Of Dungeons and Dragons, Dice, and our Adventure into Complex (for us) Math
Hi!
Disclaimer: we are not mathematicians. We're D&D kids. Posting to r/math is objectively terrifying.
The Problem:
We make dice. Really specific, only-useable-for-one-thing dice for Dungeons and Dragons. There are some rules in the game that are really hard to remember, so we try to put the rules in the dice so you don't have to look up values, or bonuses or how-many etc. Just roll the dice and take the result. Fun, cool, stylish. Then we came to Healing Potions which involve a multiple die roll (2d4, 4d4, 8d4 and 10d4). As you all know instantly (it took us a beat): results of multiple dice rolls come back on a curve, not equal chances of each result. You likely also know that's pretty difficult to make happen in a polyhedral.
The Solution:
We had to come up with a new way of deciding on the size of each face, and come up with a way to calculate the dimensions of a symmetrical polyhedral with faces of our exact measurements. We assumed there would be an equation or equations that would help with this ... but no, we couldn't find anything that solved for interior angles given side-measurements, or even distributing points on a sphere.
We made some prototypes and noticed in the rolling data that there was a relationship between the outcomes we were getting and, curiously, the ratio of the distance between the center of gravity and each corner to the distance between the center of gravity and each face.
As it turns out, what this relationship was pointing us towards was the relative surface area of each portion of the die, as projected onto a sphere. Why? Because this accounts for the likelihood that the center of gravity will be above each face given any random orientation of the die in 3D space. Great! So how do we figure out that shape? There is some previous work in this area by Reimer, Stoyan, Obreschkow. “Cuboidal Dice and Gibbs Distributions” 2013. Metrika, Vol 77 - they show that the probability of each face of a die landing is proportional to the potential energy of the die in that position a la the Boltzmann Distribution.
Since we're playing D&D, not plotting a route to Mars, we wrote a brute-force polyhedral-solving program to test a gazillion (real number?) combinations until it arrived at a solution within a precise limit. No big deal, brush up on our spherical trigonometry, why not, it only… well, never mind how long that took. :)
The Data
We can never analyze a set of randomly generated numbers and ever be 100% sure that it perfectly adheres to a particular data model. It turns out you need to roll dice way more than you might think to get an accurate picture of the die’s outcomes—we’re talking 3,000–6,000 times on a d20 before you can make a confident call about its bias. [Campbell, C. Warren and Dolan, William P., "Dice Mythbusters" (2019). Student Research Conference Select Presentations. Paper 48.]
However, we put in the work, did the tests, ran the statistical analysis, and… well, we are as certain as one can be that we cracked it. After more rolls than necessary, the deviation of our die from 2d4, and from the perfect desired outcomes, is statistically insignificant.
Discussion Questions:
- Is this interesting? We understand that our tolerances/requirements are only "this is fun for D&D" and likely not up to the rigorous standards of math.
- Are there papers/research/equations/math that we missed? We did a lot of searching, and consulting with our friends who are mathematicians and Physicists but we don't have access to the same libraries and knowledge y'all do.
- Would you have taken a different approach?
- Input to make our results more rigorous? Is there a test we should be performing or math we should be mathing?
- Do you say "mathing?"
- I'm Happy to go into our (lengthy) adventure in wrong-answers if you're interested at all. Some of our solutions were great but also resulted in physical hurdles such as "that would be as big as Michigan Stadium." or "this object would have a gravitational field."
Thank you!
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Renders and prototype images on kickstarter