Imagine the following: After weeks of cancellations, you and three friends have finally gotten together to continue the DnD campaign you’re running.

And things get exciting fast:  upon entering a dungeon, one friend sets off a trap. Since this friend is playing  a very squishy wizard, you decide the trap will target someone at random… ou know, to give ‘em a chance.

To select which of your friends has to dodge, you pull out your standard d6. But before you can make the roll, the wizard says, “Wait, we can use my new d3!” And pulls out this small, 3D-printed dragon. This thing can’t possibly be a fair die, right?

Well according to the mathematicians  who designed it, it totally is. And using their method, you can  design pretty much any die you want. Whether it’s for DnD, or any other game. [intro] This dragon d3 looks nothing like a normal die, but that doesn’t mean it can’t be.

Anything can be a die. It just needs a limited number of  orientations or sides to land on. In fact, dice weren’t always so regularly shaped.

In ancient Greece and Rome, people used the ankle bones of animals  like sheep and goats as four-sided dice. These asymmetric d4s, known as knucklebones, were used in games and divination. Unfortunately, these particular  dice had unequal probabilities.

Each side did not have the  same chance of landing face-up. After all, Mother Nature didn’t exactly  have dice in mind when designing bones. But the people who used them likely didn’t care, because they thought dice  outcomes were controlled by fate.

Not math. Eventually, though, people  realized probability was a thing, and looked into making dice fair. The most straightforward  approach to making a fair die is to use a symmetrical shape with  a consistent material throughout.

Many dice use this approach,  including those in

DnD: a game famous for using wacky dice to randomly determine players’  actions and story outcomes. Now, I know a lot of you dice  goblins out there have got at least one die that’s a bit fancier, with insides that are clearly  made of different materials. Like this one. But that makes it way harder  to ensure they’re balanced.

So whether you actually  use them in your campaigns, or just let them sit on your  shelf because they’re pretty, feel free to share them with us! The standard DnD dice set comes with 7  dice in 6 different isohedral shapes, meaning they have the same geometry on all sides. There’s the d20, with triangle  faces; two d10s with diamond faces; a d12 with pentagon faces; a d8 that  looks like two back-to-back pyramids, your classic d6 cube, and a d4  that’s a painfully sharp pyramid.

Totally not speaking from experience here Some of you out there may also own a  non-standard die that isn’t isohedral, but still has some symmetry. For example, this version of a  d3 is rotationally symmetric. And so is this one.

But if we want to evaluate the fairness  of something like this d3 dragon, we need a more rigorous procedure. One way is to just roll a die thousands of times. A person named Jason Mills did just that in 1987 to test the non-standard d100… a die sometimes used in DnD  in place of those two d10s.

And he showed the d100 was, in fact, unfair. Alternatively, you can use computer simulations  to virtually roll a die over and over again to get its outcome probabilities. But these simulations require  intense computing power and time.

And sometimes, they give impossible outcomes, like having a die land on an edge. I guess it’s technically not impossible that a perfect die could land exactly on its edge. But even the slightest nudge  would send it tumbling over.

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Recently, a team of researchers came up  with a new way to model dice probabilities that sticks to real-world outcomes and is  60 to 400 times faster than simulations. Their method uses math to  analyze the geometry of a die. No simulations required.

Although, it is a bit of a simplification. For example, they ignore momentum, so each die “roll” is really more  of a gentle placement of the die on a random edge, corner, or face,  and then watching it fall over. But while critics have complained  about how unhinged this is, this method can be pretty accurate.

Here’s how the math works: First, we figure out all of the  possible ways a die can land, which we call stable equilibrium points. Next, for each possible starting point on the die… every edge, corner, and face… we figure out which stable equilibrium point it’ll eventually land on by  looking at how the die will roll. Because of gravity, the die will roll in whichever direction  minimizes the height of its center of mass, or balance point.

In other words, it always tries  to get its center of mass as close to whatever surface you’re rolling the die on. Be it a table, or one of those  collapsible felt dice trays you’re using to protect all your fancy dice  from that very hard table. Now, remember this method  involves a bit of simplification?

Our next step is to simplify the shape of our die. Whatever it starts as… be it a cube or a dragon,  we turn it into a sphere. To determine the direction that the die will roll, we map each point on the die’s  surface to a point on this sphere, and then plot how far it is from  the center of mass using colors.

In a way, we’re making a  spherical topographic map, with light-colored peaks and dark-colored valleys. The jargon term for this is a Gauss map. From a given starting point on our Gauss map, we simply have to trace a  downwards path towards a valley, until the die reaches a stable equilibrium point.

Finally, to determine how  likely it is for our die to land in each possible orientation… we take our Gauss map and divide  it into these sort-of basins, each corresponding to a single outcome. The probability of landing on one outcome is   just the fraction of area that  the basin covers on the sphere. If our d3 is fair, it should have three basins each covering 33 and a third percent  of the sphere’s total surface area.

If one basin were a bit bigger  and another a bit smaller, i t would be unfair. According to this research team, the dragon they designed is indeed  fair using these Gauss maps. And to help seal the deal, they also 3D printed the die and verified its  fairness in real life by rolling it 400 times.

But the researchers' math isn’t  just useful for analyzing dice, it can also be used to design  fair dice of any shape. We only need to feed the computer a shape we like, and it can iteratively calculate the  die’s probabilities and tweak its shape until the Gauss map says it’s fair. In addition to the aforementioned dragon, the research team used their method to  design this oddly buff armadillo man and this large-headed kitten.

Both of them were also d3s. They then rolled these dice several  hundred times to verify their fairness. But this method doesn’t just  allow a person playing a warlock to design and 3D print a die  shaped like their patron!

We can also use this method to design  dice that are intentionally unfair. For example, the researchers made several d11s with the same probability  distribution of rolling two d6s, which you have to do in a bunch of board games… or at a craps table. Rolling a 2 has a 1-in-36 chance,  because you have to get a 1 on both dice.

Meanwhile, rolling a 7 has a 1-in-6 chance… which is why the robber moves  so gosh-darn much in Catan. Oh, and by the way, if you’re itching to try  any of these weird dice out, the researchers have posted the files online. That’s why we’ve got all these nifty props!

Now, we’ve talked a lot about  designing multi-sided dice. And all you dice goblins out there may be  excited to accrue the weirdest set of dice that best appeals to your personal taste, or your DM’s awkward homebrew. But what about the ultimate sits-on-the-shelf die?

The ultimate cheater’s die that throws the  entire concept of fairness out the window, because it always lands with the same face up? What if you want a d1? What could that possibly look like?

One option is a Möbius strip, a shape  famous for having only one surface. So every side is the same side. Another option is the Gömböc, which looks kinda like a cylinder  with one side pinched off.

The Gömböc only has one stable equilibrium point, so no matter how you place it, it eventually rolls back to this upright position. It also technically has one  unstable equilibrium point… that super sharp edge. But as I mentioned earlier, even if you  can technically balance it on that edge, it won’t stay there for long in the real world.

So it’s effectively a d1. As weird as the Gömböc looks, t’s not entirely unnatural. Some tortoise shells are  imperfect versions of this shape, to help the animals right themselves  when they accidentally flip over.

Neither of the Möbius strip or the Gömböc  look like a classic polyhedral die, though. If we get creative with the material, we can also make a d1 out of an isohedral shape. Recently, mathematicians made a  four-sided tetrahedron act like a d1 by adding just the right amount of  extra weight to one of its sides.

The possibilities for dice are endless. So if someone whips out a miniature  Beholder in your next DnD session and claims it’s their new d20,  don’t be so quick to discount it. It might actually be fair game. [ OUTRO ]