D&D Dice

Dice in D&D are a life or death matter so what are best dice and how do you get the best from your dice?

What Are The Dice in D&D? Platonic Solids and Trapezohedra

A platonic solid is one where each face is made up of equal sides and angles plus all the corners between the faces must be of the same arrangement, adjoining the same number of faces. So of your D&D Dice, only the D4, D6, D8, D12 and D20 are classed as Platonic solids. The D10 is not a platonic solids because you cannot make something 10 sided with all the corners are being equal. Instead it is a Trapezohedra with an equal number of sides above and below an equator. Though technically the standard D6 are also a special case of Trapezohedra.

These definitions of shape are not abstract, they are the mathematically perfect shape for an even distribution probability. But of course, reality is not perfect.

The 3 Things That Make a Difference to your Dice

Shape

Obviously the shape of your D&D dice make a difference. A good die should have an equal amount of surface area on each face. Unfortunately variations in face size that are too small to see can have a large impact. Unless you have a micrometer and a lot time, you have no way of knowing if your dice are true.

Balance

A good die has a centre of gravity at its centre. If there are any deviations, one side will end up at the bottom more often. With the types of resins used to cast dice, the density of the material in the die is unlikely to vary much. What really can make the difference is the pips or numbers cut into the dice. On a standard d6 with the pips drilled out, the ‘6’ will lose more weight than the ‘1’. This is why casino dice have their pips filled with paint equal to the weight of removed resin. With standard polyhedron D&D dice, the numbers are generally formed in casting and are much shallower. Whilst the difference in missing material between the ‘1’ and the ’20’ is notable, the actual amount of material, proportionate to the size of the dice is very small. However, on five of my seven D20, around the ‘2’ there is a cluster of single digit numbers. Whether this clustering makes a difference to the die’s balance, I don’t know.

Sharpness

The edges on your dice are critical to the physics of rolling dice. The nearer something is to a sphere, more that it will roll. Why is this a problem, other than you end up picking your D&D dice of the floor more often? The longer the dice rolls, the more impact deviations in shape and balance will play. Unlike both shape and balance, sharp edges are something the average D&D player can check.

Try this at home: Pick two dice of the same type from your D&D dice bag. One should be the roundest you can find and one the sharpest. Role them together on a hard surface with plenty of space. You will notice that the sharper one generally stops rolling first. Often by just a fraction of second and it is not every time because of the small variations in how they are rolled, but you will notice a difference.

Quality of the Dice

Whilst all D&D players, in their heart-of-hearts, want dice that roll whatever number they need, they will say that they want fair dice. Dice that gives a truly random result. This way you know that if you get a run of bad results then sooner or later things are going to average out.

What is Fair?

A lot of people have studied the fairness of the humble D6 because it is used in gambling. Gambling means money and money mean people trying to cheat. Consequently casino dice are made to the highest quality and tightly controlled. By law they have to manufactured to exacting standards with sharp edges. The weight of the material drilled out to make the pips has to be replaced exactly with the paint used to fill the holes. Even after all this, the dice are only used for 8 hours to prevent wear-and-tear effecting them. It is a good job our D&D dice don’t have to be disposed off like that!

What Difference Does It Make

Dan Murray, Associate Professor in Mathematics, tried to find out. An automated machine rolled a variety of casino dice 640934 times and captured the result. The variation in the dice were +/- 0.0010%. This means that all six possible results occurred statistically equally. He also tried a $0.10 die from a local store and rolled it 21543 times. In this case a 1 was rolled 2895 times but a 5 was rolled 4383 times! A massive difference so the quality of your D&D dice can have a significant impact on the games fairness.

Superstition and Dice

It is unlucky not be superstitious

A superstition is an irrational believe about cause and effect. E.g. believing that the sun won’t rise unless you pray to it. They occur when we do something, such as praying, and then something happens, such as the sun rise. The human brain automatically associates those two facts because more often than not they are connected. If I drop a stone, it falls to ground. A caveman doesn’t know about gravity, he cannot prove that the stone will always fall to the ground so he takes it on trust that two are connect. D&D Players are highly superstitious people, even the ones with degrees in probability will be able to point out their lucky D&D dice. But it has been proven that some dice will roll more randomly than others so are we being superstitious?

