notes for teachers

www.interactive-maths.comNotes 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

• 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/

• 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

• For more interactive resources, visit my website at http://www.interactive-maths.com/

http://www.interactive-maths.com/

http://mathsgear.co.uk/collections/dice/products/non-transitive-grime-dice

http://grime.s3-website-eu-west-1.amazonaws.com/

http://nrich.maths.org/7541

http://nrich.maths.org/7541


www.interactive-maths.com

Dice GamesIn your pairs, you are going to play a

game.You each have a coloured die, and you

are going to both throw your die.

The player with the highest score wins that round.

Play 10 rounds.

Who is winning overall?

Play a further 90 rounds (100 in total).

Is the game fair?



What did we discover?

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!



We see that GREEN beats RED.

What actually happened in the RED and GREEN games?

BEATS

BEATS

BEATS

NON-TRANSITIVE DICE



WHY IS IT THAT THIS HAPPENS? LET’S TAKE A LOOK AT THE PROBABILITIES!

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.



4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5 RED vs BLUE

RED

4

9

BLUE2

7

𝟑𝟔

𝟑𝟔

2

7

𝟑𝟔

𝟑𝟔

𝟓𝟔

𝟏𝟔

𝟓𝟏𝟐

𝟓𝟏𝟐𝟏𝟏𝟐

𝟏𝟏𝟐

So RED wins over BLUE with probability



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.



4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5 BLUE vs GREEN

BLUE

2

7

GREEN0

5

𝟏𝟔

𝟓𝟔

0

5

𝟏𝟔

𝟓𝟔

𝟑𝟔

𝟑𝟔

𝟏𝟏𝟐

𝟓𝟏𝟐𝟏𝟏𝟐

𝟓𝟏𝟐

So BLUE wins over GREEN with probability



4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5 GREEN vs RED

GREEN

0

5

RED4

9

𝟓𝟔

𝟏𝟔

4

9

𝟓𝟔

𝟏𝟔

𝟏𝟔

𝟓𝟔

𝟓𝟑𝟔

𝟏𝟑𝟔𝟐𝟓𝟑𝟔

𝟓𝟑𝟔

So GREEN wins over RED with probability



Pair up with somebody with the same colour die as you.

Now make a group of 4 by joining another pair (there should be two dice of two different colours in your group).

We are going to play the game again, but taking the total of the same

coloured dice.

Play 100 rounds as before, and keep track of how many rounds each colour

wins.



What did we discover this time?

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?

THIS IS THE OPPOSITE TO WHAT HAPPENED WITH ONLY ONE DIE OF EACH COLOUR!!!



With two dice, the rules are a little bit different!

BEATS

BEATS

BEATS

Let’s have a look at the probabilities again!



