Oct 6
God's algorithm
A team of folks using computing donated by google (35 cpu years) has run through
all positions in what's know as the "face-turn metric" (quarter and half turns
of a face count as one move), and found that any position can be solved in 20 moves.
Discuss "quarter-turn" vs "half-turn" metric.
There's a bit more discussion
here and
here.
I'm not aware of anyone completing the algorithm for the quarter turn metric
yet. Here's
a recent result.
Commutators and inverses of move sequences
What is the commutator of R F and U F ?
Answer: In commutator notation, we'd write [ RF, UF ],
which is (RF) (UF) (RF)' (UF)'
where (RF)' means the inverse of the whole sequence
clockwise-right-face then clockwise-front-face.
But to undo those, we'd need to undo them
in the opposite order: (RF)' = F'R' .
So the commutator is (R F) (U F) (F' R') (F' U') .
The F F' in the middle does nothing, so
this can be written [ RF, UF ] = R F U R' F' U' .
Be clear about the reverse order thing: (A B C)' = C' B' A' ,
where as in our notation from last week, the tick
mark means "undo" or "inverse".
2 x 2 x 2
To enter in sequences :
Some online solutions :
Work through a solution that one of you has looked at, or examine one of those in CubeTwister, discussing order of sequences, commutators, conjugacy.
3 x 3 x 3
Start talking about the "real" Rubik's cube.
Reminder about parity and how that works
with edges and corners positions.
Reminder about "twistiness" of corners.
Likewise, discuss "flippiness" of edges,
and argue that you cannot flip one edge.
(But you can flip two.)
From that, work out the total number of positions.
- What is the number of we pay attention to the orientation of the centers?
- How about if we allow slice rotations?
For next week: explore 3x3x3 in similar ways to this week's 2x2x2 assignment.
introduce some new group theory
Start a discussion of subgroups and quotient groups
(ideas only for now) and how they connect with
cube solution techniques.
Example:
The dihedral group D3 = C3 cross C2.
C2 = D3 / C3 : quotient group; order (2 = 3/6)
D3 = C3 X C2 : semi-direct product; order (6 = 3 * 2)
For technical reasons, the math
definitions don't work
the other way around in this case.
(The explanation depends on
notions of conjugacy classes and normal subgroups -
you're really going to like that part.)
When you say things like "solve only the top, ignoring
the rest", a group theorist would say that you're
looking for one element of a quotient group.
(The other elements of that quotient group have
the cubies that go on the top all over, but still
ignore all the rest, just like the quotient group
D3 / C3 only cares whether it's right side up
or upside down, and ignores which rotation.)
In a subgroup, each group element is one element
from the larger group. (Example: the right-side
up rotations of a triangle are a subgroup of D3.)
In a quotient group, we have new group elements
each of which is made up of many of the original
group elements. (Example: the three right-side
up triangle positions are all "face-up";
the three upside-down are all "face-down".
The quotient group has two elements, "face-up" and "face-down".)
in class
I talked about one way to think about
constructing move sequences :
(lots of stuff) (small) (undo lots of stuff) (undo small)
where the (lots of stuff) is designed to leave one layer alone except for a tiny change, and the (small) rotates that layer.
In class I gave a few examples:
twist two corners :
F D2 F' R' D2 R # top untouched except corner twist
U2 # rotate top
R' D2 R F D2 F' # undo mess; twist different corner
U2 # undo rotate top
swap three corners :
F D2 F' # pull a corner off the top
U # rotate top
F' D2 F # restore corner to different top spot
U' # undo rotate top
