Probability, Statistics, and the Math of Dice

From tiny six sided cubes to oversized polyhedrons with dozens of faces, Michael’s collection of dice is more than just a hobby, it’s a window into probability, design, and the strange ways we humans play with chance! Why might some dice feel luckier than others? How do they shape the games we play, the mathematics we study, and the way we've made decisions throughout history? Each die has it's uses but they all reveal the patterns, quirks, and surprises that lie in the numbers we trust and the randomness, or chance, that we don’t. Welcome to The Rest Is Science: Field Notes. Every Thursday, Hannah and Michael rummage through their personal troves of scientific treasure and source discoveries that explain our understanding the universe, oddities that scramble our brains, objects that hint at forces we’ll never see...and a few things that are essentially just plain cool. Expect deep dives into the science behind each pick, the spark that grabbed their attentions, and the sheer delight they get from sharing it all with you. They’ll also be tackling your questions, so email The Rest Is Science at [email protected] ------------------- For more information about Cancer Research UK, their research, breakthroughs and how you can support them, visit ⁠⁠https://cancerresearchuk.org/restisscience ⁠⁠ Cancer Research UK is a registered charity in England and Wales (1089464), Scotland (SC041666), the Isle of Man (1103) and Jersey (247). A company limited by guarantee. Registered company in England and Wales (4325234) and the Isle of Man (5713F). Registered address: 2 Redman Place, London, E20 1JQ. ------------------- Find The Rest Is Science all over the internet by ⁠clicking here.⁠ ------------------- Video Producer: Adam Thornton Video & Social: Bex Tyrrell Assistant Producer: Imee Marriott Producer: Becki Hills Senior Producer: Lauren Armstrong-Carter Head Of Digital: Samuel Oakley Exec Producer: Neil Fearn Learn more about your ad choices. Visit podcastchoices.com/adchoices

Speaker names marked ? are inferred from speaking time and the show’s known hosts — they are not voice-verified.

• 0:00Hannah Fry?

  This episode is brought to you by Cancer Research UK.

• 0:02Michael Stevens?

  Dinosaurs walked the Earth 180 million years ago. But you know, cancer was part of their story, too. Scientists have found tumors in ancient fossils.

• 0:13Hannah Fry?

  Well, that is a part of the reason why cancer is a big, big part of our story, right? It's the other side of evolution. It's the most complex disease that we face. There are more than 200 types of cancer in total, each with distinct characteristics, challenges and mysteries.

• 0:28Michael Stevens?

  And that complexity demands scale. Cancer Research UK is the world's largest charitable funder of cancer research, with more than 4,000 scientists, doctors and nurses working across more than 20 countries in the search for answers. And then sharing their discoveries beyond borders.

• 0:45Hannah Fry?

  And the impact of this collaboration is clear, because over the last 50 years, the charity's pioneering work has helped to double cancer survival in the UK. That is more people who are living longer, better lives.

• 0:57Michael Stevens?

  Fossils can show us the past, but research is shaping the future. And for more information about Cancer Research UK, their research, breakthroughs and how you can support them, visit cancerresearchuk.org/restisscience.

• 1:14Speaker 02

  Observe the North American dad. 50s, great polo collection. He enters his vehicle information on Carvana. An offer appears in minutes. Remarkable. He squints, reads it again, smiles and says, "Dang, that's a good offer." The highest form of approval. No haggling, no hassle. No one called him "boss." He's still on the couch. Astonishing. Sell your car today on carvana.com. Pickup fees may apply.

• 1:45Speaker 03

  This episode is brought to you by Indeed. You're ready to level up your business, but the pressure to find the right person to make that happen is intense. Indeed sponsored jobs can help carry the load. It boosts your job post to help it reach quality talent fast. Cut through the chaos with Indeed sponsored jobs. Get a $75 sponsored job credit at indeed.com/podcast. Terms and conditions apply. Hiring now? Then this is a job for Indeed sponsored jobs.

• 2:21Hannah Fry?

