.4, II.5, II.6
Labelled structures II
don’t know without looking at a diagram. However, with this new labeling, facet 41 is facet dlf , which you
Combinatorial
Combinatorial
know
is
the
dlf cubicle.
As for which
of the three sides
5
Oct on
5
III.1, III.2
Asst #1 it
Dueis, this is denoted by the first letter in the name:
parameters
Parameters
FS A.III
d for down. So facet dlf
is the down side of the dlf cubicle.
6
12
IV.1, IV.2
Multivariable GFs
(self-study)
If 7you 19are wondering
how the order ofComplex
the other
two letters were chosen (i.e. why didn’t we call it df l?), the
IV.3, IV.4
Analysis
Analytic Methods
answer is simple: we wrote
them
in
the
order
the
faces
appear when moving around the corner in the clockwise
FS: Part B: IV, V, VI
8
26
Singularity Analysis
Appendix
B4 labelings
direction.
You
can
check
all
the
in
Figure
4
to
verify this is the convention.
IV.5 V.1
9
Nov 2
Stanley 99: Ch. 6
Handout #1
(self-study)
Asymptotic methods
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
13
30
IX.5
Quasi-Powers and
Gaussian limit laws
Sophie
14
Dec 10
10
11
12
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Sophie
Presentations
Asst #3 Due
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
Figure 4: Facet labeling on the 3 × 3 × 3 Rubik’s cube.
(a) View of front (red),
right (yellow) and up
(blue) faces, labelled with
Singmaster notation.
(b) View of back (orange),
left (white) and down
(green) faces, labeled with
Singmaster notation.
Figure 5: Rubik’s Cube with classic colouring scheme: blue opposite green, red opposite orange, white opposite yellow.
Each cubicle is labeled using Singmaster notation.
Jamie Mulholland, Spring 2011
Math 302
19-4
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
L ECTURE 19
Part/ References
Topic/Sections
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
from FS2009
Cubicle notation:
A 1cubicle
identified by faces itSymbolic
touches.
For example, the cubicle that touches the up, right and front
Sept 7can
I.1, be
I.2, I.3
methods
Combinatorial
faces can be denoted by
urf . In particular, we can denote a cubicle by the labeling of any of the 3 facets that
Structures
2
14
I.4, I.5, I.6
Unlabelled structures
FS: Part A.1, A.2
are on the cubicle, in addition
to any of the other three orderings of the letters. For example the front-up-right
Comtet74
3
21
II.1, II.2, II.3
Labelled structures I
cubicle
can be
denotedHandout
by any
#1 one of the 6 symbols: f ur, urf , rf u, f ru, rf u, or uf r.
4
28
II.4, II.5, II.6
6
12
IV.1, IV.2
(self study)
Labelled structures II
Since a corner cubie has three facets we denote it by three letters. Similarly, edge cubies are denoted by two
Combinatorial
letters.
4 andCombinatorial
5 shows a labelling
of all the cubicles
5
OctFigures
5
III.1, III.2
Asst #1 Due(use any one of the facet labelings to denote the
parameters
Parameters
FS A.IIIbelongs).
cubicle to which the facet
Multivariable GFs
(self-study)
There
is a benefit
to labeling cubicles Complex
and facets
in a similar fashion. For the moment we focus our attention
7
19
IV.3, IV.4
Analysis
Methods
on cubies rather than Analytic
cubicles/facets.
For
example
consider the move R−1 . The cubie in position urf moves to
FS: Part B: IV, V, VI
8
26
Singularity Analysis
Appendix
B4
cubicle
df r. IV.5
However,
there
are
three
different
ways
a corner cubie can be placed in a cubicle, so just stating
V.1
Stanley 99: Ch. 6
9 urf
Nov 2moves to df r doesn’t indicate Asymptotic
Asstonce
#2 Due it gets to df r. Notice that the up face of the cubie
that
how
it
is
oriented
methods
Handout #1
(self-study)
is placed
in
the
front
face
when
it
moves
to
the
new
cubicle.
