# [RPG] Is GameScience’s non-cylindrical seven-sided die a balanced/fair die

dicestatistics

You might have come across GameScience's seven-sided die before:

There's mixed discussion of whether it might be biased toward the 6 and 7 faces (the 6 being on the opposite side of the 7 you can see in the photo). It's a GameScience die, and they tend to market themselves on making properly fair dice.

The only actual analysis I've found is a YouTube video, “Does a D7 (Seven-Sided Die) Roll Fairly?” posted by KingKool2099 on 24 April 2012. At 4 minutes 20 seconds they call their own results inconclusive, suggesting they may be introducing bias in their rolling method. (They found bias toward 6 and 7, but also toward the 2 which is on an edge.)

Has there been any decently conclusive analysis of whether this die is a fair die? Has there been a mathematical analysis, one where someone put it through a dice tower?

## The real experiment is difficult

The linked dice in the question are out of stock, so only people who already have a lot of these dice and are willing to do the statistical tests can give the "true" answer. I suspect that that population is quite small. However, I think that the existing literature and a bit of deduction can give a theoretical and historical perspective on the fairness of this d7.

## It's possible to have a fair d7 in specific scenarios

First, it is definitely theoretically possible to have a seven-sided die. The die as shown is a pentagonal prism. Geometrically, the die's fairness is most strongly affected by the size ratio of the pentagonal faces to the rectangular sides. I've made a quick mock-up of the two extremes:

• Faces are larger than sides: This corresponds to the shape on the left side. This extreme strongly favors the pentagonal faces--it's basically a coin, and it's difficult to imagine it ever landing on the edges.

• Sides are larger than the Faces: This corresponds to the shape on the right. In this case, the die is more like a pencil, and will almost always fall on the sides.

As one smoothly adjusts the size ratio between the sides and the faces, there will be a specific point where there is a transition between favoring the sides and favoring the faces. This intercept is the point at which the die is fair. Therefore, it is possible to have a fair 7-sided die.

However, this magic ratio might not be the same for all conditions. This answer on MathOverflow argues that for non-isohedral die, the fairness of the result depends on how you throw it. Likewise, this random page on the internet claims that different surfaces might affect the outcome of the roll. Neither source provides hard evidence for their claims, but it's worth considering that the intermediate value argument presented above does not prove that a single d7 can be fair under all conditions.

## The patent for this d7 shows that it was tested for fairness

So the question is, do those specific dice have the necessary geometry to be fair? The product description page that the OP links to contains a patent number: US PAT No. D-4,900,034. This number corresponds to the patent "Random gambling playing pieces and layout and game table for use with the same" filed by Bernard Bereuter in 1988. This patent, among other things, describes the construction and fairness for this particular d7 for gambling purposes:

Using playing pieces formed of hard plastic of a type such as could be used for standard dice, experimentation has shown that the desired random landing of the pieces is achieved if the regular pentagon of cross section fits precisely in a circle 1 inch in diameter (resulting in peripheral edges 3 of a length of 0.588 inch) and the length of the prism is 0.753 inch, for pieces rolled over foam-backed felt stretched over a hard horizontal surface.

...

A random gambling playing piece comprising a nonrectangular prism... having indicia spaced uniformly around its circumference, the length of said prism being different than the length of a side of the regular polygon transverse section and being equal to the length required so that the probability of said prism landing on either of its end faces is approximately equal to the probability of its landing on any of its side faces.

Thus, Mr. Bereuter has apparently done the empirical testing necessary to determine the ideal size ratio for a fair seven-sided die, at least on one particular surface.

Unfortunately, because his data isn't public and the dice aren't currently available, we can't verify or repeat his results ourselves. It's certainly possible that the dice bought from that site are not fair for other reasons, too. Still, this d7's inventor clearly put significant effort into determining the necessary dimensions to create a fair die.

If GameScience accurately followed the dimensions in the patent that they cite, then their d7 is likely fair enough for RPG purposes. After all, the original patent intended the die for gambling purposes, and in my experience, TTRPGs are far less sensitive to unfair dice than gambling is.