Notes for Teachers
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• 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-websiteeu-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/
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How did RED and BLUE compete?
We saw that RED beats BLUE.
How did BLUE and GREEN compete?
We saw that BLUE beats GREEN.
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!
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What actually happened in the RED and GREEN games?
We see that GREEN beats RED.
BEATS
BEATS
BEATS
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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.
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4, 4, 4,
4, 4, 9
2, 2, 2,
7, 7, 7
RED
𝟓
𝟔
0, 5, 5,
5, 5, 5
𝟑
𝟔
BLUE
2
𝟓
𝟏𝟐
7
𝟓
𝟏𝟐
2
𝟏
𝟏𝟐
7
𝟏
𝟏𝟐
4
𝟑
𝟔
𝟏
𝟔
RED vs BLUE
𝟑
𝟔
9
𝟑
𝟔
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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 NonTransitive.
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4, 4, 4,
4, 4, 9
2, 2, 2,
7, 7, 7
BLUE
𝟑
𝟔
0, 5, 5,
5, 5, 5
𝟏
𝟔
GREEN
0
𝟏
𝟏𝟐
5
𝟓
𝟏𝟐
0
𝟏
𝟏𝟐
5
𝟓
𝟏𝟐
2
𝟓
𝟔
𝟑
𝟔
BLUE vs GREEN
𝟏
𝟔
7
𝟓
𝟔
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4, 4, 4,
4, 4, 9
2, 2, 2,
7, 7, 7
GREEN
𝟏
𝟔
0, 5, 5,
5, 5, 5
𝟓
𝟔
RED
4
𝟓
𝟑𝟔
9
𝟏
𝟑𝟔
4
𝟐𝟓
𝟑𝟔
9
𝟓
𝟑𝟔
0
𝟏
𝟔
𝟓
𝟔
GREEN vs RED
𝟓
𝟔
5
𝟏
𝟔
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How did RED and BLUE compete?
We saw that BLUE beats RED.
How did BLUE and GREEN compete?
We saw that GREEN beats BLUE.
How did GREEN and RED compete?
We saw that RED beats GREEN.
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With two dice, the rules are a little bit different!
BEATS
BEATS
BEATS
Let’s have a look at the probabilities again!
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4, 4, 4,
4, 4, 9
2, 2, 2,
7, 7, 7
RED vs BLUE
(two dice)
0, 5, 5,
5, 5, 5
𝟗
𝟑𝟔
8
𝟐𝟓
𝟑𝟔
𝟏𝟎
𝟑𝟔
𝟗
𝟑𝟔
13
𝟗
𝟑𝟔
𝟏
𝟑𝟔
18
𝟗
𝟑𝟔
4
𝟏𝟖
𝟑𝟔
𝟗
𝟑𝟔
𝟏𝟖
𝟑𝟔
𝟗
𝟑𝟔
𝟏𝟖
𝟑𝟔
9
14
4
9
14
4
9
14
𝟐𝟐𝟓
𝟏𝟐𝟗𝟔
𝟒𝟓𝟎
𝟏𝟐𝟗𝟔
𝟐𝟐𝟓
𝟏𝟐𝟗𝟔
𝟗𝟎
𝟏𝟐𝟗𝟔
𝟏𝟖𝟎
𝟏𝟐𝟗𝟔
𝟗𝟎
𝟏𝟐𝟗𝟔
𝟗
𝟏𝟐𝟗𝟔
𝟏𝟖
𝟏𝟐𝟗𝟔
𝟗
𝟏𝟐𝟗𝟔
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4, 4, 4,
4, 4, 9
2, 2, 2,
7, 7, 7
BLUE vs GREEN
(two dice)
0, 5, 5,
5, 5, 5
𝟏
𝟑𝟔
4
𝟗
𝟑𝟔
𝟏𝟖
𝟑𝟔
𝟐𝟓
𝟑𝟔
9
𝟏
𝟑𝟔
𝟗
𝟑𝟔
14
𝟐𝟓
𝟑𝟔
0
𝟏𝟎
𝟑𝟔
𝟏
𝟑𝟔
𝟏𝟎
𝟑𝟔
𝟐𝟓
𝟑𝟔
𝟏𝟎
𝟑𝟔
5
10
0
5
10
0
5
10
𝟗
𝟏𝟐𝟗𝟔
𝟗𝟎
𝟏𝟐𝟗𝟔
𝟐𝟐𝟓
𝟏𝟐𝟗𝟔
𝟏𝟖
𝟏𝟐𝟗𝟔
𝟏𝟖𝟎
𝟏𝟐𝟗𝟔
𝟒𝟓𝟎
𝟏𝟐𝟗𝟔
𝟗
𝟏𝟐𝟗𝟔
𝟗𝟎
𝟏𝟐𝟗𝟔
𝟐𝟐𝟓
𝟏𝟐𝟗𝟔
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4, 4, 4,
4, 4, 9
2, 2, 2,
7, 7, 7
GREEN vs RED
(two dice)
0, 5, 5,
5, 5, 5
𝟐𝟓
𝟑𝟔
0
𝟏
𝟑𝟔
𝟏𝟎
𝟑𝟔
𝟏
𝟑𝟔
5
𝟐𝟓
𝟑𝟔
𝟐𝟓
𝟑𝟔
10
𝟏
𝟑𝟔
8
𝟏𝟎
𝟑𝟔
𝟐𝟓
𝟑𝟔
𝟏𝟎
𝟑𝟔
𝟏
𝟑𝟔
𝟏𝟎
𝟑𝟔
13
18
8
13
18
8
13
18
𝟐𝟓
𝟏𝟐𝟗𝟔
𝟏𝟎
𝟏𝟐𝟗𝟔
𝟏
𝟏𝟐𝟗𝟔
𝟐𝟓𝟎
𝟏𝟐𝟗𝟔
𝟏𝟎𝟎
𝟏𝟐𝟗𝟔
𝟏𝟎
𝟏𝟐𝟗𝟔
𝟔𝟐𝟓
𝟏𝟐𝟗𝟔
𝟐𝟓𝟎
𝟏𝟐𝟗𝟔
𝟐𝟓
𝟏𝟐𝟗𝟔
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BEATS
BEATS
2, 2, 2,
7, 7, 7
BEATS
BEATS
BEATS
4, 4, 4,
4, 4, 9
BEATS
0, 5, 5,
5, 5, 5
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!
