2. Into three dimensions
I often descibe the
Puzzle Square as a two
dimensional version of
Rubik’s cube.
Now it is time to move
into three dimensions,
but with all the power of
Group Theory to help.
Lars Karlsson (Keqs) / CC BY-SA
(http://creativecommons.org/licenses/by-
3. Flat world
It can be hard to visualise (and
to draw) three dimensions, so
I’m going to use this flattened
view of two sides.
Just like we focused initially
on a single row and column
of the Puzzle Square
4. Basic moves
Here’s the effect of first of all rotating the red
face clockwise followed by the blue face.
5. Labels
To be honest I found it pretty
difficult keeping track of the
colours!
So I’m going to simplify things
and use labels: A. B. C, … for
the corners and a, b, c, … on
the edges the centres don’t
move anyway.
A a B
C D
E F
b c
d
e f
g
6. Moves on labels
Now we can just keep track of the letter moves.
I’ll even drop the colours entirely and focus just
on the label letters.
C b A
D B
E F
d a
c
e f
g
A a B
C D
E F
b c
d
e f
g
A a B
b c
C d D
e f
E g F
C b A
d a
D c B
e f
E g F
7. Changing notation
Often changing notation can help make things
easier to understand in mathematics.
… and to make things even easier I’ll focus first
just on the corners, and then just on the sides.
A a B
b c
C d D
e f
E g F
A a B
b c
C d D
e f
E g F
A a B
b c
C d D
e f
E g F
8. Back to commutators
Yes we’ll use out old friend
the commutator again!
Remember this was:
1. Do something
2. Do something else
3. Do the opposite (inverse) of step 1
4. Do the opposite (inverse) of step 2
A a B
b c
C d D
e f
E g F
+
+
+
9. Corners first
See how the commutator of the red
and blue face twists ends up only
moving a few of the corners. It swops
C and E and also swops B and D, that
is the permutation (B,D)(C,E)
A B
C D
E F
C A
D B
E F
C A
E D
F B
A D
C E
F B
A D
E B
C F
A D
E B
C F
10. Now the edges
The commutator leaves the edges a,
b, f and g where they started and just
moves c, d, e in the cycle permutation
(c,d,e).
a
b c
d
e f
g
b
d a
e
g c
f
b
d a
c
e f
g
a
b e
d
g c
f
a
b e
c
d f
g
a
b e
c
d f
g
11. Putting it together
If we put this all together we get the
full effect on both corners and edges.
Remembering the [ , ] notation for
commutators we can write this:
a
b e
c
d f
g
A D
E B
C F
=
[ ]
, (B,D) (C,E) (c,d,e)
12. A a B
b e
C c D
d f
E g F
Steps to a solution
As the effect on the corners was to create two
swops: (B,D) (C,E), doing this sequence of
moves twice will move these back to where
they started, leaving just a move on the edges.
=
[ ]
, (c,e,d)
[ ]
,
+
13. Similarly, the effect on the corners was just a
cycle of three (c,d,e), so doing this sequence of
moves three times will move these back to
where they started, leaving just the swops on
the corners:
Steps to a solution (2)
= (B,D)(C,E)
[ ]
, [ ]
,
+ [ ]
,
+
A a D
b c
E d B
e f
C g F
14. A a D
b c
E d B
e f
C g F
So, we now have sequences of
moves that change some of the
corners without moving the edges
And moves that change the edges
without moving the corners
This is a great start point for solving
any cube.
Progress
A a B
b e
C c D
d f
E g F
17. Solve it!
We can now use the Z moves, I’ll call them Zcorner
and Zedge, to nearly solve the cube in two stages:
Stage 1 – use Zcorner to get the corners in the
right positions
Stage 2 – use Zedge, to get the edges nearly in the
right positions without disrupting the
corners
N.B. I say ‘nearly’, because there will be a few small, but
important tweaks left to do!
18. A
B
C
D
E
F
Stage 1a. Corners – top and bottom
If there is a corner piece on the bottom (orange
side) that should be on the top (red side):
1. Rotate the bottom so that the piece
you want to move is on position E.
2. Rotate the top so a bottom
piece is in postion C.
3. Do a Zcorner move
… and repeat until no red left on the bottom
19. The corner pieces need to be in the right order
relative to one another.
If you are careful, you can sometimes get part
way there is stage 1a, but if not,
we can combine the Zcorner
with its reflection.
We can call this reflection Scorner
Fix the corners – is the order right?
20. Reflection of a Z move
Recall that we made the Zcorner move, based on
the combinator of a clockwise twists of the red
and blue faces:
We can do exactly the same, but use an anti-
clockwise twist of the red side and yellow sides,
we get the refection of Zcorner.
[ ]
, [ ]
,
+ [ ]
,
+
Zcorner =
21. The Scorner move:
Note, we simply reflect each
basic move on the blue/yellow
edge, and got the reflected
overall move.
Check this works yourself
(on paper or a real cube)
[ ]
,
[ ]
,
[ ]
, + +
Scorner =
A
B
C
D
E
F
22. Put them together: Zcorner + Scorner
If we do a Zcorner move followed by a Scorner
we put the bottom piece (E) back where it
started and simply rotate A, B and D. The order
depends on which we do first.
