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.

Catalog Description

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.

Lecture Notes

Eventually these two sets of notes will be combined.



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!

Computer Program

Here is a computer program that produces Metapost output used to create the figures in the second set of notes.

Student Lectures

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.

If you are interested in giving a student lecture, it might be good to talk to me at least briefly before giving the talk.

Office Hours

Where to get cubes