Exotic polyhedron purveyor Dice Lab’s crowning randomizer is its monstrous, $12 120-sided die.
Creating the D120 posed an important challenge: while dice with fewer sides traditionally distribute high-low numbers on opposite sides (for example, the 6 on a D6 is opposite the 1, the 5 is opposite the 2, and the 4 is opposite the 3), the axes of symmetry in a disdyakis triacontahedron don’t lend themselves to such a simple distribution.
So the Dice Labbers consulted with Oberlin mathematician Bob Bosch, who brought his expertise in operations research to bear on the problem. He wrote a piece of software to simulate different vertex sum combinations, hill climbing through different optimizations, until it arrived at a “perfect” solution.
Their 120-sided polyhedron has 12 vertices where 10 triangles meet. Henry and Robert wanted the numbers on the 10 faces that surround a vertex of this type to add up to 605, which is 10 times 60.5 (the average of all the numbers from 1 to 120).
In addition, the polyhedron has 20 vertices where 6 triangles meet. Henry and Robert wanted the numbers on the 6 faces that surround a vertex of this type to add up to 363, which is 6 times 60.5.
Finally, the polyhedron has 30 vertices where 4 triangles meet. Here, they wanted the vertex sums to be 242, which is 4 times 60.5.
Henry and Robert did not know (nor did I) if it was possible to construct a numbering satisfying all of these conditions.
The Mind-Boggling Challenge of Designing 120-Sided Dice [Liz Stinson/Wired]