### Why do this problem?

This problem explores the familiar context of probability with
dice, in an unfamiliar way.

### Possible approach

This problem is set in the same context as the Stage 3
problem

Troublesome
Dice.

Make sure students understand the standard way of labelling
the faces of a six-sided die. They will need to calculate all the
possible edge and corner totals in order to calculate the relative
frequencies. There are different areas for exploration suggested in
the problem - one possibility is to assign particular values to the
probability of landing face, edge, and corner up. Values of
$\frac{1}{2}$,$\frac{1}{3}$ and $\frac{1}{6}$ are suggested -
students could compute the overall probabilities for each total.
Alternatively, the general case with probabilities $p$, $q$ and $r
= 1 - (p + q)$ could be explored - this would work well in small
groups with students sharing out the work.

Once students have computed the probability of each total
(either for specific values or in terms of $p$ and $q$), challenge
them to find fair ways of allocating the numbers for different
sizes of family. They could work in small groups and present their
solutions to the rest of the class.

### Key questions

I can get a 6 as a face-up, edge-up or corner-up score. Which
scores would you want to be allocated to balance this out?

A 6-sided die can be labelled in a "left-handed" or
"right-handed" manner - do the two dice give different
answers?

The probabilities are distributed symmetrically for the face,
edge and corner sums. Is this surprising? Can you explain
why?

### Possible extension

Consideration of the probabilities for dodecahedral or
icosahedral dice provide a challenging extension.

### Possible support

Spend some time working on

Troublesome Dice
to make sure students are happy with the relative probabilities of
the edge-up and corner-up scores.