In the problem Dicey
, 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
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
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
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
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
For the six-sided die, the corner totals were also distributed
symmetrically. Will the same be true for the corner totals of a
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.