A
B
C
D
E
F
A
B
C
D
E
F
B
D
C
A
E
F
+ =
23. B
D
C
A
E
F
Stage 1b. Corners – fix order
We can use Zcorner + Scorner to rotate three of
the top corners:
the permutation (A, D, B)
and simply twist the top to rotate
all four corners:
the permutation (A, B, C, D)
Recall from the 4x4 puzzle square,
if we can do both the 3 and 4 tile rotations,
we can do any permutation – done
24. Stage 2 – Edges – a few more moves
There are more edges – 4 on the sides as well as
4 on top and bottom, so there are several steps,
each using several moves:
2a. bottom edges on the the top to the sides
2b. bottom edges on the sides to the bottom
2c. fix edges on the sides
2d. fix edges on the top
25. a
b
c
d
e
f
g
Stage 2a – edges: top to sides
If there is a bottom piece on the top :
1. Rotate the cube so that there is a piece
that isn’t a bottom piece at
position e.
2. Rotate the top so that the bottom
piece is at positon d..
3. Do a Zedge move
… and repeat until no bottom bits left on top
26. a
b
c
d
e
f
g
Stage 2b(i) – edges: sides to bottom
Turn the cube upside down (orange on top):
1. Rotate the cube so that there is a piece
that isn’t a bottom piece at
position e.
2. Rotate the top so that the bottom
piece is at positon d..
3. Do a Zedge move
… and repeat until no bottom bits left on top
27. Stage 2b(ii) – edges: fix bottom
As you do the steps in stage 2b(i) try to get the
bottom edges in their right positions.
If necessary, use a Zedge move to lift a wrongly
positioned bottom edge piece up to aside and
then drop it back into the right position.
Don’t worry about twisting the bottom, you can
always twist it back so long as you only use Zedge
moves that do not alter the corner positions.
28. a
b
c
d
e
f
g
Stage 2c – edges: fix sides
If there are any side pieces on the top, use a
Zedge move to drop it into the right position.
Again feel free to twist the top as
much as you like and then simply
twist it back at the end.
If any side pieces are in the wrong
place, use Zedge moves to lift them to
the top and then drop them where they belong.
29. Is the top right?
By now all of the bottom pieces are in their
correct positions as are all of the side edges.
So, this means that all of the top pieces must be
on the top! We are close
However, like we found with the
corners, some of the top edges
may be in the wrong positions.
30. The Sedge move … just like Scorner
To finish off you may need Sedge moves, and you
make these just like we did for Scorner: reflecting
each basic move:
[ ]
,
[ ]
, +
Sedge =
a
b
c
d
e
f
g
31. a
b
c
d
e
f
g
Stage 2d(i) – edges: fix first top piece
1. Choose a piece in the wrong position, and twist the
top so that it is in position (d). Then use a Zedge
move to drop it into position (e).
2. Spin the top so that the correct location is at (d) and
use another Zedge to get it in the right place.
3. This leaves the side edge the belongs
at position (e) still on the top.
Spin the top so that you can do
either an Zedge or Sedge move to
put it back.
32. Stage 2d(iii) – number three?
So two top pieces are in their correct positions,
just the last two to go. Spin it round and check ...
… you might be lucky …
? ?
33. Stage 2d(iii) – oops
… but you might not! Just one swop to go.
No problem you think, just
do the same three steps as
for the first and second …
But no, at step 1 or 3 you’ll
find it impossible to do a
Zedge or Sedge move without
disturbing one of the ones
you’ve already put right
34. Yes, we’ve been there
before! Just like the
3x3 puzzle square, two
families.
Impossible to get from
one to the other
without disassembling
the cube.
Even and odd
+6 x 19 trillion
other cubes
+6 x 19 trillion
other cubes
even
odd
35. … and worse …
We’ve been focusing on the positions of the
pieces, they also have to be the right way round.
It turns out of a single edge is the wrong way
round, or a
single corner,
that cannot be
fixed either.
36. The odd and even
cubes each have six
smaller families.
2 x edge orientations
3 x corner orientation
Many families
19 trillion
cubes in each
19 trillion
cubes in each
even
odd
37. Prove it!
I won’t look at the edge and corner
orientations, but let’s prove that
you can’t swop the positions of
just one pair of edges.
For this, we already have all the power of
permutation groups to help us, and it will be like
revisiting old ground.
38. Corners and edges apart
Early on we worked as far as possible separately
on the corners and edges in order to make it
easier to understand.
The permutation of a basic
twist of one side is:
Corners: (A, B, C, D)
Edges: (a, b, c, d)
Both odd permutations.
A
B
C
D
a
b
c
d
39. A
B
C
D
a
b
c
d
Corners and edges together
If we look at the overall permutation considering
both of corners and edges it is:
(A, B, C, D)(a, b, c, d)
This is the combination of two
odd permutations, so it is even.
(A,D)(B,D)(C,D)(a,d)(b,d)(c,d)
40. Odd and even – proved it!
So very basic move is an even permutation, but
swopping two edges is a single swop, that is an
odd permutation.
We know that any combination
of even permutations is still even,
and hence no number of basic moves
can ever swop just one pair of edges.
QED
41. Wow!
That was hard work, when I read back I lose track
sometimes, and I wrote it … however …
1. This is precisely the reason for writing things down!
2. We’ve seen the power of simplifying and separating
concerns (corners and edges)
3. More packaging of moves
4. Families are not just about odd/even
5. … and one last thing …
42. Not the fastest!
Just like the puzzle square, this is a systematic way to
solve the cube, but by no means the fastest way!
There are optimisations we can do.
For example, stage 2 puts the edges right, so stage 1 need
not worry about them and use commutators rather than full Z
and S moves (4 instead of 12 twists each time)
In general …
Mathematics cares about what is possible
Computing worries about how long it takes