  Welcome to The Rest Is Science. This is Field Notes, a kind of podcast expedition diary where Michael and I, we trade stories about objects, thrilling discoveries, big questions, anything that's been occupying our minds.

• 2:33Michael Stevens?

  Yeah, each week, one of us is going to bring in something to either show or tell. I mean, that's what it is. It's The Rest Is Science's very own version of show and tell.

• 2:42Hannah Fry?

  Do you think people have noticed yet that the two episodes that we put out a week are not the same thing?

• 2:46Michael Stevens?

  I don't know. I mean, I think ultimately, it's you and I chatting about stuff. Like, yeah, I haven't brought that up with the producers. I don't mind. I mean, it's obviously, it's very fun.

• 2:57Hannah Fry?

  I still feel like we put so much more effort into the main ones and then people haven't even noticed that these ones, we just have a load of fun.

• 3:04Michael Stevens?

  These are different.

• 3:04Hannah Fry?

  We just have a load of fun. It just takes a tenth of the amount of time to work. But anyway, welcome. This is Field Notes. We're gonna be, we're gonna be sharing some objects later. Michael, you've got something to share with us, haven't you?

• 3:18Michael Stevens?

  I do. I've got some things.

• 3:20Hannah Fry?

  Things, yes. But first, what we thought we would do is we would, we would empty our science mailbag because we've had some really interesting questions in. One from Lou Fern, who wants to know, Michael, is it math or maths?

• 3:33Michael Stevens?

  I don't know. I don't know. Wait, math or maths? You guys across the Atlantic call it maths, plural with an S. And I didn't even know that until I moved to the UK.

• 3:45Hannah Fry?

  Didn't you?

• 3:46Michael Stevens?

  It's not, I don't think it's a thing that Americans confront all the time. Like, oh, that's the British word for mathematics.

• 3:53Hannah Fry?

  Aluminium.

• 3:54Michael Stevens?

  Yeah, exactly. I think a lot of Americans don't experience aluminium either because the way the whole world of entertainment especially is like made for the American local audience. And then the rest of the world, it just gets the American version and they get it. They're aware of these differences. I mean, I'm only really speaking for myself and as a child, I never confronted, oh, it's an elevator or a lift, it's a truck or a lorry, it's a, it's math or maths. But I am curious to know, because I've heard the justification. From people like you going, but there's more than one kind. It's maths.

• 4:30Hannah Fry?

  I don't.

• 4:31Michael Stevens?

  You don't say that?

• 4:32Hannah Fry?

  No. I mean, I agree that that is, that's what some people say.

• 4:35Michael Stevens?

  Okay. So then I take that back. I feel, I feel very bad that I put words in your mouth. But, but it's definitely a thing that I've heard as a justification for maths. And then I'm thinking, I get it. There's geometry, trigonometry, group theory, but they're all mathematics. It feels like it's one unified thing.

• 4:53Hannah Fry?

  Also, people don't say, people say maths is, they don't say maths are.

• 4:59Michael Stevens?

  Oh, that's right.

• 4:59Hannah Fry?

  You know?

• 5:00Michael Stevens?

  That's right.

• 5:01Hannah Fry?

  It just, the plural thing doesn't work for me at all.

• 5:04Michael Stevens?

  So how is, what is your feeling about it?

• 5:07Hannah Fry?

  I say it because I want to be patriotic and, and care for my, my, my country, men and women. But, but ultimately I actually think it's probably should be math.

• 5:18Michael Stevens?

  Fascinating. Now, look, I don't, I don't think one's more right than the other. So I, you know, I don't have a dog in the fight. I'm not like, no, Americans are right. But it's interesting to hear that take from you.

• 5:29Hannah Fry?

  I mean gymnastics, like you wouldn't say gyms, would you?

• 5:35Michael Stevens?

  We don't say math. I'm doing some math, specifically trigonometry right now. That's why I said math. But if I was doing trigonometry and a little bit of algebra, then of course I'd have to say that I'm doing maths.

