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Given a perfectly formed d8, or d10 or any d dice in Dungeons & Dragons (D&D), are they all fair dice? Is it equally possible to roll any number on any given dice?
I am writing a text based, online D&D engine that would allow a DM to create their own world and invite their friends to play that world online. I am writing a cryptographically secure random number generator to roll the dice, but knowing nothing about D&D, I don't know if all the dice are fair.
Of these, the d4, d6, d8, d12, and d20 are regular polyhedrons.
The d2 and d10 are not regular polyhedrons, but each face is nonetheless equally-likely.
Perfectly formed is a bit tricky, but yes.
If a solid is rotationally symmetric such that one side being face up is the equivalent of any other side being face up, then there are no differences, and as such have the same probability.
If you're interested in testing the fairness of real life dice, I'd suggest looking here: https://stats.stackexchange.com/questions/3194/how-can-i-test-the-fairness-of-a-d20
It is very easy to see if a die is fair, just see if you can deduce anything about the outcome without the faces. For example, a coin without faces is just a disc and a d6 without faces is just a cube. You can't tell anything from a resting cube or disc, so they are both fair. The same goes for all dice commonly used in rpg games. Technically a coin can stand on its edge since it is not a perfect disc, but if we ignore those cases it is fair. In the same way we can create all kinds of dice by just rolling a pen with the amount of sides we want. For example we can get a fair d5 by rolling a pentagonal pen.
Yes, all the standard polyhedral dice used in D&D (d4, d6, d8, d10, d12 and d20) are "fair".
In particular, the following assumptions are sufficient to guarantee that a polyhedral die will land on each of its sides with equal probability:
In particular, assumptions 1–3 essentially guarantee that the physics of rolling the die will not intrinsically favor any side above the others, while assumption 4 says that the initial orientation of the die also doesn't favor any side.
The extra assumption 4 is needed because I have not assumed anything about how the die is rolled — for all we know, it could be just dropped straight down or even simply placed down on the table. In practice, of course, we hope that the bouncing motion of a properly rolled die is itself sufficiently chaotic (i.e. highly sensitive to even tiny variations in initial conditions) as to render the final orientation effectively random even if the initial orientation of the die is not. But it's hard to determine exactly how much shaking and rolling is actually enough to achieve that, so it's easier to just dodge that issue by assuming that the initial orientation of the die is already random and just show that the shape of the die itself won't introduce any bias to the results.
As noted in the MathOverflow thread mentioned by Mark S. in the comments, the assumption of isohedrality isn't strictly necessary; there's a mathematical argument that there must be fair non-isohedral dice too. Well, sort of…
For a gist of the argument, consider taking a thin round coin (with two identical flat sides and a very narrow cylindrical edge) and stretching it out into a long cylindrical rod with the same diameter. Clearly the coin will almost always land on one of its two flat sides and almost never on the edge, while the long cylinder will almost always land on its round side and almost never on one of the flat ends. In particular, as the probability of the coin/cylinder landing on its round edge/side is a continuous function of its thickness/length, there must be some halfway point (looking like a very thick coin or a rather short and stubby cylinder) where the probability of it landing on its edge is exactly equal to that of landing on one of the two flat sides, making the "coin" a fair non-isohedral three-sided die.
However, the problem with this argument is that it only proves that such a halfway point must exist for any particular way of rolling the die — it does not prove that the same halfway point works regardless of how the die is rolled, and indeed there's no reason to expect that it would. So, in effect, the "fair" non-isohedral die constructed in this way will only be fair if rolled with a particular amount of force on a particular surface using a particular rolling motion. Basically, if the die is fair for you, it probably won't be fair for me, and vice versa. Which isn't really very useful or fair at all, when you think about it.
For isohedral dice, on the other hand, the symmetry of the shape forces a corresponding symmetry on the physics of the roll — in effect, it ensures that the sides of the die are indistinguishable as far as the physical laws governing the motion of the rolled die are concerned. Thus, a uniform isohedral die will be fair regardless of how it's rolled.
FWIW, we can relax the isohedrality requirement slightly and still maintain fairness regardless of the rolling method. In particular, it's not quite necessary for all sides of the die to be interchangeable by an isometric symmetry — it's enough that this holds for all sides that the die can land on in a stable orientation, or at least for all sides that are actually counted as valid rolls.
For example, a dreidel is an example of a highly non-isohedral object which, nonetheless, is shaped so that it can only come to rest on a flat surface (with a non-negligible probability) in one of four symmetric orientations, and thus behaves like a fair die. The same is true for other kinds of teetotums and long dice, including the "barrel dice" sometimes used in RPGs.
Also, as noted above, even a simple coin used as a two-sided die technically also has a third side (the edge) that it can land on, which is not isometric with the two flat sides. But the edge is narrow enough that this is very unlikely to happen, and if it did happen, one would generally just flip the coin again until it came to rest one one of the two flat side, which are (approximately) isometric with each other.
Fairness of a die depends a lot on fine details (are the edges rounded exactly the same? is the die from a homogenous material? this kind of thing), but from a theoretical, purely geometric point of view, all classical dice are "fair", in the following sense: each face of the die could be sent to any other without deforming the die, and keeping the contour of the die globally unchanged. This is enough symmetry to ensure that no face is favored over any other. For the mathematically inclined: the group of isometries of each solid acts transitively on the faces.
The classical Platonician volumes (D4, D6, D8, D12, D20) have even more symmetry than that, but even the usual D10 has enough symmetry for that.
A die will have equal likelihood of each outcome if (a) each face has the same surface area and (b) the center of gravity of the actual die aligns with the center of the polyhedron. The reason dice have a set pattern for number faces (for example, in a D20 1 is opposite 20, and neighbor to 19, which is opposite 2, which in turn is neighbor to 18) is that if the center of gravity is misaligned due to imperfections in the manufacturing process, the arithmetic mean of the outcomes over a large number of throws will still aproach (1+number of faces)/2.
If you're writing a program to generate uniformly random numbers, and you do it correctly, every number is equally likely, whether or not the physical dice are equally likely to roll a given result is moot.
You could create a number generator with a quadrillion possible results and provided your coding was on point and your algorithm for generating uniformly random numbers was correct, each face would be equally likely to occur.
The randomness of dice is in part due to their shape, but someone skilled at throwing dice a specific way could still end up with a much more favorable result than someone throwing them randomly.
Dominic LoRiggio was a man who went to a casino and played Craps with d6's, but threw the dice in such a way that the dice stuck together when thrown, and barely tapped against the far wall of the craps table, which let him control which number the dice would end up on.
There are also flaws in the dice that are made using specific manufacturing processes that create inherent flaws in the center of gravity. More about this method and how it can be detected can be found here.
You can also use a cylinder with equally wide squared off triangular facets to make cylindrical dice with any amount of numbers on them that will produce a random result when thrown. These dice look like this.
As KRyan said, most of the dice are in fact, balanced and when thrown will land on a random result provided their manufacturing method doesn't create any deviance in its center of gravity.. but if you're going to be coding a random number generator, that doesn't matter in the slightest.
