1. www.interactive-maths.com Notes for Teachers This activity is based upon Non-Transitive Dice, and is an excellent exploration into some seemingly complex probability. The three dice version has been around for a while, but with different numbers on the dice. The version here is so it fits with the 5 dice version. (If you have a three dice set, the probabilities in each case are the same, it is just the numbers on the dice that need changing in the Tree Diagrams) It is best done using the Non-Transitive Dice, which you can buy from http://mathsgear.co.uk/collections/dice/products/non-transitive-grime-dice http://mathsgear.co.uk/collections/dice/products/non-transitive-grime-dice You could also make the dice as a starter activity, and a recap on nets (just use different coloured card, and remember to put the correct numbers on each die). The slides talk the students through what they need to do, and I have put some comments on ideas for questions and practicalities in the notes box. The Grime dice (5 dice set) were discovered by James Grime of the University of Cambridge, and his video description and article can be found at http://grime.s3-website- eu-west-1.amazonaws.com/http://grime.s3-website- eu-west-1.amazonaws.com/ This slideshow is an attempt at a teacher friendly, usable in the classroom, way of presenting this information. The spreadsheet calculates all the probabilities and allows users to change the values on the dice. There is another great way to introduce Non-Transitive dice at http://nrich.maths.org/7541 http://nrich.maths.org/7541 For more interactive resources, visit my website at http://www.interactive-maths.com/http://www.interactive-maths.com/

2. www.interactive-maths.com

3. We saw that RED beats BLUE. How did RED and BLUE compete? We saw that BLUE beats GREEN. How did BLUE and GREEN compete? What do we expect in the RED vs GREEN games? We expect that since RED beats BLUE and BLUE beats GREEN, then RED will beat GREEN. This is called a Transitive Property – the win is transferred through the blue! Numbers are transitive: if 5 > 3 and 3 > 1, then 5 > 1!

4. www.interactive-maths.com We see that GREEN beats RED. What actually happened in the RED and GREEN games? BEATS

5. www.interactive-maths.com First we need to know what numbers are on each die. 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 Now we can use our knowledge of probabilities to calculate the probability in each battle. We shall use a tree diagram to consider the multiple outcomes.

6. www.interactive-maths.com 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 RED vs BLUE RED 4 9 BLUE 2 7 2 7

7. www.interactive-maths.com 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 Use the values on the three die to make two further Tree Diagrams to show that the Dice are indeed Non- Transitive.

8. www.interactive-maths.com 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 BLUE vs GREEN BLUE 2 7 GREEN 0 5 0 5

9. www.interactive-maths.com 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 GREEN vs RED GREEN 0 5 RED 4 9 4 9

10. www.interactive-maths.com

11. We saw that BLUE beats RED. How did RED and BLUE compete? We saw that GREEN beats BLUE. How did BLUE and GREEN compete? We saw that RED beats GREEN. How did GREEN and RED compete?

12. www.interactive-maths.com With two dice, the rules are a little bit different! BEATS Let’s have a look at the probabilities again!

13. www.interactive-maths.com 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 RED vs BLUE (two dice) 8 18 4 14 4 13 9 9 4 14 9

14. www.interactive-maths.com 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 BLUE vs GREEN (two dice) 4 14 0 10 0 9 5 5 0 5

15. www.interactive-maths.com 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 GREEN vs RED (two dice) 0 10 8 18 8 5 13 8 18 13

16. www.interactive-maths.com BEATS Remember the word lengths get bigger: RED (3) -> BLUE (4) -> GREEN (5) How to Use this Game Place the three dice out, and get a friend to play. Ask them to choose a die to use, and you then pick the one which will beat it. Role the dice 20 times, and you should win. Once they think they have worked it out, agree to take the die first. When they pick a die, if you are to win, leave it be, but if you are to lose say that you want to “double the stakes” with a second die each. This reverts the order! 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5

17. www.interactive-maths.com This is a set of 5 Non-Transitive Dice What do you notice about the dice? The 3 dice set is included within the 5 dice set. The numbers 0-9 appear on exactly 1 die. 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 3, 3, 3, 3, 8, 8 1, 1, 6, 6, 6, 6 This set of dice are called Grime Dice, after their discoverer, James Grime at the University of Cambridge

18. www.interactive-maths.com As with the 3 dice set, we can work out the probabilities in each pairing. How many different ways could we pair up the different coloured dice? RED with each of BLUE, OLIVE, YELLOW and MAGENTA BLUE with each of OLIVE, YELLOW and MAGENTA OLIVE with each of YELLOW and MAGENTA YELLOW with MAGENTA 4 3 2 1 We use OLIVE and MAGENTA instead of green and purple for a good reason we shall see!!!. So there are 10 possible pairings! We need to look at all of them!

19. www.interactive-maths.com Each pair has been given a colour pair to look at. Use a tree diagram to calculate the probabilities involved, and which colour will win. We already know three:

20. www.interactive-maths.com What do you notice? There are 2 chains that work for the 5 dice How do the names relate to the chains? Colour names get longer Colour names are alphabetical How do the probabilities compare?

21. www.interactive-maths.com BEATS

22. www.interactive-maths.com Notice that we can make several sets of 3 Non-Transitive dice by following paths on this graph. Each of these 5 subsets of dice will produce a valid set of 3 Non- Transitive Dice. They are obtained by taking 3 consecutive dice in the Word Length list.

23. www.interactive-maths.com We can also make sets of 4 Non- Transitive Dice! Each of these 5 subsets of dice will produce a valid set of 4 Non- Transitive Dice. They are obtained by taking 4 consecutive dice in the Alphabetical list.

24. www.interactive-maths.com

25. What do you notice? The Word Length Chain is reversed as expected. The Alphabetical Chain is in the same order Colour names get longer Colour names are alphabetical How do the probabilities compare?

26. www.interactive-maths.com This line is 50:50 either way

27. www.interactive-maths.com How to Use this Game Place the three dice out, and get a friend to play. Ask them to choose a die to use, and you then pick the one which will beat it. Role the dice 20 times, and you should win. Once they think they have worked it out, agree to take the die first. When they pick a die, if you are to win, leave it be, but if you are to lose say that you want to “double the stakes” with a second die each. This reverts the order! 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 3, 3, 3, 3, 8, 8 1, 1, 6, 6, 6, 6 Word LengthAlphabetical

28. www.interactive-maths.com Colour Title Background Info Some of the Maths Challenges Presentation Succinct Layout

29. www.interactive-maths.com A Special Game We can now use the set of 10 dice to play two players at once, and improve our chance of beating both of them Invite two opponents to pick a die each, but do NOT say whether you are playing with one die or two. If you opponents pick two dice that are next to each other on the alphabetical list (not next to each other around the circle), then play the one die game, and use the diagram to choose the die that will beat both most of the time. If you opponents pick two dice that are next to each other on the word length list (next to each other around the circle), then play the two dice game, and use the diagram to choose the die that will beat both most of the time.