# Fair Dice (Part 2) – Numberphile

Somebody made a dice rolling machine. There’s a YouTube video, and I bet you can find it, to test the claim that I made about the dice changing because of the drilling the holes in their faces. And he was a undergraduate at University of Chicago’s Statistics Department, and he did it as a final project for a course. And he had a camera he had an automated, Brady, and he rolled six dice at once. I think. And there was a table underneath it, and there were little springs, and, and a pusher, and so that pusher would push it, and then the springs would go like that, and the dice would go all over the place. And then when they finally came to rest, the camera would take a picture. And then he had something that parsed that picture, and would record it. And so, in that way he could get hundreds of thousands of rolls of a die. And, and that was one way of making a dice rolling machine, which was to test random rolling. There are schools in Nevada which claim to, which purport, to teach you how to roll dice in a casino and control them. And, now, it’s pretty questionable, but very roughly speaking, what they try to teach you is to roll the die so that it slides around without tumbling, and then of course, if you slide a die on the table, but just rotate it this way, the top face would just stay on top. You can imagine somebody learning how to slide dice in that way, so that they spun around. And maybe even if just one of the dice slides, that’s a pretty big advantage. The casinos, in order to combat that, make sure that when you roll dice they hit the backboard. And the backboard has ridge rubber, which are diamond fingers that look like that, and if, I promise you, if a die hits ridge rubber it gets bounced back, and tumbles back, and, and, and it’s, it’s a fascinating psychological and sociological setting. Suppose that you think you can control the dice. Of course you’re not perfect at it, but you don’t have to be perfect at it in order to get some kind of an advantage. You could trick yourself, even if you were watching a random sequence, into thinking that you were changing the odds a little bit. I’m sure that when you throw a die in a casino, and it hits the backboard with ridge rubber and comes off, that there’s no, no control is possible. Brady: “To be a fair dice, is really the only criteria that all your sides are the same shape and area?” So, in our theorem and, and the ones that you suggested, the ones with a two-thousand-gon here, all the fair dice in our families always had an even number of faces, okay? So now here’s a question. Is that, well there isn’t because I proved there isn’t, is there a fair five-sided die? Well, our theorem says no, but I’m gonna build one for you in the air. You know what a Toblerone is? They’re pretty good. They are good. So if I had a Toblerone, if I threw it up in the air and it landed on the floor, any of those three sides is equally likely. That’s certainly true. Now suppose we start eating it, and it gets thinner and thinner until it’s a thin triangular coin, right? I’ve just eaten until it’s just. Now, it never is going to land on these edges, but it could land here or here. That’s a two-sided thing. Okay, so it started out as a three sided thing. Now, by continuity, there’s some place, it has to be, where the chance of these three is the same as the chance of these two, and that would be a fair five-sided die. Now what’s wrong with that argument? Why does that argument, you know, not contradict our theorem? How much chocolate you would have to eat depends on what kind of surface you’re throwing the die on. If I was throwing the Toblerone up in the air and catching it in my hand, that’s one scenario. If I’m dropping it in sand or if I’m dropping it on a table so that it bounces, those are all very different physics, and very different dynamics. And how much chocolate you would have to eat in order to make a fair five-sided die would depend on the dynamics. Brady: “Mathematically, even in an abstract world of mathematics, that dice is not is not fair?” Is not, well, it’s not, it, you see, so so so suppose, let me try to, let me try to, I’ll answer with one more, one more example. How thick does a coin have to be so that it has probability a third of landing on its edge? Okay? Obviously, if the coin is very very thick so it’s a cylinder, well, it always lands on its edge. And if the coin is very very thin, it never lands on its edge, or almost never, and then it’s fifty-fifty. So how thick does a coin have to be so that it has probability a third of landing here, or here, or here. Well, by the argument I gave, you know, there’s some point in between where it’s true. Now, you might think, well, suppose I make the surface area of the band, of the edge, equal to the surface area of this face equal to the surface area of this face. Maybe that’s the right answer. When you cut broom handles to that thickness, it doesn’t work. It has to do with the dynamics and the physics in a rather complicated way, you could just see that. And, the argument that does work, take the thick coin, okay, embed it in a sphere, so there’s a unique sphere with the center of gravity of the coin in the middle of the sphere, and imagine that that sphere hits the table at a random point on its surface, okay? And then put any point on the surface, and then let the coin settle down to wherever it settles down. You want the area of the three spherical caps, this cap, this cap and the cap around the edge, you want those spherical areas to be equal. And that, it’s quite a different calculation. That seems to be the answer, in practice. Actually doing the physics nobody’s ever succeeded. It’s pretty hard physics problem. So physics has to come in, and, I’m sure that if we actually tried it, it would matter if the thick coin lands in the hand, or bounces on a table, or it bounces on glass, or bounces on a rug, the thicknesses would probably change. Brady: “You’re a mathematician. You’re in a statistics department. You’re a math guy.” Yeah. Brady: “Yet whenever I’m talking to you, “I can never quite tell where the boundary is between thinking purely “mathematical in the abstraction in a perfect world, and in a real world where, you know, “carpet is coarse and, and dice wear away, and you have to drill holes.” Well, that’s, that’s applied mathematics and applied statistics, and that’s what makes us happy. That is, it’s not only beautiful math, it actually says something about the world. And, the, it comes back. Once you look at the world, you think, oh my God, I cleaned that problem up to make me able to do the math, it’s completely unrealistic. Maybe I can go back and make it a little bit more realistic, in this trade-off between actually throwing dice, or, you know, working in a casino and, and doing the math, and doing group theory is what makes me happy. Brady: “It’s sort of imperfect math, though, isn’t it? It’s like you, you come up with some perfect math, and then you say, but that’s not gonna work.” Right, but sometimes the math is very robust, and then that’s the best. That’s, sometimes the math works wonderfully well, and, but, and you just, we have to live with that. Many of my colleagues don’t want to hear about the real world. And, and some of my friends that work in casinos couldn’t care less about the math. Don’t tell me about all of that stuff. You know, what’s the bottom line? Well, both parts of it make me happy.

