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Not all dice are created equal: A survey of the debate over dice fairness


Not all dice are created equal. At least that is the claim of the few manufacturers that produce "precision" dice. And they appear to be right, although how meaningful this is in the context of D&D is debated. What are we talking about here?

A die's fairness is measured, at least in one understanding, by how close its roll result is truly random. For example, you roll a d20. There should be a five percent chance of getting a 20. Not six percent, but 5.000 percent. Each number on a d20 should have a five percent chance of showing on a roll. This perfect randomness exists only in the pure world of abstraction. You could say that the Platonic solid icosahedron is the eternal form upon which inferior earthly material copies are based. Whew.


So how do mortals muck up perfect randomness? One issue is how dice are rolled. We'll get to that later. Another issue is the material and manufacturing process used to make dice.


Material, Manufacturing & Precision Dice

Whether metal, plastic, stone, or glass, if the material of a die has air bubbles, impurities, or some other cause of unevenness in weight, then the die is not balanced. Even if perfectly shaped, if one side of the die is heavier than its other sides, then the heavier side will have a greater chance of being on bottom when the die comes to a rest. Or at least common sense suggests this. One academic appears to disagree.

Another issue is shape. In theory, the distance between every opposite pair of faces of a die must be identical. The faces of a die must of course also be identical. If these conditions are not met, then the die will be asymmetrical and will not roll with perfect randomness. This brings us to the great debate between Louis Zocchi, founder of Gamescience, and almost everyone else in the dice business. Well, it's not really a debate (except among bloggers). Zocchi has pugnaciously but entertainingly asserted the superiority of his dice for over 40 years. The rest of the industry (and there's a lot of them) appears bemused and more or less silent.

Most dice are injection molded molten material is injected into a mold. Here is the problem according to Zocchi. Most dice companies place the molded die in a tumbler to polish off the sprue and the flash. Sprue is the excess material that solidifies in the passage through which molten material is introduced into a mold. Flash is the excess material attached to a molded item usually at the seam of the mold. Then the entire die is painted and put in a tumbler again, this time to rub off the paint except for the portion trapped in the number indentations. The die is then tumbled again with extra fine abrasion to polish off the scratches caused by the prior tumbling. Here's the rub. Tumbling softens the edges of a die, and does so irregularly. Some edges are rubbed down more than others. Voilà – an irregularly shaped die!

Zocchi does not use a tumbler, hence his dice have sharp edges, are produced unpainted, and have a small sprue mark. The yellow die is Gamescience (yes, it's pretty old). It's hand-inked. Even given its age, the edges are relatively sharp. The green die is a basic tumbled die, with significantly rounded edges. Zocchi's claim is that such irregularly worn-down edges and the often oblong shape of tumbled dice are not as symmetrically shaped as his dice and thus do not roll as true as his dice. Some manufacturers of metal dice, also usually injection molded, reiterate Zocchi's sermon about the sinfulness of tumbling. For example, see Die Hard Dice.


Precision Dice are Fairer

Zocchi's claim appears to be true. See the Daniel Fisher video, dated July 29, 2015, where he applies the saltwater test to dice to measure balance. He also measures the width of the dice. Gamescience dice prove more balanced and symmetrical.

In this Awesome Dice Blog post of April 18, 2019, the authors review the results of a randomness test using Gamescience and Chessex d20s. The dice were rolled 10,000 times by hand! Gamescience rolled truer. The authors question the meaningfulness of the difference in the context of gaming, but are confident of their results. They do provide the raw data of the test so others can do their own statistical analysis.

In this 1000d4 blog article, dated February 14, 2013, Alan De Smet and Eva Schiffer measure the distance between opposite faces of d20s from Crystal Caste, Chessex, Koplow, and Gamescience. It is a thorough effort using a digital caliper. Gamescience proves more symmetrical, although the authors admit to not knowing how this actually affects the rolls. The authors also have links to other tests and results of dice shape and randomness when rolled.


This is enough evidence for me to conclude that Gamescience dice are more symmetrical and balanced and roll more randomly that other brands. Zocchi's rationale for this makes sense on its face and the experiments referenced above appear reasonable in formulation and execution. Of course, as the authors of the Awesome Dice Blog article mention, the difference here may be so small as to be of little significance, at least in the world of a role playing game. But even if small, I think it is worthwhile as a Dungeon Master to assure players that your are trying to be as fair as possible when it comes to dice. Even if the physical reality is small, I think the emotional benefit, message to the players, and magical fun of it all is not.


Really, Really Precise Dice


I love the fact that Zocchi dedicated a good portion of his working life to promoting more precise dice. I have to acknowledge here, though, that a company has exceeded Zocchi in the quest for perfection. And that company is Gravity Dice. The company's website says it all: "Our dice are precision machined out [of] aircraft grade aluminum, featuring chamfered edges, which allow the dice to have a more complete, random roll. The dice are then anodized and coated in our signature coating, to protect from scratches and fading. Lastly, the pips are drilled to perfectly calculated depths, meaning each die has a perfect center of gravity." That is a lot of effort in polyhedral dice production.

I am not aware of any randomness tests with Gravity Dice, but my guess is that they win hands-down. I couldn't resist and bought two d20. Here's an image of one of them. They make the usual variety of polyhedral dice used in gaming. They look and feel beautiful and I bet they are close to perfect. They are, understandably, more expensive than plastic dice, but they are a work of art and science.

And of course, near perfection in a d6 is old news. Casino quality d6 are machined out of quality resin and are intended to be made as close to the Platonic standard as possible. Not a surprise: what's at stake in Las Vegas is billions of real dollars exchanging hands every year. A mechanical engineer conducted a fascinating experiment on Games Workshop and Chessex d6s to test their randomness. He also consulted a physics department friend as well as reviewed results from a test conducted by Caesar's Palace. Not surprisingly, he found problems with the game company dice. At the end of the day he recommends using casino dice in gaming. This is common sense. I think they are not commonly used in D&D because, to the extent players know about them, they clash with the aesthetics of the RPG industry and the gain in accuracy is not worth it to most players. I have 16mm casino-quality dice, which are smaller than most regular casino dice, and like them very much.


Dice Rolling Technique


Our focus has been on the physical dice themselves. Obviously, how dice are rolled is important, when it comes to fairness. In short, a die (and we are talking about the all-important d20 for most role playing games and the d6 for many wargames and other board games) should be rolled from a height of at least 6 inches so as to bounce around and change face at least a few times before coming to a rest. You should not touch the die until the result is confirmed. You should not spin or slide it, hold it with two fingers, or know what number is facing up when it rests in your palm. For more information on this topic, see the interesting discussions here and here. There are also common sense protocols you should consider. See for example the mildly scolding but humorous discussion by Guy of How to be a Great GM.


Esoterica


For those of you who can't get enough about dice, here are a number of videos and articles by scholars that delve into the theory of shapes as applied to dice. Heavy stuff, and fascinating.

Here is Part 1 of a video where Stanford University Professor Persi Diaconi explains theoretical and practical issues of fairness in polyhedral dice. Fascinating and an unusual foray for gamers, to say the least. Here is Part 2 of video. You can also read a paper ("Fair Dice") by Professor Diaconi and Professor Joseph B. Keller in The American Mathematical Monthly.


There you go. May you earn the blessings of the gods of randomness. Happy rolling!

Photographs of dice in this post are by Ken Hardy.

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