Mathematics of the Rubik's Cube
This Sophomore Seminar will use the Rubik's cube as a
tool for studying group theory, which is the branch
of algebra concerned with symmetry and transformations.
- Class will meet Mondays and Wednesday at 12:15
in 380D, basement floor of building 380.
- Although this is a Sophomore preference
seminar, it should be of interest to math
majors in general.
The seminar will introduce students to Group Theory, which
is an important branch of algebra, in a "hands-on" way.
Important topics in group theory which can be illustrated
with the Rubic's Cube include subgroups, homomorphisms and
quotient groups, the symmetric and alternating group,
conjugation, commutators and Sylow subgroups. There is a
beautiful construction which can be illustrated using the
subgroup generated by rotations of two adjacent faces, namely
the construction of the outer automorphism of the symmetric
group of degree six. Finally, if time and the interests of
the audience permit, we may consider the question of how
quickly a sequence of random moves will scramble the cube,
a mathematically interesting problem.
Eventually these two sets of notes will be combined.
- January 12, 2009: Read Chapter 1 of newcube.pdf, then start Chapter 1 of
rubik.pdf. Let me know if you have any problems.
The handouts that were posted here before January 9, 2009
were left over from the previous time the class was taught
and will be revised. Please stand by!
Here is a computer program that produces Metapost output
used to create the figures in the second set of notes.
- drawrubik.tar.gz. Extract
the files, then see README.
- I recommend TeXmacs
for mathematical writing. It was used to produce the second
set of notes.
In this seminar students will have opportunities
to speak. The purpose of the student lectures will be to make
the class more interactive. The same goal could be accomplished
if we had a lively informal discussion of every topic. You
are not required to give a student lecture.
- Impossibility of a single corner twist.
- Impossibility of a single edge flip.
- Conjugation from a theoretical point of view: conjugation is a
homomorphism, the index of the centralizer is the number of conjugates.
- Conjugation from a practical point of view: conjugation is a
standard tool for the Rubik's cube. How conjugation can change the
effect of an operation.
- Your cube algorithm.
- Other topics of your choice.
If you are interested in giving a student lecture, it might
be good to talk to me at least briefly before giving the talk.
- My office hours: I will be available before or immediately
after class (Tu, Th 11:15-12:15 or right after class). I
can also find time MWF.
Where to get cubes
- In the past, Toys 'R Us in Redwood City has had cubes.
There may be other places.