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  • 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.

Somebody made a dice rolling machine.

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フェアダイス(後編) - Numberphile (Fair Dice (Part 2) - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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