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This is a set of 5 Non-Transitive Dice
4, 4, 4,
4, 4, 9
2, 2, 2,
7, 7, 7
0, 5, 5, 3, 3, 3, 1, 1, 6,
5, 5, 5 3, 8, 8 6, 6, 6
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.
This set of dice are called Grime Dice, after their
discoverer, James Grime at the University of Cambridge
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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
4
BLUE with each of OLIVE, YELLOW and MAGENTA
3
OLIVE with each of YELLOW and MAGENTA
2
YELLOW with MAGENTA
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!
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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
7
12
BLUE > OLIVE with probability
OLIVE > RED with probability
7
12
25
36
𝑃 𝑹𝑬𝑫 > 𝑩𝑳𝑼𝑬 =
7
≈ 60%
12
𝑃 𝑩𝑳𝑼𝑬 > 𝑶𝑳𝑰𝑽𝑬 =
7
≈ 60%
12
𝑃 𝑩𝑳𝑼𝑬 > 𝑴𝑨𝑮𝑬𝑵𝑻𝑨 =
2
≈ 67%
3
𝑃 𝑴𝑨𝑮𝑬𝑵𝑻𝑨 > 𝑶𝑳𝑰𝑽𝑬 =
13
≈ 70%
18
5
𝑃 𝑶𝑳𝑰𝑽𝑬 > 𝒀𝑬𝑳𝑳𝑶𝑾 = ≈ 56%
9
25
𝑃 𝑶𝑳𝑰𝑽𝑬 > 𝑹𝑬𝑫 =
≈ 70%
36
5
𝑃 𝒀𝑬𝑳𝑳𝑶𝑾 > 𝑴𝑨𝑮𝑬𝑵𝑻𝑨 = ≈ 56%
9
13
𝑃 𝑹𝑬𝑫 > 𝒀𝑬𝑳𝑳𝑶𝑾 =
≈ 70%
18
5
𝑃 𝑴𝑨𝑮𝑬𝑵𝑻𝑨 > 𝑹𝑬𝑫 = ≈ 56%
9
2
𝑃 𝒀𝑬𝑳𝑳𝑶𝑾 > 𝑩𝑳𝑼𝑬 = ≈ 67%
3
There are 2
chains that
work for the
5 dice
What do you notice?
How do the probabilities compare?
How do the
names relate
to the
chains?
Colour names are alphabetical
Colour names get longer
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BEATS
BEATS
BEATS
BEATS
BEATS
BEATS BEATS
BEATS
BEATS
BEATS
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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 NonTransitive Dice.
They are obtained by taking 3
consecutive dice in the Word
Length list.
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We can also make sets of 4 NonTransitive Dice!
Each of these 5 subsets of dice
will produce a valid set of 4 NonTransitive Dice.
They are obtained by taking 4
consecutive dice in the
Alphabetical list.
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𝑃 𝑴𝑨𝑮𝑬𝑵𝑻𝑨 > 𝒀𝑬𝑳𝑳𝑶𝑾 =
𝑃 𝒀𝑬𝑳𝑳𝑶𝑾 > 𝑶𝑳𝑰𝑽𝑬 =
16
≈ 60%
27
56
≈ 70%
81
𝑃 𝑩𝑳𝑼𝑬 > 𝑴𝑨𝑮𝑬𝑵𝑻𝑨 =
5
≈ 56%
9
𝑃 𝑴𝑨𝑮𝑬𝑵𝑻𝑨 > 𝑶𝑳𝑰𝑽𝑬 =
7
≈ 60%
12
85
𝑃 𝑶𝑳𝑰𝑽𝑬 > 𝑩𝑳𝑼𝑬 =
≈ 60%
144
625
𝑃 𝑶𝑳𝑰𝑽𝑬 > 𝑹𝑬𝑫 =
≈ 48%
1296
85
𝑃 𝑩𝑳𝑼𝑬 > 𝑹𝑬𝑫 =
≈ 60%
144
7
𝑃 𝑹𝑬𝑫 > 𝒀𝑬𝑳𝑳𝑶𝑾 =
≈ 60%
12
𝑃 𝑹𝑬𝑫 > 𝑴𝑨𝑮𝑬𝑵𝑻𝑨 =
The Word
Length Chain
is reversed
as expected.
56
≈ 70%
81
𝑃 𝒀𝑬𝑳𝑳𝑶𝑾 > 𝑩𝑳𝑼𝑬 =
What do you notice?
How do the probabilities compare?
5
≈ 56%
9
Colour names are alphabetical
Colour names get longer
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The Alphabetical
Chain is in the
same order
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This line is 50:50
either way
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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
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!
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Colour
Title
Background
Info
Presentation
Some of
the Maths
Challenges
Layout
Succinct
A Special Game
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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.