Nice Dice: Teachers' Notes

This problem provides an interesting application of probability and systematic thinking. It also has some surprising generalisations and some worthwhile extensions. It is useful for students to explore in class or even at home.

 

These notes are not intended to be a sequential description of how to use this activity with students but they should provide you with further information that may help you to better understand the problem and how it might be used to provide a rich mathematical experience for your students. You should also read the Students’ notes for this problem, as they provide specific hints on completing the Bright Sparks activity online.

Before starting this problem students should have some basic idea of probability. We would expect them to know how to determine the outcomes of rolling two dice by constructing a table. They should also know how to determine the probabilities of each of these outcomes.

 

Now the problem is about taking two cubes and turning them into ‘dice’ that produce the sums 1, 2, 3, … and 12 with equal probabilities. One bright spark once said that all you had to do was to put very high numbers on the cubes and then the probability of getting each number from 1 to 12 would be zero – problem solved! There are actually several other sneaky ways of solving this problem. It is possible to get the probabilities of 1 to 12 being 1/36, say, while having other numbers appearing as possibilities. Let us say right at the start, that the only possible sums should be the numbers 1 to 12 and that each of these should be equally likely.

 

And as good a way as any to get started is to play. Tell the students to try whatever numbers come into their heads and see where that leads them.  However, there are 36 possible outcomes for a nice dice and the probability is the same of summing to each of the numbers from 1 to 12. So we would expect each possible sum has to appear 36/12 = 3 times.  It would make good sense to use this information in any experimentation.

 

There are two further things to note about the computer. First it only allows the numbers 0 to 12 inclusive to be placed on the faces of the dice.  Second the computer knows before the start that if you generalise this problem you get four infinite families. So it will refuse what looks like a perfectly good ‘new’ pair of nice dice if one from the same infinite family has already been entered by the student.

 

Click to view a summary of how to show that there are only four essentially different pairs of nice dice on the nice dice proof page.

 

Finally lead students to think about the possible infinite families of nice dice that might exist here.

The question now is can we take the problem any further. What Nice-Dice-like problems can we build around this one? It’s very important to have this discussion with students at all levels. Extending problems is a fundamental way of working for mathematicians and student should know that. Let us suggest some related problems.