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Non-transitive Dice

Age 11 to 14
Challenge Level Yellow starYellow star
Secondary curriculum
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Non-transitive Dice printable sheet
 

Here are three dice that are used to play a game for two players:
 

 

The red die has the numbers {1, 1, 6, 6, 8, 8}
 
The green die has the numbers {2, 2, 4, 4, 9, 9}
 
The blue die has the numbers {3, 3, 5, 5, 7, 7}


Each player chooses a different die.
They roll their dice.
The winner is the person whose die shows the bigger number.

Alison and Charlie are playing the game. Charlie wants to go first so Alison lets him.

Was that such a good idea?
 
Can you advise Alison on which die to choose once she knows which die Charlie has selected?


 

Notes and background
 
Dice of this sort are known as non-transitive dice. You can read more about transitivity in this article or have a go at creating your own in Dicey Dice. This spreadsheet might be useful if you want to create your own.

After you've had a go at the problem, you may be interested to read more about dice in the Plus articles Curious dice and Let 'em roll.

You may be familiar with another non-transitive game known as 'rock, paper, scissors'.

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In a race the odds are: 2 to 1 against the rhinoceros winning and 3 to 2 against the hippopotamus winning. What are the odds against the elephant winning if the race is fair?

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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