Our good friends at Maths Gear have sent us a tube of “unique polyhedral dice” to review. The description on mathsgear.co.uk says they’re “made from polyhedra you don’t normally see in the dice world”. My first thought was that we should test they’re fair by getting David to throw them a few thousand times but — while David was up for it — I’d have to keep score, which didn’t sound fun.

So instead we thought of some criteria we can judge the dice on, and sat down with a teeny tiny video camera. Here’s our review:

https://www.youtube.com/watch?v=prFc3Vv0lXs

The question of what makes a die fair is more complicated than you might think, and still open. The simplest thing to be is *isohedral*, which we just about described in the video: it means every face is the same, and every face can be mapped onto any other by a symmetry of the whole object. Three of the dice we reviewed were isohedral, and two of them pretend they are by only having numbers on one kind of face. The makers of the dice could’ve paired up each blank face with a numbered face and still had a fair die.

The dice we reviewed were unique among mass-produced dice. There are more than a few pages on the internet showing off all sorts of weird but fair dice, usually made as one-offs: here’s one by Loki3, while Alea Kybos’s page has loads more. Ed Pegg, Jr., of mathpuzzle.com fame, titled his master’s thesis “a complete list of fair dice”, by which he meant all isohedral polyhedra, which he has listed on his webpage. Finally, Gérard P. Michon has a comprehensive page answering all sorts of hard mathematical questions about dice.

It’s not clear if there can be a fair die which isn’t isohedral. The late Bill Thurston wrote a good answer to a MathOverflow question on this subject a few years ago. The short answer is that it depends on how you throw it, which probably isn’t the kind of “fair” you’re looking for.

### More information

Unique polyhedral dice from Maths Gear, £9.95.

Maths Gear sell quite a range of dice: they’ve got some funky-looking skew dice, dice to decide who goes first, and the famous non-transitive Grime dice.

The d60 seems to roll fairly straight, so I wonder if the great circles all have reasonably close stats. If not, maybe it is possible to cheat by pre-setting the orientation of the dice in your hand before rolling.

I HAVE found out about a piece of die that isn’t isohedral and is still very fair. A lot of the dice on this website ( http://loki3.com/poly/unique-dice.html#5d-jack ) that aren’t isohedral, but my personal favorite is the jack die. It looks kind of like someone stuck a prism inside of a star, and it is fair because it has an equal chance of landing on either one of the sides on the prism or on two of the legs of the star, and how it’s numbered makes it so it has five probabilities with equal chance for each.