### Why do this problem?

This problem offers students the opportunity to consider the
symmetry of dodecahedra, and to develop insights from reasoning
about dodecahedra to help them to analyse symmetry in
icosahedra.

### Possible approach

The problem could be given to students to work on
independently (perhaps as a homework task) but could also be used
as a whole class activity as follows:

Start by making sure students are happy with the idea that
opposite faces of a six-sided die sum to 7, and then challenge them
to come up with a dodecahedral die where the opposite faces sum to
13. Some students may wish to visualise the die, some may be happy
working with a two-dimensional representation, and some may prefer
to create a net and actually make the die in three
dimensions.

There is opportunity for rich discussion about the number of
distinct dodecahedral dice it is possible to make (with the
constraint that opposite faces sum to 13), and different groups of
students could be encouraged to work with distinct dice.

Once students have created (or visualised) their die, ask them
to create a frequency table showing the frequency with which the
different edge scores occur.

"Does everyone get a symmetric distribution?"

"Are you surprised that everyone got a symmetric distribution
even if they started with different dice?"

"Can you explain why it was symmetric?"

This could lead into rich discussion about the symmetry of the
distribution and how it relates to the constraint that opposite
faces sum to 13.

Finally, once students have explanations of where the symmetry
in the distribution comes from, they could be challenged to do a
similar analysis for corner totals of the dodecahedral die, or for
edges or corner totals on an icosahedral die.

### Key questions

If a face of the die is numbered $n$, what would the opposite
face be numbered?

If adjacent faces were numbered $n$ and $m$ (with an edge
total between them of $n+m$, what would the opposite edge total
be?

### Possible extension

Write up a really clear proof of the result concerning the
symmetric distribution of the edge totals.

I have a 100-sided die. To the eye, it looks pretty regular,
but it can't be exactly regular because there is no such thing as a
100 sided Platonic solid. Consider the issues which might arise in
an analysis of the various totals.

### Possible support

Spend time considering the edge totals of the six-sided die to make
sense of the symmetry there before working on the
dodecahedron.