Similarly, the right face stays on the right face. It
9
VI.1
Sophie
10
−1
would
be
more
descriptive
to
say
that
R
takes
the
cubie
in
position urf to position f rd. We can write
12
A.3/ C
Introduction to Prob.
Mariolys
11
18
IX.1
Limit Laws and Comb
uf r
R−1
−−Marni
−→ f dr.
Random Structures
20
IX.2
Limit Laws
Sophie
This indicates
that cubie
in cubicle
uf Discrete
r moves
to cubicle
f dr, and the stickers moved as follows: u-facet moves
and Limit
Laws
FS:
Part
C
Combinatorial
to f -facet,
f -facet
moves to d-facet, and
r-facet moves to
r-facet.
23
IX.3
Mariolys
(rotating
12
instances of discrete
presentations)
Cubie25notation:
IX.4
Continuous Limit Laws Marni
A cubie is identified by its home cubicle. We use capital letters to denote cubies, and lower case to denote
Quasi-Powers and
13
30 ForIX.5
Sophie location is the urd cubicle. It may seem that using
cubicles.
example, U RD denotes the
cubie
home
Gaussian
limitwhose
laws
the
same
notation
to
denote
cubies
as
cube
moves
is
a
bad idea, however, we’ll see that this doesn’t cause any
14
Dec 10
Presentations
Asst #3 Due
trouble at all. We just need to be aware as to whether we are talking about cube moves, or cube pieces.
Table 2 summarizes the terminology introduced here.
Terminology
Definition or Abbreviation
cubies
The small cube pieces which make up the whole cube.
cubicles
The spaces occupied by the cubies.
facets
The faces of a cubicle
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
types of cubies:
A corner
corner, edge, and centre:
cubie has three facets.
An edge cubie has two facets.
A centre cubie has one facet
home location - of a cubie
The cubicle to which a cubie should be restored.
home orientation - of a cubie
The orientation in the home location to which a cubie
should be restored.
positional names
for cube faces
Up (u)
Right (r)
Front (f )
Notation for cubicles
- shown in italics
Lower case initials. For example, uf denotes
the Up-Front edge cubicle, dbl denotes the Down-BackLeft cubicle.
Notation for cubies
- shown in italics
Upper case initials. For example, URF denotes
cubie whose home position is in the the Up-Right-Front
corner
Down (d)
Left (l)
Back (b)
Table 2: Summary of terminology and notation
With all this notation now in our tool box, we are ready to investigate Rubik’s cube.
Jamie Mulholland, Spring 2011
Math 302
19-5
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
19.2
Sections
L ECTURE 19
Part/ References
Topic/Sections
from FS2009
Impossible
Moves
I.1, I.2, I.3
Combinatorial
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Complex Analysis
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
Through
previous
investigations
we’veUnlabelled
found structures
that there are some moves that are impossible to do on the cube.
Structures
2
14
I.4, I.5, I.6
FS: Part A.1, A.2
Figure 6 shows five moves
that
are
impossible.
This will be helpful when coming up with a strategy to solve
Comtet74
21
II.1, II.2, II.3
Labelled structures I
Handout
#1 is impossible to do, will prevent us from going on a search we would never come
the3 cube
since
knowing
what
(self study)
4
28
II.4, II.5, II.6
Labelled structures II
back
from.
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
10(a)
single edge flip
9
IV.5 V.1
VI.1
12
A.3/ C
18
IX.1
20
IX.2
23
IX.3
25
IX.4
13
30
IX.5
14
Dec 10
11
12
Asst #1 Due
Multivariable GFs
Singularity Analysis
Asymptotic methods
Asst #2 Due
Sophie
(b) edge swap
(c) single corner
Mariolys
twist
Introduction to Prob.