Next video for Matt Parker, how long is a Toblerone that is also a fair 5 sided dice?

Matt Parker is working on the coin problem right now.

T o b l e r o n e y

see also: Maths Parker

What about a pentagonle prism

why Is he saying tobleronie, it's pronounced toe-bler-own. It's not lunchmeat

I still have this question hanging from the last video, is the 30 sided die fair or not? I got lost watching this at 2am and with all the kinds of symmetry 😛

For a 5 sided die just have a 10 sided one and pairs of numbers = 1 2 3 4 5

if the cylinders height is equal to its diameter would it have equal chances for each side to land on?

I'm sure many people managed to make a coin land on its edge. That possibility alone makes a coin toss probability different than 50/50

tobleroni

Tobleroni tobleronie tobleroney toblerone-E

I don't care about this knowledge for casinos (I don't gamble). It's D&D where this is going to come in handy for me! I want nat 20s! Lol

6:25

– "You are a math god"

+ (Puts sunglasses) "

Yeah."LULI don't gamble because I'm a Christian. Christians are banned from gambling although they're of age (21+). All other kinds of saints are banned from such mature activity. Leave the Asiatically Indian meat for Muslims and atheists!

i know it's not the point of the video but if you want a random number between one and five (ignoring websites) just roll a tall pentagonal prism

In the case of the 3-sided cylinder coin, what if we weighted it somehow to make it more fair?

A long cylinder but with the ends weighted 'unfairly', so that the ends have the same chance as the the long side…

I know, older video… but I wanted to mention that I have some odd sided die that I believe to be fair. They are made with a polygonal cross section that has a curved taper to each end. Three or five identical faces with the same number of identical edges and two identical nodes – and can't sit on its end.

someone should make a coin that cant land on its edge. the edge would need to be extruded so that the top and bottom surfaces approach the thickness of a point. you could also make a coin that is conical, and color heads/tails two diff colours, then base the flip upon up/down states. I imagine the conical coin would be perfectly fair, although it's dimensions would be based on some weird symmetry based on pi

5:10 try doing that by taking some of the old British 1 pound coins they made 20 years ago and glue a few of them together to see when it'll land on the edge.

Am i the only person who has never heard of a toblerone?

Interesting series. That said, I dispute the professor's claim that all fair dice have an even number of sides. This is true of polyhedra, where the sides are planar, but it's entirely possible to make a convex three-dimensional shape with an arbitrary number of identical,

curvedfaces that can be rolled fairly.Take, for example, a pentagon, with points pulled from the center of each side as shown in the video to form a fair 10-sided solid. As shown, this will have a planar pentagonal "equator". If you evenly smooth each side at that equatorial division, you will end up with a five-sided solid with symmetry that can be rolled fairly.

Wait. A fair d5 is just a d10 with only 1-5 on both sides

Well, that was a whole lot of nothing.

what if i take a golf ball and put a number in every divot ? would that be a fair dice?

My solution for an odd number die (n) is to take the even sided die (n+1) and change the highest value to "roll again."

That's why a d5 is just a 5 sided toblerone where the ends come to a tip instead of staying flat.