• 5:48Hannah Fry?

  I'm absolutely fine with it. What I normally do is I write math and then put a little S in the brackets.

• 5:52Michael Stevens?

  Sure. Yeah. I mean, what I like to do is just, just elongate it to mathematics and then everyone's happy.

• 5:58Hannah Fry?

  Anyway, you'll hear both from us and we're not ashamed of it. Okay?

• 6:02Michael Stevens?

  No, not at all. I think we all know what we're talking about and that is the point of language. Speaking of language, I'm going to use some more to describe a question from Max Sebastian. And by describe, I mean verbatim read it. Max asks, can you talk a bit about mosh pits and fluid dynamics and how that links to crowd safety at large concerts?

• 6:23Hannah Fry?

  Absolutely, I can, Max. Um, when this question came in, I had completely forgotten that there was a paper that was published in about mid 2010s, uh, which was the most wonderful paper, and it was called Collective Motions of Humans in Mosh Pits and Circle Pits at Heavy Metal Concerts. Uh, it was by some proper, good mathematicians and physicists. Um, and what they did is they attended a number of heavy metal concerts and also watched videos of them on the internet. And it is written as a proper academic paper would be. Um, here's a sentence from it, right, just to give you a sort of flavour of how it reads. "The mood is influenced by the combination of loud, 130 decibels, fast beats

• 7:04Hannah Fry?

  exceeding 300 beats per minute, music synchronized with bright flashing lights and frequent intoxication."

• 7:11Michael Stevens?

  Yeah. All right. It's a physical system and they're describing it properly. I appreciate that.

• 7:15Hannah Fry?

  What a way to distill the joy of a good night out with your friends. Um, anyway, here's the thing. If you stop thinking of people as people and you start thinking of them as particles, actually what you see in mosh pits is this behaviour that is common across sort of systems of fluids, essentially. And each person effectively is like a particle, right? They're propelled, they're constantly colliding, and they're reacting to what's going on around them locally, not the whole global system. They're not sort of all following a series of rules. They're just reacting to what's going on around them. So these physicists, what they did is they made this mathematical model, this computer simulation. I mean,

• 7:58Hannah Fry?

  adorably good sense of humour here. They called it the Mobile Active Simulated Humanoids Model, otherwise known as MASH's, um, queue. Uh, and there are two different tendencies that people have when they're in this sort of crowd situation. Either we tend to copy what people are doing around us, flocking behaviour, which is what you get when you see big, um, big flocks of birds, where they're sort of copying the average speed and direction of their neighbours. And humans are doing the same thing, right? If the crowd is moving in a particular direction, we tend to copy what's going on immediately around us. Um, but then we also, you have sort of more random movement, unpredictable movement, when we're acting as an individual, you know, say somebody spots a friend or whatever it might be.

• 8:44Hannah Fry?

  And what they found is that actually these mosh pits, they do have this gas like state. They they essentially form the same patterns that you would see if you were looking at a box of atoms, right, this sort of disordered gas. This Maxwell Boltzmann distribution is what it's known, but people are sort of pinging around from each other in this in this disordered way. But then, when you get more people going in, people organize into this vortex like state, this sort of circular motion that you see precisely as you do in fluids, where people are sort of rotating with the audience. And and this is like something, the these these circle pits that that emerge from nowhere.

• 9:29Hannah Fry?

  It's this emergent property. Nobody ever says like, OK, off you go, start rotating in this direction.

• 9:34Michael Stevens?

  These are people with their own personal wills and they're not, they haven't organized any of these patterns.

• 9:41Hannah Fry?

  That is just something that happens when you stop behaving like people and start behaving like particles. Have you have you ever been in a mosh pit, Michael?

• 9:48Michael Stevens?

  No, I never have.

• 9:49Hannah Fry?

  No.

• 9:50Michael Stevens?

  Have you?

• 9:51Hannah Fry?

  No, absolutely not. This is I am very much not a mosh pitter.

