You may also like

problem icon

The Lady or the Lions

The King showed the Princess a map of the maze and the Princess was allowed to decide which room she would wait in. She was not allowed to send a copy to her lover who would have to guess which path to follow. Which room should she wait in to give her lover the greatest chance of finding her?

problem icon

A Dicey Paradox

Four fair dice are marked differently on their six faces. Choose first ANY one of them. I can always choose another that will give me a better chance of winning. Investigate.

problem icon

Nines and Tens

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

Non-transitive Dice

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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'.