Abstract Algebra

Not long ago, I received some advertising for class sets of Rubik’s cubes, which are sold along with a solving method. Soon afterwards, a correspondent wrote:

I see on your website that you have done some work with kids on Rubik’s cubes and group theory. I have purchased a class room sets of the Rubik’s Cubes and of the 15 sliding puzzle. I have saved the last two weeks of school to make a lesson out of this. […]

My goals for this gig: 1) learn the basic operation properties with some context. 2) create some understanding that mathematics can explain a thing like the cube. 3) put them in touch with some of the historical context for group theory. 4) have some chances for kids to go as deep as they can. 5) most importantly, have fun.

Do you have any lessons or resources for making this kind of thing happen?

My response:

I haven’t actually used Rubik’s cube in the classroom in a very long time. My sense is that it can only do what you are after if you introduce much smaller groups concurrently. The lessons on my Web site have been extraordinarily effective with a wide range of kids in middle and high school to introduce the concept of a group. You should work through them yourself and see how you like them.

The one lesson in that packet that I usually skip is “Magic Carpets”, which for some reason hasn’t seemed to click with high schoolers. (I used to have fun with it with 4th and 5th graders.)

Anyway, that sequence is far more effective than starting with the definition of a group. By the end, kids have worked with enough examples, and the examples look different enough from each other, that they can make sense of the formal definition, which I do introduce at the end of the packet. By then, students are more likely to have some sense of why the Rubik group transformations form a group, gigantic as it may be, and what that means.

You might have luck continuing on with some version of the remaining lessons on that same page on my site (isomorphisms, fields, and connection with fields students are already familiar with). I originally wrote those for teachers, but have successfully introduced isomorphisms to high school kids using that approach. (11th and 12th graders in my Space class.) Anyway, you can decide whether to forge ahead with this after you’ve gone through the initial packet.

The Rubik’s cube PDF on that same page is just my own “how-to-solve it” set of moves, assembled from various sources.

In any case, good luck with this project!


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