• 9:55Michael Stevens?

  OK.

• 9:57Hannah Fry?

  One of the things, so I actually, this is one of the things that I did after, so my PhD is in fluid dynamics, but after that, I then started working on complex social systems, right? So including things like crowd behavior, this is one of the things that I was doing, you know, mathematician modeling this stuff. And the thing that's really interesting about crowds is that you get this, you know, this behavior, like a box of atoms or a fluid that's moving around. But there comes a point where crowds become so dense, and it's normally when you get about five to six people per square meter.

• 10:31Michael Stevens?

  OK.

• 10:32Hannah Fry?

  When they start to actually become quite dangerous, when it stops flowing like a a fluid and instead becomes, it's called granular flow. So it's much more like sand in a sort of collapsing.

• 10:44Michael Stevens?

  Yeah, right. OK, so there's a lot more squeezing and cleaving.

• 10:50Hannah Fry?

  Exactly, exactly. And, and this is the, the fact that people turn into that state where they're acting like granular flow, that's one of the reasons why crowd crushes are so, are, are really dangerous because they're quite hard to stop once they begin. If you think about trying to control a pile of sand, sort of move it effectively, and apart from making it less dense, you, it's really difficult to control that flow. It's also, I don't know if you, if you've ever been in a, in a really busy crowd, I've been in a couple, you know, at football games, and sort of people start shouting like, stop pushing, stop pushing. And it's not that people are pushing, it's that the movement sort of ends up driving itself. You end up getting these, these dynamics that flow through the crowd.

• 11:32Michael Stevens?

  Oh, interesting. Yeah, that's happened to me one time in New York City on New Year's Eve.

• 11:37Hannah Fry?

  Oh, right.

• 11:37Michael Stevens?

  We were all waiting to get into Times Square, and I found myself being moved just by the people around me moving. But you're right, no one was pushing. Like, it's, it's, no one was on the outside going, I'm gonna push everybody.

• 11:51Hannah Fry?

  Yeah.

• 11:51Michael Stevens?

  It was an emergent property of everyone's motion together. And I was like, being lifted and moved like this, like I was in an ocean. And I was like, this is, it was very scary.

• 12:01Hannah Fry?

  It's very scary, because you, because I think at that moment, you don't feel like you have the autonomy of a person anymore. I mean, because you don't. But the thing that's really interesting about this, using these fluid dynamics and physics equations and, and models to, to look at crowd behavior is that you end up getting really counterintuitive results. So for example, one of my favorites is if, if you imagine that you have a doorway, right, and a huge number of people who are trying to get through it, so you have this sort of granular flow, where people are kind of like, bunched in together and pushing, everyone desperately trying to get through. You, you would expect to see that clumping that you would have if you were, you know, if you're trying to push, uh, kind of grains of, of wheat through a, through a, a really small, um, opening.

• 12:43Michael Stevens?

  Yeah.

• 12:43Hannah Fry?

  You kind of would get these clumps where it clogs up for a bit, and then maybe you kind of release the pressure, and then you get a big burst, and then it stops again. You get the same pattern of behavior when you have humans all pushing through towards, um, getting through a small opening. But one way that you can solve it, right, which is something that was discovered by mathematical modeling and then, um, demonstrated in experiments since, is that it's completely counterintuitive. If you put a bollard.

• 13:08Hannah Fry?

  Immediately in front of the doorway. So where the crowd would be standing, you sort of you have your doorway and then a couple of steps backward, you have a bollard, right? Effectively kind of blocking the doorway in a way. What that actually ends up doing is it forces people rather than bunching around that that single opening, it forces people to form these lanes around the side. And then if you think about it as though it was sand or grains of wheat or whatever it might be, if you stop that clogging from happening immediately above the opening, actually you would get this much faster flow, this much smoother flow of people.

• 13:48Michael Stevens?

  I was gonna ask, is that possible with pouring grain?

• 13:52Hannah Fry?