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
(f) single edge flip
Limit Laws and Comb
Marni
Discrete Limit Laws
Sophie
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Sophie
Presentations
(g) edge swap
(d) double corner
twist in same direction
(e) corner swap
(i) double corner
twist in same direction
(j) corner swap
Asst #3 Due
(h) single corner
twist
Figure 6: Five different moves that are impossible to perform. The image in the bottom row is a face-on perspective of the
top face of the corresponding cube in the top row. The thin rectangular boxes on the sides indicate the colour of the side
facets, and the long rectangular box indicates the side face colour.
We’ve given permutation-parity arguments to show why it is impossible to (i) flip an edge, (ii) swap two edges,
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
and
swap two corners. The impossibility of the corner twist configurations were investigated using SAGE.
Version(iii)
of: 11-Dec-09
In a later lecture we will come back to these configurations and give mathematical proofs that they are indeed
impossible. Therefore confirming the computations done by SAGE. Since we were relying on group theoretic
algorithms in SAGE, that we don’t know/understand, providing an independent proof will provide us will some
closure on this topic.
19.3
A Catalog of Useful Move Sequences
Over the previous few lectures we have built some useful moves using commutators. These were move sequences that affected only a few pieces, while returning everything other piece to the position it started. Using
conjugation we are able to modify these move sequences to produce other useful moves of the same form. Below
is a list of the moves we’ve created for convenient reference.
Notice that for each type of cubie (corners and edges) we can (i) 3-cycle any three cubies of the same type, and
(b) twist/flip a pair of cubies of the same type. Knowledge of these moves is enough to solve the cube: first
place cubies in their home locations (using 3-cycles), then orient the cubies in their home orientation (using
twist/flip moves).
Reminder: [x, y] = xyx−1 y −1 is the commutator of x and y and y −1 xy is the conjugate of x by y. In the following
tables, the move labeled C/E# is created using commutators, and the corresponding move denoted by C/E#’ is
the conjugate of it by the indicated move sequence y.
Jamie Mulholland, Spring 2011
Math 302
19-6
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
19.3.1
Sections
name
I.1, I.2, I.3
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
11
12
13
14
C1
Combinatorial
effect Structures
2
10
Part/ References
from FS2009
Corner
Moves
IV.5 V.1
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
L ECTURE 19
Topic/Sections
Notes/Speaker
Symbolic methods
move-sequence
Unlabelled structures
Labelled structures I
Labelled structures II
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Complex Analysis
Multivariable GFs
2 −1
[LDAsst
L #1 ,Due
U]
= LD2 L−1 U LD2 L−1 U −1
Singularity Analysis
Asst #2 Due
Asymptotic methods
9
VI.1
12
C1’
A.3/ C
Introduction to Prob.conjugate
Mariolys
18
IX.1
Limit Laws and Comb −1Marni 2
Sophie
20
IX.2
23
IX.3
25
IX.4
30
IX.5
B
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
C210
Dec
R UBIK ’ S C UBE : B EGINNINGS
Discrete Limit Laws
[LD L−1 , U ]B
Sophie
= B −1 LD2 L−1 U LD2 L−1 U −1 B
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Presentations
C1 by y = B:
Sophie
[F
−1
−1
D #3 F
R
Asst
Due
−1
D2 RF −1 DF, U ]
= F −1 D−1 F R−1 D2 RF −1 DF U F −1 D−1 F R−1 D2 RF −1 DF U −1
C3
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
C3’
[L−1 D2 LBD2 B −1 , U ]
= L−1 D2 LBD2 B −1 U BD2 B −1 L−1 D2 LU −1
conjugate C3 by y = B −1 :
B[L−1 D2 LBD2 B −1 , U ]B −1
= BL−1 D2 LBD2 B −1 U BD2 B −1 L−1 D2 LU −1 B −1
Jamie Mulholland, Spring 2011
Math 302
19-7
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
19.3.2
Sections
Part/ References
fromMoves
FS2009
Edge
I.1, I.2, I.3
4
Combinatorial
Structures
name I.4, I.5, effect
14
I.6
FS: Part A.1, A.2
Comtet74
21
II.1, II.2, II.3
Handout #1
(self study)
28
II.4, II.5, II.6
5
E15
Oct
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
2
3
IV.5 V.1
L ECTURE 19
Topic/Sections
Notes/Speaker
Symbolic methods
move-sequence
Unlabelled structures
Labelled structures I
Labelled structures II
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Complex Analysis
Multivariable GFs
2
[S`RAsst
, U#1
] Due
2
= S`R U 2 S`−1
R U
Singularity Analysis
Asst #2 Due
Asymptotic methods
9
VI.1
12
A.3/ C
Introduction to Prob.
