Are HexaCubes Fair?

Fairness and Accuracy

Most of the traditional polyhedral dice are examples of Platonic solids. The two 10-sided dice are the exception. Theoretically, there is no die that is as fair as a Platonic solid. In all situations, shapes such as the 8-sided octohedron are as fair as dice can be. I should point out that no die is utterly fair. Experts, such as Stanford University professor Persi Diaconis, have proven that a rolled die is a tiny bit more likely to land on its original orientation. However, to say this is splitting hairs would be an understatement. A perfect Platonic solid would be practically fair. Fairness has slightly different meanings in various fields of study. I will define it here as the systematic bias of an instrument. A perfect cube has virtually no bias among sides.

Though 10-sided dice are not Platonic solids, all of their faces are the same shape and size. This means that they are as fair as Platonic solids in any reasonable situation. An example of an exception to this is if you drop a d10 through a very thick fluid. In that situation it is more likely for a d10 to land on its middle round rim, rather than one of its points. People use d10s anyway, because exceptions like this do not matter. They are fair in any situation where tabletop gaming happens.

Accuracy, however, we will define as a slightly different concept. Accuracy is a measure of how reliably an actual tool works in practice. A perfect cube is a nearly perfectly fair die, but it is impossible to make a physically perfect cube. The accuracy of any die is subject to the tiny imperfections that affect any die. It has been long established that mass manufactured dice are imperfect enough to affect the outcome of a die. In many cases, particularly with d20s, this is detectable with normal use. In the world of tabletop gaming, there really are “lucky” and “unlucky” dice.

The HexaCube Solution

HexaCubes are shaped like chamfered cubes. Chamfered cubes are made of two different kinds of faces. There are six square sides, like a normal cube. Also, there are 12 six-sided faces that are identical in shape and layout with each other. Chamfered cubes are theoretically less fair than Platonic solids and d10s. However, it is not a given that a die that is unfair in some situation is not fair when used as a gaming die. Recall the d10 that is fair in any situation that gaming dice would be used, even through it is not fair in every conceivable situation, as a perfect d20 (icosahedron) would be.

HexaCubes are like that. HexaCubes have been tested rigorously under a wide variety of conditions. Throughout their development, various HexaCubes were rolled over 6,000 times in total and the results recorded. They were thrown hard into a shoe box. They were rolled on glass. They were rolled on mouse pads. The results are in, and as long as the dice do not actually roll (not counting air travel) for more than three feet or so, there is no detectable difference. Rolls longer than three feet might slightly bias the results in a statistically significant manner towards landing on square faces. The biggest difference found in testing was about 2/3 of 1 percent variance per square face. This was accomplished by gently rolling the dice in a curved path down a two foot half-pipe like device. HexaCubes are practically fair under normal gaming conditions.

As an additional safeguard, the numbers on HexaCubes are distributed in a manner to minimize the effects of any difference between the odds of getting a square face result or a hex face result. Regardless of any bias, d6 and the binary die are completely immune to any effects of a bias toward one side type or the other.

Through redundancy, HexaCube dice provide an accuracy advantage. On a typical die, the accuracy of a result is dependent on how perfect each side of the die is, and no die is perfect. This is also true of HexaCubes, but many of the outcomes on each die are on more than one side. Each face will have a random amount of error, but several faces will conform to the Central Limit Theorem and will be closer to the designated value. If, for example, there is a flaw on the ‘4’ face of a normal d4, it will be inaccurate by +/- a percentage. In the case of HexaCubes, any variance of a “4” face would be averaged with the variance of the other “4” faces. It is more likely that an average of multiple random errors is closer to average (zero, in this case) than one sample.

A Typically Numbered Die: % of each outcome = % +/- error 1 side

Not every number on every HexaCube die has this redundancy. On dice where single results appear, they are assigned values in the middle of the dice. This means that any anomalies are concentrated on the more average results, further bolstering the amount of confidence a player can have in the overall outcome.

Numbered 1-4, Twice: % of each outcome = Avg. of % +/- error 2 sides

The d20 is the most problematic of the polyhedral dice. Due to its small faces and potential warping when cooling, it is very likely that the d20 any particular player is using is biased towards certain outcomes. HexaCube d20s are superior. The binary die has two outcomes that are on 9 faces each. Any typical imperfection of a face, or even an axis, of the die is going to make very, very little difference. With 3 ones, twos, nines and zeros on the d10, players can be sure that their criticals and fumbles are well earned. Also, they can be slightly more sure of any other result because the single numbers are on the 6 larger faces. These partial cube faces are easier to accurately manufacture than small icosahedron faces.

The HexaCube Solution: Avg. % +/- error 4 to 5 sides

By trading a little theoretical fairness for a great deal of practical accuracy and coupling design with quality manufacturing, HexaCubes will give players the results they deserve, for better or worse (HexaCube Dice takes no responsibility for catastrophic rolls. Sometimes your space knight is just going to have a rough day at the office).

Whew! That’s everything and more that about 30 people worldwide wanted to know about HexaCube fairness and accuracy.