  Yeah, exactly. It absolutely is with pouring grain. If you stop the clogging by kind of creating a little barrier immediately in front of the exit, then yeah, you get much, much smoother flow. The only problem is that even though it works mathematically and in experiments, no one wants to put a bollard in front of an exit because it sort of feels like you're doing something quite dangerous.

• 14:14Michael Stevens?

  Yes, it does. It sounded dangerous. But in the context, I'm like, oh, I see how this is a safety mechanism, actually.

• 14:21Hannah Fry?

  Yeah. So what you have instead with newer designs of stadiums and buildings that deal with large crowds, they do things like they'll have offset barriers or they'll have railings in particular way or curved approaches. But the the the general idea is think of lots of humans together as though they are grains of rice or wheat or whatever it might be. And and the way that it will clog up and design the system so that they can flow as smoothly through as possible.

• 14:48Michael Stevens?

  That is all really cool. I love that. I love knowing that I'm a human with with with a conscious will, but at the same time, I can also become a piece of sand or a molecule of gas. Thank you for that question, Max. Coming up after the break, we're gonna look at some other bizarre behaviors and see what emerges. That's right, I'm talking about.

• 15:11Speaker 03

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• 15:41Speaker 04

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• 16:10Michael Stevens?

  Dice.

• 16:18Michael Stevens?

  I have my 10 dice. A D1 to a D10. I'll show this off later when I've earned my turn. But here's a two sided die. It's basically just a coin. It's rounded enough that it will not land on its edge. I'm gonna flip and or toss this. I need you to pick a number. Do you want to be one or two?

• 16:42Hannah Fry?

  I want to be number two.

• 16:44Michael Stevens?

  OK, here we go. OK, I'll flip it, catch it. It's over and the answer is?

• 16:50Hannah Fry?

  One. It's number one. You are up first. What on earth possessed you to make a die that was just a coin?

• 16:59Michael Stevens?

  We had to because we wanted to do. A D1 to a D10. And D1 means a die that allows you to make one decision. Like, this die right here, I'll pull it out of the foam. It's a D1. It's not a one-sided die because it has more than one side. But because of the way it balances, it can only ever fall on one. This is its shape, okay?

• 17:23Hannah Fry?

  Looks like a piece of pen A.

• 17:24Michael Stevens?

  It looks like a piece of pen A. And it only has one stable equilibrium point, which is this way with the one facing up. And so it's perfect for if you have no choice but still can't make up your mind. Okay? If it rolls a one, I'll do it. It's always gonna be one.

• 17:43Hannah Fry?

  Hang on, I need to ask more questions about this, right? What is wrong with just normal, a normal dice? Like, what is wrong with the six sides? You know, you know where you stand. This wasn't enough for you?

• 17:54Michael Stevens?

  Nothing's wrong with it. It's perfect. I love these things. We all know them. This is a version with numerals instead of pips, the little dots. But sometimes you've got more than six decisions you need to make. Sometimes you're working with a percentage, right? You're playing some tabletop game where you need to figure out, okay, what's my, you know, likelihood of having a critical strike? And that should be out of 100. So you could use a 10-sided die to do, you know, 100%, 90, 80, and so on percent to decide that randomly. You rarely need a three, or a five, or a seven, or a nine. But that's what we do. The rare, the unnecessary stuff. The stuff that's a conversation piece.

• 18:35Hannah Fry?

  Presumably, you can use these in situations that don't just involve tabletop games, like Dungeons and Dragons, or whatever it might be. Are you a Dungeons and Dragons fan, by the way?

• 18:43Michael Stevens?

  I'm a fan, but I have never played.

• 18:45Hannah Fry?

  You've never even tried to play a game?

• 18:48Michael Stevens?

  I've never had any friends who played.

• 18:51Hannah Fry?

  Michael, I'm sorry.

• 18:52Michael Stevens?

  What about yourself? I bet you went through a phase that's still going on.

• 18:59Hannah Fry?