18
IX.1
Limit Laws and Comb 2 Marni
−1
20
IX.2
23
IX.3
25
IX.4
13
30
IX.5
14
Dec
E210
10
E1’
11
12
Sophie
Mariolys
conjugate E1 by y = DR2 :
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Discrete Limit Laws
R D
[S`R , U 2 ]DR2
Sophie
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Sophie
Presentations
E2’
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
19.4
R UBIK ’ S C UBE : B EGINNINGS
−1
2
Asst
#3 Due
[S`−1
DS`
S`−1
RD
R
R D S`R , U ]
conjugate E2 by by y = B −1 R−1 :
2
−1 −1
−1
R
S`−1
RB[S`−1
R D S`R , U ]B
R DS`R D
Strategy for Solution
Our primary goal is in understanding the cube. With that goal in mind we should come away with a strategy
for solving the cube. We will not find an optimal strategy, nor will will look for a large collection of moves
to tackle all sorts of configurations. Instead, we will be content with a method that systematically solves
the cube and uses the tools we have developed in this course. Ideally the method should not involve lots of
memorization, but should rely on a solid understanding of the mathematics of permutations (i.e. commutators
and conjugates).
If you haven’t already tried to use the moves listed in Section 19.3 to find a strategy yourself, try it now. The
fun of discovering a solution on your own may be lost if you read the strategy described below.
More efficient methods than the ones described here, all of which require memorization, are left for the reader
to find. A simple google search can keep you busy for weeks.
Jamie Mulholland, Spring 2011
Math 302
19-8
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
19.4.1
Sections
L ECTURE 19
Part/ References
FS2009 Method
Thefrom
Layer
I.1, I.2, I.3
Topic/Sections
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
Symbolic methods
Combinatorial
The
method
we will use
to solve the cube
is known as the layer method. We begin by solving the top layer,
Structures
2
14
I.4, I.5, I.6
Unlabelled structures
Part A.1, A.2
followed by the middleFS:
layer,
and
finally
the
bottom layer. A sketch of the steps involved in implementing this
Comtet74
3
21
II.2, II.3
Labelled structures I
#1 7.
strategy
areII.1,
shown
in Handout
Figure
(self study)
4
28
II.4, II.5, II.6
Labelled structures II
6
12
IV.1, IV.2
(self-study)
Multivariable GFs
7
19
IV.3, IV.4
26
9
Nov 2
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Complex Analysis
8
You may begin by solving any colour, and it is best to choose a colour that stands out to you from the rest. This
Combinatorial
Combinatorial
5 itOct
III.1,to
III.2find the pieces on the scrambled cube.
AsstIn
#1 these
Due
way
is5easy
notes we’ll begin by solving the blue layer, in
parameters
Parameters
A.III will be green.
which case the bottomFSlayer
10
11
12
IV.5 V.1
Singularity Analysis
Asymptotic methods
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
Sophie
Mariolys
18
IX.1
Limit Laws and Comb
Marni
Random Structures
20
IX.2
(a)
Step 1: Solve
edges
in top
and Limit
Laws
layer.
FS: Part C
23
IX.3
25
IX.4
13
30
IX.5
14
Dec 10
(rotating
presentations)
Discrete Limit Laws
Sophie
(b) Step 2: Solve
corners in top
layer.
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Sophie
Presentations
(c) Step 3: Solve edges in middle layer
Asst #3 Due
(d) Step 4: Flip over, solve remaining corners (first permute
then orient).
(e) Step 5: Solve remaining
edges (first permute then orient).
Figure 7: The Five-Step strategy for solution.
