# Grime Dice

‘Rock, paper, scissors for dice.’

Here are three dice with some amazing properties.  To discover them you are going to play a simple two-player game. Inspect the dice and choose which you think is the best dice (that will give you the highest score when you roll it).  Your opponent will do the same and whoever rolls the highest score wins.  Simple!

Really? So which is the best dice? One you’ve played a few rounds of the game you will want to discover the truth behind these dice and analyse the probabilities of winning.

### Resources

• You can make yourself a set of these dice by cutting out some sticky labels with the correct amount of dots on three ordinary six-faced dice.
• Otherwise you can use the following  dice simulation (you will need to enable macros to use this file).
• You can't beat having some actual physical sets of these dice.  You can buy them from  mathsgear

### Description of the Game

• Player 1 - choose the dice which you think is the best (that will give you the highest score when you roll it).
• Player 2 (or perhaps your teacher) now chooses one of the remaining dice.
• Both players roll their dice. The player with the highest score wins.
• Play the best of 10 rounds with these dice.
• Who won? Which dice won?
• Now player 1 can select another dice (he/she may choose the same dice or select a different dice depending on the results of the first game).
• Player 2 chooses one of the remaining dice.
• Play 10 rounds with these dice.
• Which dice won this time?
• Is there a best dice?
• Is there an advantage in going first or second?
• Clue:
There is an advantage in going second! Whatever dice player one chooses player two should be able to choose a stronger dice.  If player one chooses the red dice then player two should be able to choose a dice that is better. However, if player one chooses blue then player two should choose a different dice to win.  Similarly, if player one chooses green then player two should be able to find a better dice. Can you find the correct strategy?

### Analysing the Game

• Students should have chosen a best strategy for winning the game.  Perhaps they have done this by observation.  Now it is time to use some probability theory to work out which truly is the best strategy to win this game.  It is best to attempt this worksheet AFTER playing the actual game!

• The following  Crazy dice worksheet is designed to guide students through calculating probabilities of winning.

• The second worksheet  Crazy dice structured gets students to do exactly the same thing as above but offers more structure and hints.
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