Get Mat Parker and Persi Diaconis together! Maybe we'll see a fair 3 sided coin happen.

what about dice with non-planar faces? like imagine a tennis ball, where the grooves extrude outwards to become edges. This would be a two sided round die, with 50/50 probability, even more fair than a coin flip. Couldn't this make basically any n-die conceivable, even oddly numbered ones?

Im baked and i want to eat these casino dice.

Perreroni Toblerone. Tohbleroni

no casino lets you throw dice directly because of those tricks, they always use cups

@30:00 … Does it sound to anyone else like there is some confusion to how this device actually works 🙂

His pronunciation of Toblerone triggered me

Why not curved surfaces?? A Toblerone that curved to points on either end could then be 5 sided and fair.

What of the d20?

Stop saying Tobleronie.

Can't you just say triangular prism no need to say toblerone because you pronounce it wrong

Also why were you like so so so so so

I've got a d5. This guy should hang out with D&D gamers. We have lots a dice.

Also, that guy with the automatic dice rolling machine should have put in a bunch of different colored dice, to check for individual bias of a specific die. "Oh, the green one is on a 2 almost 50% of the time, it must be off-center."

Would something like the Toblerone be considered a fair 3 sided die? Why not just have a similar shape but with pentagons on the ends instead of triangles or is that not considered a die anymore?

just make a pentagon prism

Tobleroni x'D

But those on the machine weren't casino dices.

Ok, sure. But what about this big Toblerone?

The three sided coin on Matt Parker is a nice tale

Can't you just take a D6 and round off three edges so that it's effectively made from three panels of exactly the same shape and size?

The math vs real world discussion have me the most out of this.

Geometry and quantitative sciences are somewhat pretentious this way. They find comfort in the perfection of concepts. My worry is that a concept is mostly never accurate to reality, it transforms reality in some direction.

Really great and informative video! Thanks for making it and sharing this knowledge! ^^

There can be a fair five sided die. If you take a pentagonal prism and put pentagonal pyramids on the ends, you would have a die that, although it has more than 5 sides, it only has 5 plausible outcomes, because if it was to land on one of the ends with a pyramid, the center of gravity would not allow it to remain in that position.

Using curved sides you can make odd sided fair dice. Like a 3 sided toblerone with the ends of the prism nibbled down to opposing points like an American football.

Wouldn’t a cylinder with the diameter equal to the length be equally likely to land on any of its 3 sides?

"6:42 that's applied mathematics" , it's the department at the other end of the corridor 😉

This guys so smart that no one’s ever felt they could tell him he’s been pronouncing Toblerone like an idiot

It wouldn't be a real dice, but a sphere with coloured regions of equal area could potentially work as a die for any number of regions and wouldn't have problems associated with multiple bounces although the "sides" might be different shapes.

This guy must have INT:18

What if, you put the center of the sphere at the center of mass of the object and project the sharp edges onto the sphere using rays from the center of the object/sphere to the edge of the sphere?

Please talk to him about shuffling cards.

Skew dice!

If two shapes with the same area but different in shape make a polyhedron, will it be a fair die?

Take a rod with a pentagram cross section, cut it to length of about twice it's thickness, then taper the ends to points. Voila; 5 sided die

It's easy to make a fair odd sided die. Similar to the d10, but instead of a hard edge, the faces curve from one point to the other.

I don't understand, why it wouldn't be possible to construct a fair dice with odd number of faces. Take the toblerone shape, and cut off the edges at each vertex at an agle, which makes it impossible to stand on it's end (center of gravity outside of the facet's area), and you have a fair 3 sided dice. You can even write all three numbers on all three faces in different sequence (123,231,312) first meaning the rolled digits, second and third painted in a color same of a dice. I bet i'd be more fair than a (non-casino quality) 6 sided dice.

What id you make a shape like a 5 sided cylinder and then roll it?

Genuine Q,

What about a 5 sided die, made from a pentagonal prism, except the ends of the prism taper towards eachother.

Similar to the 10 sided die made from pentagonal pyramids, accept the "top" pyramid, and "bottom" pyramid don't have an edge between eachothers equivalent faces.

Sorry, the above Q is badly, worded. But I'm not sure how else to explain :/

My kids play Dungeons & Dragons. I saved these videos too show them. Great job!

What if I eat all of the chocolate and pass out from a sugar crash instead of testing the fairness of a d5?

Well dice controlling is a thing and a guy has made millions of it

Which die is the most fair (the most difficult to influence in throwing)?