  I'm really sorry to disappoint. I have played it, but I wouldn't say that it's a regular occurrence in my life. Again, it's because I don't have the friends. My friends are all Catan people, you know? They're snobby about Monopoly, right? But not snobby about Catan. They're just, they're not, they're not the extreme end of the spectrum.

• 19:15Michael Stevens?

  Well, we've got to find some better friends, Hannah, because Dungeons and Dragons, I think, would be fantastic for us, especially armed with all this randomized mathematical knowledge and tools.

• 19:28Hannah Fry?

  You could use this beyond just board games, though, right? I mean, in your day-to-day life, anytime you come across, I don't know, a crossroads, you can, you can get out the three-way die to choose which path you're going to go in.

• 19:41Michael Stevens?

  Who's going to pay the bill? Well, there are five of us, so I will use the five-sided die. You can't use a regular six-sided for that. Now, I don't want to get ahead of myself. I'll show off the three. Now, the three is basically a kind of barrel die, and I'll explain that for those who weren't watching the video. It's basically just a barrel with rounded tops and bottoms, so it won't land on its top or bottom. And then the sides are faceted, so there's three sides, and it can fall on them to get a three, or a two, or a one.

• 20:18Hannah Fry?

  Have you checked whether these are fair? Have you checked whether you are equally likely to get a one, two, or three?

• 20:23Michael Stevens?

  Yes, and that's why they took so long, because we 3D-printed all of the prototypes, and then realized that the weight was different than it was going to be when they were injection molded. So, we had to have an actual factory make an entire set, send them to us, and then, I didn't do it. I had our science writer, Scott, do it. I said, "Today, you're going to be rolling these die and making tally marks." And we actually found that they were all fair, right off the bat. But that was because of all the prep we had done. And when I say "we," I mean Scott. He did all the mathematics behind, okay, how is this going to be fair? How thick does a shape need to be to land on its edge? Just as frequently as it lands on its, you know, top and bottom.

• 21:07Michael Stevens?

  So the four-sided die is just a tetrahedron.

• 21:12Hannah Fry?

  Triangle-based pyramid.

• 21:13Michael Stevens?

  Standard. You can find these in any game shop, any dice shop. The five-sided die, that's the oddball, and there's a lot of different ways to do it. Have you ever seen it done?

• 21:24Hannah Fry?

  Five-sided? No, I mean, it's really tricky, right? Because you can't do, well, you can't just use a pentagon. Because then you've got the other ends, right? That if it falls on either of those, that's sort of, that's seven sides. So you've got, I mean, it's tricky.

• 21:39Michael Stevens?

  It's tricky, and the cop-out, the one I don't like, is to just keep making barrel dice, where you take a cylinder, you round the top and bottom, so it won't stay on them, and then you just slice a regular pentagon around the side. So it's got five facets there. And now you roll it, and it rolls like a log, and it's going to wind up with one of those facets down, one of them up, or whatever, you know? I thought that was boring. It was just too easy. So instead, we took a triangular prism, and we made it thick enough that it will land on its edge, one of its edges, just as often as it lands on its top or bottom. If it lands, say, with this edge down, the two is at the top.

• 22:24Michael Stevens?

  So you've rolled a two. See that?

• 22:26Hannah Fry?

  So hang on. So whereas the four-sided die is like a triangle-based pyramid.

• 22:31Michael Stevens?

  Yeah.

• 22:32Hannah Fry?

  And then the five-sided die, it's more like a chunk of dairy leaf, you know? Laughing cow. That's where we're talking.

• 22:39Michael Stevens?

  That's right.

• 22:40Hannah Fry?

  A little bit of squeezy cheese, yeah.

• 22:42Michael Stevens?

  The six is just the standard cube. A cube gives us six sides, and it is a regular platonic solid.

• 22:52Hannah Fry?

