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More Dicey Decisions

Age 16 to 18 Challenge Level:
In the problem Dicey Decisions, we encouraged you to consider the possible edge totals by adding up the numbers that meet on the different edges of a six-sided die.
If you haven't already done this, why not try now?
Imagine that instead of a six-sided die we had a dodecahedron numbered 1-12.
There are different ways to arrange the numbers from 1-12. A standard six-sided die has opposite faces that sum to 7, so perhaps our dodecahedral die should have opposite faces that sum to 13.
Can you create a net for a dodecahedral die whose opposite faces sum to 13?
For the six-sided die, the edge totals were distributed like this:
Edge total  3   4   5   6   7   8   9   10  11
Frequency 1 1 2 2 0 2 2 1 1

The mean edge total is 7, and the edge totals are distributed symmetrically about the mean.
What is the mean edge total for your dodecahedral die?
Are the edge totals distributed symmetrically?
Ignoring rotations and reflections, there is only one way to number a cube to create a six-sided die with the constraint that opposite faces sum to 7, but there are multiple ways to create a dodecahedral die with opposite faces that sum to 13.
Can you make any general statements about which dodecahedral dice will have edge totals with a symmetric distribution? Can you prove your statements?
For the six-sided die, the corner totals were also distributed symmetrically. Will the same be true for the corner totals of a dodecahedral die?
Now use your insights to make and justify some statements about the edge and corner totals of an icosahedral (20-sided) die with opposite faces that sum to 21.