The Dual Dice Set 2

Another collectible dice set with a pleasing duality 

One roll of the numbered die gives you a number from 1 to 12, and one roll of the die with pips produces a number from 1 to 8. How can this be? Challenge your friends to figure this one out.

These dice draw on the idea of geometric duality. When we roll dice, we usually use the fact that a cube has six faces - but it also has eight corners (vertices). The cube’s dual solid, the octahedron, has eight faces and six vertices!

The D8

For the die with pips, we use the fact that a cube has eight corners (vertices). When you roll the die, you can identify one of the eight vertices as being the closest to you. If you add together the pip values of the three faces surrounding it, you'll get a unique total.

Add the three faces meeting at the vertex closest to you:

The D12

For the numbered die, the trick relies on the fact that a cube has twelve edges. When it lands, one of the faces will be the closest one facing you, and another face will be visible on top - and the edge between them defines the dice roll you’ve made.

It would be great if you could add the faces either side of the edge - but it's impossible to build a D12 this way!

Instead, we subtract the smaller number from the larger and there you have it: a unique number from 1 to 12 is produced.

Find the two faces neighbouring the edge closest to you, and subtract the smaller one from the larger one:

If you want to use this die in a different way, you can take into account the orientation of the cube. When you’re looking at the front edge, the dice might land in either orientation - if you consistently subtract the front face from the top face, the differences will range from −12 to −1 and 1 to 12. So maybe this is really a D24!

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!