A commenter on a recent blog post suggested that the boys might like playing around with non-transitive dice. I loved the suggestion and was super excited when our set arrived yesterday. Today’s project didn’t really have a lot of structure, rather we just played around and looked at the non-transitivity.

These dice were discovered / invented by James Grime, who the boys know from the Numberphile videos about Alan Turning and the Enigma Machine.

First – nearly an unboxing. I’d looked at the dice yesterday but the kids hadn’t seen them yet. What would they notice about these dice?

Before jumping into some of the properties of these dice, I wanted to remind the boys what the word “transitive” meant. We talk about the transitive property in math just for a little bit at the beginning of arithmetic, but it sort of disappears from the radar screen since just about everything you see in school math has the transitive property. What are we talking about again?

Now back to the dice. We wanted to see what would happen if we rolled one of the dice against one of a different color. Which die would we expect to roll a higher number? Our initial competition was red vs. yellow. After that we studied red vs. purple.

Not surprisingly, playing around with these dice is not just an opportunity to talk about transitivity, but also a nice lesson in probability (and fractions!) for kids.

Also, sorry for my total inability to write while reaching through the tripod in this project. Sheesh!

Now we had two colors left to test – green and blue. We first tested green against the purple, red, and yellow dies. We found that it would win against the red one 25 / 36 times. That led my younger son to suggest the greed would also win against purple since purple had beaten read only 20 / 36 times.

Interesting conjecture – let’s test that idea and see what happens!

Next we introduced the blue die to the competition and found that it could slot in between the red and the green. That meant we had ranked the colors in this way:

Purple > Red > Blue > Green > Yellow

But we never tested the purple die against the yellow one. The ordering above might lead you to believe that such a test isn’t necessary, though. How could the yellow die win against the purple given the current rankings . . . . 🙂

For the grand finale we have the purple versus yellow battle. It seems that there’s no way that yellow can win, but . . . . surprise!

So, thanks to the comment from Joshua, we ended up with a nice project. I really like projects like this that are (i) fairly simple, and (ii) very much non-intuitive. Made for a fun morning (despite the rather unenthusiastic sign off) – ha ha.

Just recently found and started following your blog. You are doing great work! As a math teacher and mother of 3, I’m getting some great ideas to try with my own kids. Thanks for sharing your journey.

I wonder if they can come up with any cool games to play involving the NTGD or whether any existing games become interestingly different when played with these?

In case it doesn’t come up, i thought this was a good discussion of non-transitive relations on stack exchange.

Before I read that, I could come up with many non-mathematical examples, but no “interesting” mathematical examples (only contrived versions). I’ll be interested to follow-up the reference to Poincare’s work on tolerance relations. Also, I enjoy saying it like, “I tolerate you, you tolerate your friend, but I can’t tolerate your friend!”

Lastly, the link to economics is interesting to consider: do people have transitive preferences or are there non-transitive examples? A related idea is, if A is preferred to B, is A+C preferred to B+C? For me, blueberries are pref to ketchup, but I prefer hamburgers+ketchup to blueberries plus hamburgers.

FWIW, I think your dice ordering was a little off. We get so used to transitivity that we need to be ultra careful trying to write a non-transitive relationship in a chain like that!