  Okay, I've got a couple of things to say about the six-sided die, if I may. Right, the first thing is, I'm extremely in favor of switching over all your decision-making to dice-rolling, rather than actually making any choices. And there's a company, a restaurant in the UK, that has gone for this approach. They're called Dishoom. They're like this Indian restaurant. They're really popular. Um. But there was one point where they they decided they wanted to get more people to come during the day. So they ran this promotion where if you like ate enough times in their restaurant, you qualified to be given a teeny tiny little die, right? With, you know, standard six size, the standard, you know, cube based die. And and what happens is if you go in and

• 23:34Hannah Fry?

  you eat a meal between, I think it's one and six, something like that, and you roll the die, right? If you land on a six, you get your whole meal for free. Okay. Which is like, hmm, that's smart. That's kind of like a cute idea. But the thing is, is that when you actually work out the sort of mathematics of this, it would have been exactly the same as if they had just said, if you're a regular customer, if you eat between one and six, you get 15% off your meal. Right? It's exactly the same thing. But how much more fun is it to do it where it's it's based on the roll of a die?

• 24:05Michael Stevens?

  Okay, so if you rolled a one, two, three, four, or five, you paid full price.

• 24:09Hannah Fry?

  Full price. But there's a one and six chance that you are walking away free and easy.

• 24:13Michael Stevens?

  Yeah. So would you rather have 15% off every time you went, or a one in six chance of getting it for free?

• 24:19Hannah Fry?

  Obviously the latter.

• 24:20Michael Stevens?

  Obviously the last one, though in the long run, they're the same thing. You'll wind up spending the same amount. But in the short term, you could roll a six three times in a row. And, you know, that's good.

• 24:31Hannah Fry?

  And and you know, it's it only needs to average out like, I mean, if you're the one individual, you could only go once and then just, you know, never pay them. Some other poor sucker who's rolled a one many hundred times in a row is is the one that's struggling and effectively paying for your meal.

• 24:45Michael Stevens?

  Do you know what the dice were made of?

• 24:47Hannah Fry?

  I don't. I don't.

• 24:48Michael Stevens?

  Because I wouldn't I wouldn't do that as a store like manager or owner, because people could game it. If you take a regular die and you just put it in a like a warm oven, like a 200 Fahrenheit oven for a few hours, it'll soften and gravity will cause it to get imperceptibly, but truly a little bit heavier on the bottom. So you put it in there with the six side up, and the one side will fatten in a way that you cannot see. But it's so much denser there that you're gonna roll a six. I mean, not all the time, but much more often than you should.

• 25:22Hannah Fry?

  Well, anyone who's got a dishoom little dice knows how to game the system now. I have, I have a second thing to say about the six sided die, if I may. Which is a critique of your decision to go for the bog standard boring die.

• 25:35Michael Stevens?

  Oh, well, what else should I have done?

• 25:36Hannah Fry?

  So here's the thing, I think there's redundancy. I think there's more going on in the standard six sided die than, than you need, because it has like extra levels of symmetry that are just totally irrelevant. Right? You take a die, yeah, of course you want it that it lands on the numbers one to six. You know, there's equal chance of landing, landing on each one. But you don't really care that you can rotate it on the number one and the number one still come up, right?

• 26:00Michael Stevens?

  Right.

• 26:01Hannah Fry?

  I'm not interested in that. I think that's just extra redundancy that I don't need. What I think you should have gone for instead, Michael, and you know, good as your amazing curiosity box of tender and die are, is like a wonky die like this.

• 26:15Michael Stevens?

  Oh, my gosh.

• 26:16Hannah Fry?

  Have you seen these before?

• 26:17Michael Stevens?

  The wonky die? Can I, let me go look in my dice collection and see which kind I have. I have some like that. I may not have them. But that, so that looks like a regular cubic D6. And yet it's all slanted and skewed, but it's still fair?

• 26:36Hannah Fry?

  Still totally fair. So this is like exactly right. It looks like a normal die, but it looks like someone sat on it. Or left it in the oven for too long and kept turning it, basically. It's sort of like, it's sort of like an Alice in Wonderland version of a, of a normal die.

• 26:49Michael Stevens?

  That's right. It's like modern art meets a regular die.

