Is GameScience's non-cylindrical seven-sided die a balanced/fair die?

Question

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?

Answer 1

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.

Answer 2

No

(With 99.5% Confidence)

I purchased 5 of these dice to test their fairness. I rolled the dice 2250 times (450 sets), by hand in the dice bowl pictured above, and recorded the results.

Face	Count	Deviation
1	306	-0.7%
2	343	1.0%
3	281	-1.8%
4	307	-0.6%
5	298	-1.0%
6	369	2.1%
7	346	1.1%

Performing a Chi-Square Goodness-of-Fit Test, we can test how likely these results are from a fair die. I found a 0.5% chance of observing these results from a fair die.

The results are even more evident if we group all the edges (1-5) and the faces (6, 7). We can see that the dice lands on the 6 and 7 3.2% more often than a fair dice.

Face	Count	Deviation
1-5	1535	-3.21%
6-7	715	3.21%

Can we extrapolate the fairness of a single dice with pooled results?

Yes

These dice are very close; weighing them with a mg sensitive scale, they are all within 1.5% of one another and purchased at the same time (presumably from the same batch).

Mass (g)	Deviation
6.500	-1.43%
6.534	-0.92%
6.599	+0.07%
6.534	-0.92%
6.689	+1.43%

I rolled them each individually 200 times, and found no evidence to reject the notion that the dice have the same distribution as the pooled results at the 95% level.

Results

Face	Die 1	Die 2	Die 3	Die 4	Die 5
1	28	34	25	26	29
2	35	22	30	35	30
3	21	22	18	26	21
4	26	27	38	30	25
5	30	32	32	21	31
6	25	31	28	35	26
7	35	32	29	27	38
Total	200	200	200	200	200

Deviations

Face	Die 1	Die 2	Die 3	Die 4	Die 5
1	-0.29%	2.71%	-1.79%	-1.29%	0.21%
2	3.21%	-3.29%	0.71%	3.21%	0.71%
3	-3.79%	-3.29%	-5.29%	-1.29%	-3.79%
4	-1.29%	-0.79%	4.71%	0.71%	-1.79%
5	0.71%	1.71%	1.71%	-3.79%	1.21%
6	-1.79%	1.21%	-0.29%	3.21%	-1.29%
7	3.21%	1.71%	0.21%	-0.79%	4.71%

Below is a plot of the portion of rolls that came up on each face for each die, and the distributions measured from the pooled result above, as well as 95th percentile error bounds for the portion of rolls.

Answer 3

No, they are not fair unless you ignore the sides

There are a few requirements for a uniform solid die to be fair.

The active sides must be face-transitive.

Dice are only fair if all of the sides that are being used are equally likely to be landed on. In order for this to be true, it needs to be face-transitive, meaning all the sides are the same shape. More specifically...

Isohedral Figure

In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

Isotoxal Figure

Regular Polyhedra are isohedral (face-transitive), isogonal (vertex-transitive) and isotoxal (edge transitive).

This 7 sided die is neither of those things. But it is if we ignore every result on the pentagonal sides.

Put another way, given a face on the die, there must be a rotation (at least one) that results in every other face, edge, and vertex being mapped onto the same place as a different face, edge, and vertex, respectively. Let's try it in 2-d.

This makes a good 2 dimensional die. Rotating the triangle 120 degrees around the center maps every vertex and edge of the triangle to another one. Let's take that to 3 dimensions, say a cube. A d6. We're all familiar. A d6 is a fair die because there exists at least one rotation that results in each face, edge, and vertex being mapped onto the location of a different one. One of those rotations would obviously be a rotation that can be represented by "90 degrees on one axis, and 90 degrees on another". Or, in Euler Angles, 90, 90, 0. Or, if it helps, 90 degrees pitch, and 90 degrees yaw. Or any combination of pitch, yaw, and roll.

All other fair dice have this property. A rotation exists that maps every face, edge, and vertex of a d4 onto a different face, edge, and vertex. There exists one for a d20. There are in fact, many rotations that do this for these fair dice. But there is no rotation that does this for a d7. You could spin it 180 degrees around the "up" axis (sitting on neither 6 or 7), but then the top edge would not have been translated into the position of another edge. You could lay it flat on 6 and spin it 72 degrees, but then the pentagonal faces would not have been translated into another face.

The center of each face must be equidistant to the center of mass.

When it comes to (fair) dice, the centre of mass is in the exact centre of the object. This means all the faces are equidistant from it. The result of this is, after a roll, every face has equal opportunity to come up. However, if the centre of mass is moved from the geographical centre of the die, then the axis of rotation is changed, and the die is no longer fair. source

Changing the center of mass is known as weighting the die. As the centre of mass is moved further from the middle of the die, the effectively lighter face will roll upwards more often than not.

Making fair dice by ignoring faces

Dice with an odd number of flat faces can be made as "long dice".[26] They are based on an infinite set of prisms. All the (rectangular) faces they may actually land on are congruent, so they are equally fair. (The other 2 sides of the prism are rounded or capped with a pyramid, designed so that the die never actually rests on those faces) Source

That last sentence is the most important part. This 7 sided die is fair for ranges 1-5, provided you ignore the 6 and 7th face. As we read above, any prism can be fair provided the ends are "capped" or ignored (see Long Dice). So, a real d7 would be made of a heptagonal prism. So, ignoring the ends, there exists a rotation that maps every face, vertex, and edge onto the location of a different face, edge, and vertex. Lets go back to that example above. We lay it flat on the 6th edge and spin it 72 degrees. Voila! Each of the faces is now in the location of where a face used to be, each edge is in the place of where a different edge was, and each vertex is in the place of a different vertex. Except for the caps, which we've ignored.

More recently, you may have noticed barrel dice. They use the same basic principle, except their sides are triangles rather than rectangles.

Why don't non-symmetric unorthodox shapes work?

The result of the die being face-transitive and having a center of mass equidistant from the centers of the faces is that it requires the same amount of force in one direction to turn it over no matter what face it has landed on. When we look back at the d7, we can easily guess that applying force to go from face 1 to face 2 is the same amount of force that will change it from face 2 to face 3 as it rests on the table. This is due to the fact that the angles between the faces are the same, and because the faces are the same on those sides. There is as much surface area touching the table when "1" is up as there is when "2" is up. Let us consider faces 6 and 7.

When face 6 is up, face 7 is down. There is now a greater surface area on the table. Moreover, the angle between face 6 and any other face touching it is greater (90 degrees versus 72 degrees). Both of these mean that it requires more force to push it onto one of the other faces. So when the die is tumbling and face 6 or 7 hits the table near the end of the tumble and loses some of its velocity and rotational velocity, it is more likely that X amount of force will not result in the die tumbling over that face to land on 1-5.