The idea of Non-Transitive Dice has been around for a while. The basic premise is that Red beats Blue, and Blue beats Green, so we expect that Red will beat Green. However, as the Non-Transitive suggests, this is not the case, and actually Green beats Red.
There are many sets of 3 Non-Transitive Dice, and one way to introduce them would be to use the trusty NRICH. This introduces the idea to students, and lets them play around with them a little bit.
However, as interesting as the Dice are in themselves, we want to get at the maths behind them.
The video below is of James Grime of the University of Cambridge. He starts by explaining a 3 Non-Transitive Dice game, and goes on to look at a 5 Dice game that he has invented. There is also his full article on the Grime Dice as well. And you can buy an amazing set of the dice from mathsgear.
As soon as I saw these dice, I immediatly thought about the lesson I could do on Probability using them. As seen in the video, all the probabilities are just combined independent events, so we can use Tree Diagrams (or sample spaces) to calculate the theoretical probabilities.
But we want to enjoy the practical nature first, so get students to play around in pairs with the dice and record how many times each dice wins. We can then bring all this experimental data together, before comparing it to the theoretical values.
Of course, the most amazing thing for this set of Non-Transitive Dice (which is not true of all such dice) is that if we use two of each colour, the order of winning reverses. Again, this can be explored experimentally, before going on to look at the theoretical probabilities. This time the Tree Diagrams have 3 branches, and you need to do a smaller tree diagram to calculate the probability of the different results of the two same colour dice. Although more complex, the principles are the same, and still very achievable.
This set of 3 Non-Transitive Dice then extends nicely into the Grime Dice, the set of 5. The methods are the same, but more calculations need to be done. Perhaps split the calculations amongst the class.
I have prepared a powerpoint presentation that can be used with a class for looking at these amazing dice. There is also an excel spreadsheet that can be used to work out all the different possible probabilities, for the 3 dice and 5 dice game, as well as the single die and two dice variants.
I am looking forward to giving this activity a go in the classroom, and seeing how it works. If you have a go with it, then please comment to let me know how it went!
I am a maths teacher looking to share good ideas for use in the classroom, with a current interest in integrating educational research into my practice.