Solving the top and middle layers are pretty straightforward. You should be able to do this with a little practice
won’t be needed until the end-game, which is when we
reach the bottom layer.
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version
of: 11-Dec-09
and using
general heuristics. A theory based strategy
19.4.2
Solving the Top Layer
Solving the top layer is a straightforward task. You can send pieces to the bottom layer, then bring them back
to the top layer, to achieve desired twists. That is, make use of conjugation.
Step 1: Solve the edge cubies in the top layer.
Keep in mind that centres remain fixed, so there is only one proper home orientation for an edge cube. Use
the centres as a guide. This is indicated in Figure 7a where the centres are shown, and the facets of the edge
cubies must match the centres.
Step 2: Solve the corner cubies in the top layer.
Let α be any of the moves R, L, F , B. This will bring one corner cubie into the down layer. Rotating the down
layer will then bring a new cubie into the cubicle whose contents are moved back up to the top layer by α−1 .
In other words, αDα−1 allows you to change a corner cubie in the top layer without affecting any other cubies
in the top layer. This should help you finish the top layer completely.
Jamie Mulholland, Spring 2011
Math 302
19-9
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
19.4.3
Sections
Part/ References
from FS2009
Solving
the Middle Layer
I.1, I.2, I.3
Combinatorial
L ECTURE 19
Topic/Sections
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
Symbolic methods
Step
3:
Solve
the edgeStructures
cubies in the middle
layer.
2
14
I.4, I.5, I.6
Unlabelled structures
FS: Part A.1, A.2
Comtet74
We3 could
modify
of the move sequences
in Section
19.3 to solve edges in the middle layer. However, this
21
II.1, II.2, some
II.3
Labelled structures
I
Handout #1
may
be
overkill,
since
at
this
stage
there
is
plenty
of
”wiggle
room” in the down layer so we should be able to
(self study)
4
28
II.4, II.5, II.6
Labelled structures II
find a general heuristic that works. Try to find one yourself.
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
9
VI.1
18
IX.1
Combinatorial
Combinatorial
(self-study)
Multivariable GFs
Asst #1 Due
parameters
Parameters
One method that is pretty
straightforward
is described here.
FS A.III
If the cubie that is to be placed in the middle layer is currently in the bottom layer then rotate the bottom layer
IV.4 edge
Complex
Analysisthe centre cubie of the same colour. For example, see Figure 8
so 7one19facet IV.3,
of the
cubie
is directly
beneath
Analytic
Methods
FS: Part B: IV, V, VI
where
the cubie to be moved
in
the
middle
layer
has a red facet on the side layer, so it is placed directly under
8
26
Singularity
Analysis
Appendix B4
IV.5 V.1
Stanley 99: Ch. 6the colour of your cubie is, rotate the entire cube so the cubie is now in the f d
the9 red
center
cubie.
Whatever
Nov 2
Asst #2 Due
Asymptotic methods
cubicle, and the colourHandout
of the#1facet in the f face matches the centre cubie of the f face right above it.
(self-study)
Sophie
10
Depending
on
the cubie is to Introduction
be moved
to theMariolys
right of the left we can apply one of the two sequences:
12
A.3/whether
C
to Prob.
11
right:
−1
Limit −1
Laws and
Comb
[D
,R
][D, FMarni
] = D−1 R−1 DRDF D−1 F −1
Random Structures
Discrete Limit −1
Laws
Sophie
left:
[D, L][D , F −1 ] = DLD−1 L−1 D−1 F −1 DF
and Limit Laws
FS: Part C
Combinatorial
23
IX.3
Mariolys
(rotating
of discrete
12
Notice
that in either presentations)
case the move instances
sequence
is a product of Y and Z commutators
25
Continuous Limit Laws Marni
discussion
ofIX.4these commutators).
20
IX.2
13
30
14
Dec 10
Quasi-Powers and
Gaussian limit laws
IX.5
Presentations
Sophie
Asst #3 Due
Figure 8: Moving an edge piece into the middle layer.
