The Dual Dice Set 1

A collectible dice set with an elegant duality 

One roll of the four-sided die gives you a number from 1 to 6, and one roll of the eight-sided die gives you a number from 1 to 12. All with uniform probability! How can this be? Challenge your friends and see if they can figure it out.

These dice draw on the unusual idea of geometric duality. The four-sided die (a tetrahedron) has six edges and the eight-sided die (an octahedron) has twelve edges.

When you roll either of these dice, a triangle shaped face always lands on the table, with one vertex closest to you.

This means as you look at the die, you can always identify one edge and its corresponding two faces closest to you - and if you add together the numbers on those faces, you'll get a unique sum for each edge:

So this D4 (four-sided dice) can actually produce six different totals when you throw it.

And this D8 (eight-sided dice) can produce twelve different sums.

The Dual Dice were designed by Ivo David. To read more about the maths behind the dice, take a look at chapter 4 ("Dicey Dice") of "Mathematical Puzzles and Curiosities" by Ivo Fagundes David de Oliveira, Tanya Khovanova, and Yogev Shpilman (published by World Scientific in Feb 2026).

You can get hold of your own set of Dual Dice right here on Maths Gear!