Rubik’s Cube Anniversary

Woke up today to see this fun article about Rubik’s cubes in the NYT:

Since my kids love playing with these cubes, I thought there must be a great Family Math project hiding in here somewhere.  After thinking about what might be a fun project for kids, I decided that counting the number of different arrangements of a 2x2x2 cube might work well.

We  really haven’t done much counting / combinatorics yet, so it seemed like the best place to start was with a much more straightforward problem – counting the number of arrangements of blocks in a line.  That meant instead of Rubik’s cubes, the math starts with counting arrangements of snap cubes.  I think kids will enjoy trying to figure out the two problems we pose in this video – it was definitely fun for me to hear my son’s reasoning. Also, sorry for the glare in the first two videos, I didn’t notice it until I was turning off the camera after the 2nd video.

The next step before we get to the cube is counting the arrangements of snap cubes in a square rather than in a line.  This problem allows us to talk a little about symmetry.  The problem here is still pretty easy to understand, but understanding the symmetries makes it a little bit harder than counting arrangements in a line.  My son struggled a little bit here, but hopefully those struggles are actually helpful in understanding why this problem is a little harder:

With that background we moved on to counting the possible arrangements in a 2x2x2 cube.  We used the snap cube counting as the a starting point, and were also lucky enough to have a broken cube handy to help us see how to build one from the pieces.   Counting the symmetries here is a little bit more difficult here, and made even more complicated by the fact that some of the arrangements of a 2x2x2 cube cannot be solved:


The very last step is figuring out which arrangements can be solved and which can’t.  A rigorous solution to this problem is a little bit outside of what I think my kids can understand, but from playing around with these cubes they to have some idea about the answer.  If you build up a 2x2x2 cube from scratch, you’ll either be able to solve it, or the best you’ll be able to do is nearly solve it with just one piece being rotated.  Since there are three possible rotations for each piece, 1 out of 3 arrangements is solvable, so for our final answer for the number of arrangements, we have to divide by 3.

This was a really fun project.  I’m a little sorry that I had to squeeze it in quickly since I’ve leaving for a short trip in 10 minutes, but we still had fun.  Lots of neat math hiding in these cubes!