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.

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