Talking through the Rubik’s Cube episode of the Mathematical Objects podcast with kids

Yesterday I listened to the new episode of the Mathematical Objects podcast:

I love listening to the discussions that Katie Steckles and Peter Rowlett have on this podcast. This episode made for an especially great project for kids, I thought. So, I had the boys listen to it after breakfast and then we talked about some of the ideas that they thought were interesting:

Now we dove into a tiny bit more detail about groups and modular arithmetic. Here I wanted to show the boys that the idea of an identity element was pretty important even though it seems like a pretty simple requirement when you see it for the first time:

Finally, we moved on to talking about some of the group theory ideas that relate to Rubik’s cubes. The specific idea we talked about from the podcast episode was “commutators”. We tried out three examples that – honestly by accident – turned out to be nice illustrations of the idea:

I really enjoyed the podcast and also learning about what the boys found interesting listening to it. It definitely is fun illustrating some basic ideas from group theory with Rubik’s cubes!

Studying symmetry with Rubik’s cubes and Zometool

Over the last week we’ve been looking informally at symmetry. The topic came up because we were studying counting with symmetry in Art of Problem Solving’s “Introduction to Counting and Probability” book. Questions such as “how many different ways can 5 people sit around a table, if ways that differ only by rotation are considered the same?” let do some nice discussions, but the slightly different question: “how many different ways can 4 keys be put on a key chain, if ways that differ only by rotation and flips are considered the same?” was a little more challenging to talk about. Wanting to talk about that second question in a little more depth led to the fun diversion into symmetry that we’ve been doing this week.

We started with symmetries of simple polygons and moved on to the symmetries of a cube yesterday. That jump was much more difficult than I expected. We talked a little bit about cubes in the morning, they boys thought more about it during the day, and we had this nice conversation last night when I got home from work:

Today I wanted to move on to symmetries of an octohedron and got a nice surprise when they boys remembered looking at octohedron’s with our Zometool set earlier this summer.

As background, we go on a little vacation with college friends every year. This year the weather forecast for the first couple of days was lots and lots of rain (I think it was the remnants of a tropical storm), so I brought our Zometool set along hoping that it would be a fun way to pass the time with all of the kids. We also brought along this book that Patrick Honner had recommended on Twitter:

Book Pic

We did a few projects from the book as well as some other projects that probably would be best described as having the kids just play around. One of the neat projects that came from the kids playing around was this one from a 12 year old girl who assured me that she “hates math.”  It was really fun to talk to her about how her creation showed that you can fit 6 pyramids inside of a cube:

Shape

One of the projects from the book that we did involved building some Platonic solids.  I have to confess that I didn’t remember this project until the kids reminded me today if you want to build an octohedron, you start by making a “starburst.”  Bet you didn’t know that!

I’m glad that that the activity from the Zome Geometry book had stuck with them.  It was a much better starting point to talking about symmetries of an octohedron than what I had planned.   The “starburst” idea really helped them understand why those symmetries are the same as a cube:

I’m happy to see one more example of how the Zometool set helps kids understand shapes and geometry.  Happier still to see that they are also a great tool to help kids understand symmetries.  The list of useful recommendations from Patrick Honner is getting to be quite large!

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!

 

 

A little fun and a little math wtih Rubik’s cubes

Sometime last year my kids became fascinated by Rubik’s cubes.  Not sure why or how it happened, but once they started playing around a little they were hooked.  So much so that learning more about how to solve the cubes seemed like a fun topic to include as part of the school year, so we’ve been studying some of the 2×2 and 3×3 speed solving techniques for fun since September.   Even though I’m not practicing the speed solving with the boys, I’ve gotten a little hooked, too 🙂

Cubes

Following a few folks on twitter also led to some interesting Rubik’s cube related reading in the last year.  Christopher D. Long (@octonion) tweeted about the book “Adventures in Group Theory:  Rubik’s Cube, Merlin’s Machine & Other Mathematical Toys.”  Definitely a fun read if you are into math, though it’ll be a while before I can pull much from it for the boys.

I also ran across Cathy O’Neil’s (@mathbabedotorg) old post about math contests :

Math contests kind of suck

This comment really struck me  – “I have never been particularly fast at working out the details of something from the conceptual understanding (for example, it takes me a long time to solve a 7x7x7 Rubik’s cube) but it turns out the Rubik’s cube doesn’t mind. And in fact mathematics in real life isn’t a timed tests- the idea that you need to be original and creative really quickly is just a silly, arbitrary way to select for talent.”  (as an aside, you should definitely follow her blog and follow her on twitter – you’ll not find a more interesting blog.)  I agree with Cathy O’Neil’s point that there’s not anything special about solving the cubes fast.  The kids seem to like it and enjoy learning the techniques, but it is mostly just a matter of practice.  That said, the world record solve times (~5.5 seconds for solving the 3×3, for example) really are  mind blowing.

So, what can kids learn from these cubes?

There is quite a bit of interesting math related to the cube solving algorithms (see the book mentioned above).  A simple introduction to these algorithms probably has some benefit, but I’m aiming a little lower right now.

One interesting advanced topic is parity.  This position on the 3x3x3 is impossible to solve:

Cube Reverse

You can not create a position with just one middle reversed with legal moves.  In order to make this postion, you have to take the cube apart.

However, this position on the 4x4x4 cube, which seems pretty similar to the picture above, is solvable:

4x4 Reverse

At least for my (very, very very slow) solving technique, figuring out how to solve the 4x4x4 from this position was the final obstacle to overcome in learning how to solve the larger cubes.

For my younger son, it turned out that the cubes were also fun tools for learning about topics like fractions:


ratios:


and exponents:


 

** Update **  Imaginary and non-commutative numbers!!


It isn’t hard to believe that kids will be more excited about learning when they are having fun, but it great to see that excitement in practice.    Of course, it has also been really fun for me to use the Rubik’s cubes to help teach a bunch of different math topics.  Maybe one day we’ll even be able to replicate something like this 🙂