### Why do this problem?

This problem offers students the chance to visualise in three
dimensions (using the context of dice), list possibilities
systematically, and work on probabilities in a less familiar
context.

### Possible approach

Begin with some visualising of dice:

"Imagine a six-sided die, where the opposite faces add up to 7.
Rotate your die so that the top face shows 1. What does your bottom
face show?"

"Now rotate it round so that the front face shows 5. What will the
back face show?"

"What numbers are on the other two faces? Think about what's on the
left and right faces of your die. Are you all picturing the same
die?"

These questions tease out an understanding that all 6-sided dice
whose opposite faces sum to seven are essentially the same, the
only possible difference being that you can make a "right-handed"
and "left-handed" version that are mirror images of each other
(read more

here.)

The problem

Right
or Left? explores the differences between right- and
left-handed dice.

Now introduce the context of the problem and set the class to
working out all the possible edge scores and how these could be
distributed so that everyone has an equal chance of getting the
pie, when there are two, three, four... people sharing the
meal.

After everyone has shared their work on this task, the same
analysis can be done with the die that always lands
"corner-up".

### Key questions

Numbers on opposite faces sum to 7. Does this imply that numbers
NOT on opposite faces DO NOT sum to 7?

The smallest edge total is 3, and the largest is 11. How do I know
that at least one of the numbers between 3 and 11 can be made in
more than one way?

The smallest corner total is 6, and the largest is 15. How do I
know that at least two of the numbers in between must be impossible
to make?

### Possible extension

Discuss whether the probabilities can be combined, and whether it's
equally likely that the die will land "face-up", "edge-up" or
"corner-up" - these ideas are developed in the problem

Dicey
Decisions.

### Possible support

Offer students dice to use while working on the problem.