[D, L][D
−1
,F
−1
(see Lecture 13 for a
To move right apply [D−1 , R][D, F ], to move left apply
].
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
19.4.4
Solving the Bottom Layer
We now have one layer left to solve. This is the end-game of Rubik’s cube since it is here where things get a bit
more difficult. Trying to place the remaining few pieces while leaving previously places pieces alone requires
a collection of strategic moves: ones that move only a few pieces at a time. Luckily, the theory of commutators
and conjugates has provided us with such moves (see Section 19.3).
Flip the cube over, so the bottom layer is now the top layer. This will allow us to see everything we need to
solve.
Step 4: Solve the remaining corner cubies.
We’ll do this in two steps:
Step 4a: Place the remaining corner cubies in their home locations. Don’t worry about twisting them into
their home orientations just yet.
Look at the facets of each of the remaining corner cubies. The colours that appear will tell you exactly where
its home location is. For example, the corner cubie with green, white and red facets belongs to the location
which is the intersection of the green, white and red faces. Recall the colour of a face is given by the colour of
the centre cubie.
Jamie Mulholland, Spring 2011
Math 302
19-10
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
L ECTURE 19
Part/ References
Topic/Sections
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
from
FS2009where each corner cubie must be moved, see if a simple rotation of the up face will restore
Now that you
know
all1 corners
not, then
we are in one of the following cases:
Sept 7 to
I.1,their
I.2, I.3 proper locations. IfSymbolic
methods
Combinatorial
Structures
2
14 It isI.4,
I.5, I.6
Case
1:
possible
to
exactly
oneUnlabelled
cornerstructures
cubie in its correct location, and have the other 3 out of position.
FS:put
Part A.1,
A.2
Comtet74
Use
the
3-cycle
move
sequence
C1’
in
Section
19.3,
3
21
II.1, II.2, II.3
Labelled structures I or its inverse, to move the remaining 3 corner cubies into
Handout #1
their correct positions.(self study)
4
28
II.4, II.5, II.6
Labelled structures II
For example, if we need
to swap two corner
cubies as in the following diagram
Combinatorial
Combinatorial
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
10
11
IV.5 V.1
parameters
FS A.III
(self-study)
Parameters
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Complex Analysis
Asst #1 Due
Multivariable GFs
Singularity Analysis
Asymptotic methods
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
Sophie
Mariolys
18
IX.1
Limit Laws and Comb
Marni
Random
20 canIX.2
Limit Laws
then we
first rotate
theStructures
face so 1 Discrete
is home,
and 2, Sophie
3, 4 are out of position, then we just need to perform a
and Limit Laws
3-cycle (2, 4, 3).
FS: Part C
Combinatorial
12
23
IX.3
Case 2: Up to a
25
IX.4
13
30
IX.5
14
Dec 10
(rotating
presentations)
physical
rotation
instances of discrete
Mariolys
of the whole cube, we are in either one of the two following positions.:
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Sophie
Presentations
Asst #3 Due
The first case can be taken to the second case by rotating the face counterclockwise 90◦ . So assume we are in
the second case. Apply C1’ to produce the 3-cycle (1, 4, 2), which produces the following position.
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
Now use C1’ to produce a 3-cycle (1, 3, 2).
Therefore, to restore the corner cubies to their correct locations at most two 3-cycles need to be applied.
Step 4b: Orient (twist) the remaining corner cubies into their home orientations.
Repeated applications of C3 will be enough to orient the corners. (Note, we already used SAGE to discover it
is impossible to have exactly two corners twisted in the same direction.)
For example, denoting a corner cubie that must be rotated clockwise to be restored by +1, and one that must
be rotated clockwise by −1, here are a couple of possible scenarios that we could be faced with:
Jamie Mulholland, Spring 2011
Math 302
19-11
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
from FS2009
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
9
Nov 2
Part/ References
L ECTURE 19
Topic/Sections
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due
Multivariable GFs
In the first case, applying C3’ will solve the corners. In the second case, we can apply C3 on corners 1 and 4
7
19
IV.3, IV.4
Analysis
Methods 1 toComplex
to solve corner 4, and Analytic
take corner
−1. Then applying C3 to corners 1 and 2 will solve the remaining two
FS: Part B: IV, V, VI
8
26
Singularity
corners. Other
scenarios
areB4possible and canAnalysis
be dealt with similarly.
