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Dicey Dice

Age 16 to 18
Challenge Level

Why do this problem?

Along with X-Dice, this is a great problem for thinking about conditional probability.

Possible approach

Simply pose the challenge and encourage experimentation and calculation to begin with. There is no strictly 'linear' way into this problem: creativity and tinkering is required to end up with a valid set of dice. However, all students should eventually find a solution. 
Once a set of dice with the correct properties has been invented and assessed move onto the question of the existence of a totally fair set of dice. You might begin this by asking who does or doesn't think that such a set of dice exists, using their intuitive reasoning. At the end, see who was correct.

Key questions

How can we determine whether one die is likely to beat another?

Possible extension

Investigate the variances of your set of dicey dice. 

Possible support

Simply invent pairs of dice and see which is better than the other.
You can read about some of the issues which might arise when teaching probability in this article.