"Unfair" Dice

Okay, I've got a weird question regarding oddly-shaped dice.

So imagine a polyhedron where the faces are made up of two different types of shapes (pic related). If you rolled a die shaped like this, then I assume there would be a greater chance that it would land on one of the larger sides, right? So I'm wondering: is there any way to determine the probabilities for rolling oddly-shaped dice like these one?

I ask because these sorts of dice could potentially produce some really bizarre and unique probabilities.

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mathartfun.com/thedicelab.com/DiceDesign.html
hermetic.ch/misc/dice/dice1.htm
youtube.com/watch?v=G7zT9MljJ3Y
youtube.com/watch?v=8UUPlImm0dM
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I made a couple charts to help illustrate what I mean.

For instance, the die pictured to the left has 14 faces total, made up of 8 triangles and 6 squares. For the sake of argument, let's assume that you're twice as likely to land on a square side (since they're larger). The results would vary depending on which numbers you choose to put on the triangular sides and which ones you choose to put on the square sides. So I made these graphs to show the results for multiple difference scenarios.

Yes, these results are basically just a wild guess since I have no idea what the actual chances of landing on a square side compared to a triangular side. But my goal here is just to explain why I I think these sorts of dice could potentially be interesting.

I doubt even the world's greatest geometrician can do better than just rolling a few thousand die and determining the approximate odds empirically.

The empirical way (rolling it 10000 times and recording it) always works but other than that direct correlation of sizes is skewed (and I can't imagine it beeing linear)

Bullshit, just calculate the relative surface areas of the sides and calculate the probabilities from there. In a regular dice each side consists of one sixth of the total surface area and thus has 1/6 chance to be landed on.

But user, that's higher math. It's easier to roll it a bunch of times.

It don't really know anything about this, but wouldn't the results vary a lot depending on what material the dice was made of and what surface you were rolling on? I would think that standard dice work under any circumstances because all the sides and edges are the same.

What I mean is that when it comes to standard dice, it doesn't matter if you roll them on a carpet or on the moon, because the part that hits the ground will always be the exact same faces and edges, and hence the same probabilities. Does that make some sense?

I get the feeling weight distribution and angles are going to affect rates.

That was my first thought, there should be a good way to add a bit of weight to the small faces so they have a more equal chance, but then you need to calculate the odds to find the best weight.

Or would having a bit of weight added evenly across the die actually affect it?

The problem is that these dice can't be symetrical (not possible with non equal surfaces) and as such have different weights and kinetic energies above the surface

Sure the approximation over surface area might work depending on the specific dice but in general you can't assume it to be right

Empirical is just the better way of doing it.

I've always wondered if "twice" d20 gives better randomization than a normal d10 because it actually is a platonic solid.

Only if you roll it through a straight drop with minimal movement, and even then, it's questionable because it might tilt on the way down.
Cup it between your hands and give it a good shake so that you can't know what point is up or down and there shouldn't be any functional difference in probabilities.

Yeah d10 is kind of my least favorite of the standard dice cause it's so imperfect...

however the d4 is perfect but isn't very good as a die cause it doesn't really roll. you have to flip it like a coin.

lol! The differance exists, but its usually small enough to not matter!

I still prefer the doubled icosohedron, however! Lol! I have strict principles!

Here's some VERY anecdotal evidence: mathartfun.com/thedicelab.com/DiceDesign.html

>A drawback to the truncated tetrahedron (4 triangles, 4 hexagons) is the fact that it can land on a small triangular face, which we've found happens about 3% of the time.

>A drawback to the truncated octahedron (6 squares, 8 hexagons) is the fact that it can land on a square face, which we've found happens about 2% of the time.

But I think that the main thing that matters is the location of the center of gravity. And because the shapes are so symmetrical the center of gravity is the exact center (ignoring all the imperfections iaterials and the markings)

Why not make a small face have some sort of special function in-game? If it's about a 2% landing, have half the squares be 'good' and half be 'bad' critical effects or something.
It's not enough to justify buying new dice, I think, but it's a neat idea.

>drawback
Eh, call it a critical.

The shape in OP's picture is a cuboctahedron. Here's something I found online:

> An interesting problem is: When a cuboctahedral die is thrown what is the probability of a square face resulting?

>If the probability of a face of a certain type resulting depends on the relative area of that face (square vs. triangle) then one might reason as follows: Assuming a unit edge for the cuboctahedron the area of a square face is 1 and that of a triangular face is sqrt(3)/4 = 0.433, so the probability of a square face turning up is 6 / ( 6*1 + 8*sqrt(3)/4 ) = 0.634. But it might be that the probability depends on the physical properties of cuboctahedral dice in motion and how exactly a die comes to rest on one face, in which case the question cannot be answered so simply.

So his estimate based solely on surface area is actually relatively close to OP's estimate:
>Square = 63.4% total (10.566% for each face)
>Triangle = 36.6% total (4.575% for each face)


Source: hermetic.ch/misc/dice/dice1.htm

>dat probability for getting an 11 with 2 prime dice

Well that's some fucking A grade heresy and witchcraft.

>Bullshit, just calculate the relative surface areas of the sides and calculate the probabilities from there.

There's more that goes into it than that. How you roll the dice and how they bounce are pretty important. I can't remember the details at this point, but I remember that Lou Zocchi was talking about the results he got out of either the gamescience d5 or d7, and he had tested the thing and gotten a pretty good distribution, but somebody had said that his results weren't very evenly distributed, and it turns out that the surface he was rolling on didn't give much bounce to the dice, which apparently made all the difference.

>I've always wondered if "twice" d20 gives better randomization than a normal d10 because it actually is a platonic solid.
What does that have to do with anything? All the faces on the d10 are equal.

Just build a physics simulation that throws the die at a random angle with random amount of spin and run it until you get bored of this shit.

I'm trying to calculate probability of particular numbers being rolled on exploding/poisonous d12s; anyone know some formulas I might be able to use?

This picture makes me very sad and I can't put my finger on why. I think it's because the puppy is just so happy and oblivious and tiny and vulnerable, and the soldier will probably die.

Numberphile did some videos on this, asking actual mathematics and physics professors.
youtube.com/watch?v=G7zT9MljJ3Y
youtube.com/watch?v=8UUPlImm0dM

They decided that a die is "fair by symmetry" iff it is "transitive upon the faces", i.e. for any of the faces, the die can be placed on any of its faces, and the rest of the die turned so that the die fills the same 3d space. For example, a d6 is fair by symmetry because you can pick it up and then place it down back on its spot, on any of its faces, and it will be filing the exact same cubic centimeter of space as it did before. You can't do that with the dice in the OP pic, because if you turn it, for example, from a triangle-shaped face to a square-shaped face, you won't be able to fill the exact same 3d space with that die.

There were other criteria for the fairness of dice though, relating to how many difficult it was control the dice, as well.

You don't need oddly shaped dice to produce results like that. You could just have dice which have certain numbers repeated or something. For example, I've got some six sided dice which are numbered 2,3,3,4,4,5.