Observer Bias

The problem with your average D&D dice is that we don’t track the results. We remember some rolls and forget others. You remember that the red D20 rolled the ’20’ you needed to make the vital saving throw but forget the 3 times it rolled a ‘1’ on unimportant skill checks. Some players can even give the impression of being lucky by making a big thing out of good results and quiet moving on if they roll badly.

Does Really Matter?

Unless you have some very poor quality D&D dice, the variation in the roles is not going to make the slightest bit of difference. Only after thousands of roles with a D6 can you see clear bias. With a D20 you need even longer. The simple fact is we don’t role dice that much. In a casino game, the dice are rolled every minute for hours and hours. In D&D, hours can pass between dice rolls.

Believing some dice are lucky and some are unlucky is fun. It is a way of handling the vagaries of probability and it adds to the atmosphere. So go and sort your D&D dice, pick out your best, charge that dragon and may all your rolls be lucky.

Sources and Notes

This article was inspired by the post Lou Zocchi and the Science of Dice. You can buy the dice in question from Game Science.

Anyone seriously interest in the statistics of dice rolling, have a look at Experimentally obtained statistics of dice rolls and the excellent Are my dice random?.

I combed the scientific literature available on the science and physics of polyhedron D&D dice but it is extremely thin on the ground. Even studies on casino dice are rare. However there was one paper that was written on the subject that I cannot get online. So If anyone has access to the Mathematics Gazette, there is this article I would love to see this:

K. Robin McLean. Dungeons, dragons and dice. MG 74 (No. 469) (Oct 1990) 243-256. Considers isohedral polyhedra and shows that there are 18 basic types and two infinite sets, namely the duals of the 5 regular and 13 Archimedean solids and the sets of prisms and antiprisms. Then notes that unbiased dice can be made in other shapes, e.g. triangular prisms, but that the probabilities are not obvious, citing Budden and Singmaster, and describing how the probabilities can change with differing throwing processes.

As referenced by: 10.AA. NON-REGULAR DICE


Image Credit – New Dice by Niriel – CC-BY-NC-2.0

 

7 comments

  1. ” Only after thousands of roles with a D6 can you see clear bias. With a D20 you need even longer. The simple fact is we don’t role dice that much. In a casino game, the dice are rolled every minute for hours and hours. In D&D, hours can pass between dice rolls.”

    I’m sorry but this is just patently false. A standard course in statistics and probablitiy will show that if there is bias in a set of all available outcomes, there is bias in any infinitesimally small trial.

    That biased d6 is just as biased, whether you roll it once or 10 million times. You may not be able to *observe* bias until you roll it 10 million timses, but the bias is nonetheless present.

    The best thing you can do is buy some high quality dice, or use a random number generator with a good algorithm.

  2. Apologies. The sentence before the block quote is the one that sparked my comment. I’ll re-read before posting next time.

    “Unless you have some very poor quality D&D dice, the variation in the roles is not going to make the slightest bit of difference.”

  3. Jeffrey

    Whilst technically a biased die is always a biased die, the sample size relative to the size of the bias dictates whether that bias is important.

    I have a loaded die that always roles 6s. Even on a sample size of one role, that is a significant bias but this is an extreme case.

    In the $0.10 die test by Dan Murry, the 5 appear twice as often than the 1. This would be apparent (or at least suspected) after 100 rolls. Yet that bias has only changed the average result of the die to approximately 4.5 instead of 3.5.

    In D&D having dice as poor quality as the $0.10 die (which is unlikely) is only like to shift the average result of a skill roll, attack, damage or saving throw by 1 or 2. True this might get you killed but more likely is that you will slightly over or under perform across the session.

    However most D&D Dice are not poor quality. The variation to the average result is going to +/- 0.01 or even less.

    Can you honestly say that having a d20 that rolls on average 10.501 instead of 10.5 is going to make the slightest bit of difference in a game as complex as D&D?

    Chris

  4. on the astronomical occurance that a 10.501 gets you killed vs the 10.5 has you killing the “dragon of doom”, yes, that would be a soab moment.

  5. hey thanks for the info i picked some up at a hobby store like 5 years ago cause they looked cool but i didnt know they were for d&d till i played with my friends one day. they are pretty good quality. thanks for the info.

  6. Good to know. Now I know that when looking for a 6 on a d6, I’ll use one with pips, but when looking for a 1, I’ll use one with etched numbers.

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