Appendix
IV.5 V.1
Stanley 99: Ch. 6
Asymptotic methods
Handout #1edge cubies.
Step 5: Solve the remaining
9
VI.1
12
A.3/ C
(self-study)
10 do this in two steps:
We’ll
Asst #2 Due
Sophie
Introduction to Prob.
Mariolys
Step 5a:
Place
the remaining edge cubies
in their home
locations. Don’t worry about flipping them into their
18
IX.1
Limit Laws and Comb
Marni
11
home
positions just yet.
20
IX.2
23
IX.3
25
IX.4
Random Structures
Discrete Limit Laws
(rotating
instances of discrete
Sophie
Limit Laws
Much like the corners,and
we
can
E1’ to restore all the edge cubies.
FS:
Part
C use 3-cycles
Combinatorial
Mariolys
12
Step
5b: Orient (flip) presentations)
the remaining edge cubies into their home orientations.
Continuous Limit Laws
Marni
Using E2 and E2’ we can flip any pairQuasi-Powers
of edges and
to restore to their home orientation.
13
30
IX.5
Gaussian limit laws
Sophie
Note, it is impossible to have a single edge flipped as we’ve already discovered. Therefore, flipped edges occur
Decand
10
Presentations
Asst #3 Due
in14pairs
so E2, E2’ are the
only moves we will need.
Congratulations! Not only have we solved the cube, we built the moves to do it from scratch! Behold the power
of the theory of permutations.
19.5
Exercises
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
1. Practice solving the first two layers of your cube. Repeatedly scramble and solve until you are confident
you can easily solve the first two layers.
2. Practice with Step 5: solving edges in final layer. In each part below, a configuration of the last
layer is shown. The only pieces out of place are the indicated edge pieces. All other non-visible cubies are
in their home orientations. Write down a strategy to solve the puzzle.
(a)
(b)
(c)
(d)
3. Practice with Step 4: solving corners in final layer. In each part below, a configuration of the last
layer is shown. The only pieces out of position are the indicated corner pieces. All other non-visible cubies
are in their home orientations. Write down a strategy to solve the puzzle.
Jamie Mulholland, Spring 2011
Math 302
19-12
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
from FS2009
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
7
19
L ECTURE 19
Part/ References
Topic/Sections
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
Parameters
IV.1, IV.2
Combinatorial
parameters
FS A.III
(self-study)
IV.3, IV.4
Analytic Methods
Complex Analysis
Handout #1
(self-study)
Asymptotic methods
R UBIK ’ S C UBE : B EGINNINGS
Notes/Speaker
Unlabelled structures
Labelled structures I
Labelled structures II
(a)
Multivariable GFs
Asst #1 Due
(b)
(c)
FS: Part B: IV, V, VI
4. Impossible
Configurations.
In
each part below, a configuration of the last layer is shown. All non26
Singularity Analysis
B4
V.1
visible IV.5
cubies
areAppendix
in
their
home
orientations.
Show that each configuration is impossible.
Stanley 99: Ch. 6
8
9
10
11
12
13
14
Nov 2
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
30
IX.5
Quasi-Powers and
Gaussian(b)
limit laws
Sophie
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
(a)
Dec 10
Sophie
Presentations
(c)
(d)
Asst #3 Due
(Hint: Try showing the configuration is equivalent to one shown in Section 19.2.)
5. Practice with Steps 4 and 5: solving corners and edges in final layer. In each part below, a
configuration of the last layer is shown. Some edge and corner pieces are out of position. All other
non-visible cubies are in their home orientations. Write down a strategy to solve the puzzle.
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
(a)
Jamie Mulholland, Spring 2011
Math 302
(b)
19-13