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

𝟗𝟑𝟔

𝟗𝟑𝟔

4

14

𝟗𝟑𝟔

𝟗𝟑𝟔

𝟐𝟓𝟑𝟔

𝟏𝟑𝟔

𝟐𝟐𝟓𝟏𝟐𝟗𝟔

13

9

9

𝟏𝟎𝟑𝟔

𝟏𝟖𝟑𝟔

𝟏𝟖𝟑𝟔

4

14

𝟗𝟑𝟔

𝟗𝟑𝟔

9𝟏𝟖𝟑𝟔

𝟒𝟓𝟎𝟏𝟐𝟗𝟔𝟐𝟐𝟓𝟏𝟐𝟗𝟔𝟗𝟎

𝟏𝟐𝟗𝟔𝟏𝟖𝟎𝟏𝟐𝟗𝟔𝟗𝟎

𝟏𝟐𝟗𝟔𝟗

𝟏𝟐𝟗𝟔𝟏𝟖

𝟏𝟐𝟗𝟔𝟗

𝟏𝟐𝟗𝟔

So BLUE wins over RED with probability



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

𝟏𝟑𝟔

𝟐𝟓𝟑𝟔

0

10

𝟏𝟑𝟔

𝟐𝟓𝟑𝟔

𝟗𝟑𝟔

𝟗𝟑𝟔

𝟗𝟏𝟐𝟗𝟔

9

5

5

𝟏𝟖𝟑𝟔

𝟏𝟎𝟑𝟔

𝟏𝟎𝟑𝟔

0

10

𝟏𝟑𝟔

𝟐𝟓𝟑𝟔

5𝟏𝟎𝟑𝟔

𝟗𝟎𝟏𝟐𝟗𝟔𝟐𝟐𝟓𝟏𝟐𝟗𝟔𝟏𝟖

𝟏𝟐𝟗𝟔𝟏𝟖𝟎𝟏𝟐𝟗𝟔𝟒𝟓𝟎𝟏𝟐𝟗𝟔𝟗

𝟏𝟐𝟗𝟔𝟗𝟎

𝟏𝟐𝟗𝟔𝟐𝟐𝟓𝟏𝟐𝟗𝟔

So GREEN wins over BLUE with probability



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

𝟐𝟓𝟑𝟔

𝟏𝟑𝟔

8

18

𝟐𝟓𝟑𝟔

𝟏𝟑𝟔

𝟏𝟑𝟔

𝟐𝟓𝟑𝟔

𝟐𝟓𝟏𝟐𝟗𝟔

5

13

13

𝟏𝟎𝟑𝟔

𝟏𝟎𝟑𝟔

𝟏𝟎𝟑𝟔

8

18

𝟐𝟓𝟑𝟔

𝟏𝟑𝟔

13𝟏𝟎𝟑𝟔

𝟏𝟎𝟏𝟐𝟗𝟔𝟏

𝟏𝟐𝟗𝟔𝟐𝟓𝟎𝟏𝟐𝟗𝟔𝟏𝟎𝟎𝟏𝟐𝟗𝟔𝟏𝟎

𝟏𝟐𝟗𝟔𝟔𝟐𝟓𝟏𝟐𝟗𝟔𝟐𝟓𝟎𝟏𝟐𝟗𝟔𝟐𝟓

𝟏𝟐𝟗𝟔

So RED wins over GREEN with probability



SUMMARY

BEATS

BEATS

BEATS

ONE DIE

BEATS

BEATS

BEATS

TWO DICE

Remember the word lengths get bigger:RED (3) -> BLUE (4) -> GREEN (5)

How to Use this GamePlace 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



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



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!



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:

RED > BLUE with probability

BLUE > OLIVE with probability

OLIVE > RED with probability



𝑃 (𝒀𝑬𝑳𝑳𝑶𝑾 >𝑴𝑨𝑮𝑬𝑵𝑻𝑨 )=59 ≈56%

𝑃 (𝑩𝑳𝑼𝑬>𝑶𝑳𝑰𝑽𝑬 )= 712 ≈60%

𝑃 (𝑶𝑳𝑰𝑽𝑬 >𝒀𝑬𝑳𝑳𝑶𝑾 )=59 ≈56%

𝑃 (𝑹𝑬𝑫>𝑩𝑳𝑼𝑬 )= 712 ≈60%

𝑃 (𝑴𝑨𝑮𝑬𝑵𝑻𝑨>𝑹𝑬𝑫 )=59 ≈56%

𝑃 (𝑹𝑬𝑫>𝒀𝑬𝑳𝑳𝑶𝑾 )=1318 ≈70%

𝑃 (𝑴𝑨𝑮𝑬𝑵𝑻𝑨>𝑶𝑳𝑰𝑽𝑬 )=1318 ≈70%

𝑃 (𝑶𝑳𝑰𝑽𝑬 >𝑹𝑬𝑫 )=2536 ≈70%

𝑃 (𝑩𝑳𝑼𝑬>𝑴𝑨𝑮𝑬𝑵𝑻𝑨 )= 23 ≈67%

𝑃 (𝒀𝑬𝑳𝑳𝑶𝑾 >𝑩𝑳𝑼𝑬 )= 23 ≈67%

And now for the full list of all the probabilities………

What do you notice?There are 2 chains that work for the

5 dice

How do the names

relate to the chains?

Colo

ur n

ames

get

long

erColour nam

es are alphabetical

How do the probabilities compare?



BEATS

BEATS

BEATS

BEATS

BEATS

BEATS

BEATS

BEATSBEATS

BEATS



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.



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.



Combine two pairs to make a group of 4 people, with 10 dice!

In your group, investigate what happens in the different combinations available when each pair has 2 dice (of the same

colour).You can use a mixture of experimental

probabilities and theoretical probabilities.



𝑃 (𝑩𝑳𝑼𝑬>𝑹𝑬𝑫 )= 85144 ≈ 60%

𝑃 (𝒀𝑬𝑳𝑳𝑶𝑾 >𝑶𝑳𝑰𝑽𝑬 )=5681 ≈70%

𝑃 (𝑶𝑳𝑰𝑽𝑬 >𝑩𝑳𝑼𝑬 )= 85144 ≈60%

𝑃 (𝑴𝑨𝑮𝑬𝑵𝑻𝑨>𝒀𝑬𝑳𝑳𝑶𝑾 )=1627 ≈60%

𝑃 (𝑹𝑬𝑫>𝑴𝑨𝑮𝑬𝑵𝑻𝑨 )=5681 ≈70%

𝑃 (𝑹𝑬𝑫>𝒀𝑬𝑳𝑳𝑶𝑾 )= 712 ≈60%

𝑃 (𝑴𝑨𝑮𝑬𝑵𝑻𝑨>𝑶𝑳𝑰𝑽𝑬 )= 712 ≈60%

𝑃 (𝑶𝑳𝑰𝑽𝑬 >𝑹𝑬𝑫 )= 6 251296 ≈ 48%

𝑃 (𝑩𝑳𝑼𝑬>𝑴𝑨𝑮𝑬𝑵𝑻𝑨 )=59 ≈56%

𝑃 (𝒀𝑬𝑳𝑳𝑶𝑾 >𝑩𝑳𝑼𝑬 )=59 ≈56%

And now for the full list of all the probabilities………

What do you notice?The Word Length Chain is

reversed as expected.

The Alphabetical

Chain is in the same order

Colo

ur n

ames

get

long

erColour nam

es are alphabetical

How do the probabilities compare?



This line is 50:50 either way



SUMMARY

How to Use this GamePlace 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!

ONE DIE TWO DICE

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 Length Alphabetical



In your groups you are going to create a poster on Non-Transitive Dice.

ColourTitle

Background Info

Some of the Maths

ChallengesPresentation

SuccinctLayout



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.


notes for teachers

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