Consider a pencil sharpened at both ends. That's a fair 6-faced die. Any right "tapered ends" prism with a convex regular polygon base like the pencil, that is with any number of sides, would be a fair die in theory as well as in practice, I claim. So clearly any number of faces ≥ 2

including odd numbersis easily done by this nullifying of the end or base cases!Wow, interesting project to make samples. Looks pretty fun to build not complicated but interesting. Was that a mechanical engineering student or cs student or maybe even other than that?

How to make an odd numbered die, take a 6 or 10 sided fair die and make 3/5 sides equivalent in value to another side of the die so that there are 2 instances of each value on the cube.

So if 10-sided is definitely possible, we can take 10-sided and label it from 1 to 5, but each digit label twice

A five sided object similar to a standard length pencil would roll with an even chance of landing on any of it's sides. Just as a coin has distinct legal areas of valid chance the "Pencil" die would only count the sides it lands on as valid. The number would be read on it's ends. 🙂

The same could be don just as the previous video showed with any number of sides until you reached a theoretical roundness that would not be any sided at all.

No one solved the thick coin problem??

He says the thickness of a fair 3 sided coin depends on the material of the surfaces.

Can anyone find the range of thicknesses where coins can be fair for some real materials?

I always thought a 5 sided die was just a D6 where you reroll on 6.

You could make a fair five sided die with five curved surfaces. Like an American football with five pieces of leather.

But… But what were the results of the experiment?!

Technically if you had a long pentagonal prism and rolled it like a ball, not tossing it, it would roll and land on a random side because the sides are all congruent other than the bases, which are taken out of the equation because we are rolling it like a ball

I guess a perfectly fair 5 sided dice is possible. But its faces are not flat and it's not mandatory that they are.

I loved these two videos. Fascinating, thanks!

Yeah but what if you take a pentagon and add the points like the d10, but curve the side edges so it has 5 curved sides thst has trasitive symetry

Tobleroni, the San Francisco Sweet.

"well, there isnt because i proofed there isnt"

we should add a spherical dice also (jwan atto)

Could you make odd-sided dice the same way the long toblerone worked? Like a really long pentagon or heptagon?

so an odd number isn't fair?

no, etc.

so its not fair?

no, etc

so your saying that an odd die can never be fair,

no… probably

No control possible except if your Data

Why not a 5 sided toblerone ? You'd have a 5 sided dice

I was saddened during the discussion of the five-sided Toblerone die discussion that the moment of inertia between triangles' and rectangles' centroids are different; moreover, really that moment of inertia wasn't brought up at all in this.

But the toblerone started as a fair odd sided die. Proved quickly that odd sided die can be fair. Just dont eat them.

The probability of landing on a surface of a coin or any type of object is not only dependent on the amount of surface area but also the interaction of the edges of the object to the surfaces it interacts with prior to the 'coin' settling down. Equal areas for each chance potential is not a strict condition to make the probability equal to all others.

A fair 3 sided die? Idk, how about a fair "3" sided die? Just write 1, 2, and 3 on two faces each, opposite each other, on a 6 sided die?

I have a 5 sided die made by Gamescience, and they used that system. They rolled a die tens of thousands of times until the chance of getting an edge was the same as getting one of the flat faces.

The die worked on hard surfaces. On a hard surface it would hit, and usually start spinning. Once it was spinning, getting the edge faces was equal to the flat faces.

But if you rolled it on a soft surface (say a vinyl gaming map), then the flat faces came up a lot more often.

Very interesting talk.

Warm regards, Rick.

This channel is sick! Touching innocent numbers like that!

If you took 8 marbles, or rather small metallic spherules, like beebees, and [insert science here] so they were all magnetically repelled, then pushed them into a hollow acrylic sphere, like siblings under punishment together in a room, would they form the natural outline of a cube? As the beebees jostle to gain the farthest distance between each other, and as you add beebees one by one, giving it a jostle if you seriously used a hampster ball, would the resulting configurations identify the outlines of the best possible symmetry for each yabbadabbahedron? Sorry, I forgot the words. Or would it just make useless weird shapes where planes can never land? -Phill, Las Vegas

His pronunciation of toblerone is sinful

For approx. 45 years ago I had a math teacher who said that if we rolled a dice enough times, six would come up the most times. (same reason, holes and paint) we rolled it about. 100-200 times and five were the most common outcome, damn. About 3-4 years later (boarding school) a friend and I, for over a month to roll a dice, every time we have free time to do so, and this time the result showed that the probability increased the higher the number. One lowest six at highest.

It's easy to make a fair five-sided die. Well, technically it would have fifteen sides, but ten of those are impossible to land on:

Make a pentagonal prism. Cap the ends off with pentagonal pyramids so that when the die lands on any of the pyramid faces, it topples over to land on its side.