• 26:55Hannah Fry?

  Exactly. But the thing is, is that when you roll this, you still are equally likely to get a one or a one to six, right? Any of those numbers are equally likely to come up. But what this is doing is it is removing all of the redundant ways that the original dies are symmetrical. So this is like the, has the minimum amount of symmetry required to make the dice fair and no more.

• 27:18Michael Stevens?

  Oh, wow.

• 27:19Hannah Fry?

  Which makes me think that these are superior. I've got to be honest with you.

• 27:21Michael Stevens?

  I didn't know that about them. I just thought they were like novelties. I knew they were probably fair. But they have the minimum amount of symmetry needed for six choices.

• 27:32Hannah Fry?

  Yeah, there's no rotational symmetry on it at all. Much, much nicer, much more pleasant.

• 27:37Michael Stevens?

  Well, the seven, we got to use our little trick. We made a pentagonal prism that's just thick enough that it's likely to land on any of its edges just as much as it is its top or bottom.

• 27:50Hannah Fry?

  So it's five. Five is the weird one, then.

• 27:53Michael Stevens?

  Well, five, seven, and nine are all weird. They're all prisms that have to have a certain thickness. The nine is a heptagonal prism with just the right thickness.

• 28:04Hannah Fry?

  I like those a lot. I like those a lot. I also just like the idea of you wandering around your day to day life and using them at all opportunities. That's the thing I like the most.

• 28:12Michael Stevens?

  Yeah, it impresses people. And if it doesn't impress them, well, then they're just not for me.

• 28:17Hannah Fry?

  Then go and play Dungeons and Dragons on their own, Michael.

• 28:20Michael Stevens?

  Some people call it people repellent. I call it a friend finder.

• 28:27Hannah Fry?

  All this dice stuff, you know, using it to generate random numbers, to help you make decisions is all well and good, right? But what if you need more? What if you need more randomness? What if you need more than just six digits? And for that, Michael, I've got something to show you. You can turn to this book that I have very proudly had on my shelf for a number of years. It's called A Small Book of Random Numbers. It's an absolute page turner. Yeah, it's quite dinky. They come in larger sizes, but I'll be honest with you, I've got the cheap one. Shall I give away the ending for you? It's quite the page turner. The ending here is 53477 and then 43466.

• 29:12Hannah Fry?

  I know what you're thinking. You didn't see that coming, right?

• 29:14Michael Stevens?

  I did. Whoa, spoiler alert.

• 29:18Hannah Fry?

  Spoiler. It is quite literally just an entire book filled with pages and pages of nothing but random numbers. One of the reviews on Amazon for this book says that it relies too heavily on 10 characters.

• 29:35Hannah Fry?

  Which I quite enjoyed, but it is totally unbiased. And I know you might be thinking, why on earth would anybody ever want to have an entire book of random numbers? But there are like actually really strong reasons for this. So particularly in the 1950s, when people were starting to do big calculations, right? For like the space programme or where they were like trying to sample populations or like come up with military routes to like travel across like terrain that might be attacked by other people. In all of those situations, having random numbers, genuinely random numbers, was really, really important, right? If you are like, if you are crossing enemy territory and you, there's a route that you want to go in, you want to make yourself as unpredictable as possible.

• 30:19Hannah Fry?

  So you might want to add in a certain random amount of noise. If you try and come up with that yourself, if you try and like make up that randomness, then you are going to fall into the trap that humans are very, very, very bad at coming up with random numbers. You need something like this small book of random numbers in order to help you. And so in the 1950s and many, many times since, scientists have sat down with, with processes that create genuine randomness. So things like the static on a television or a wall full of lava lamps or looking at the radioactive decay of like very, very small, you know, atomic properties and use that to like collect the randomness that appears naturally in nature and then print

• 31:04Hannah Fry?

  it for other people to use. And so, yeah, I'm not, I'll be honest with you. I, I haven't done it start to finish. I've just flicked in, flicked in and flicked out.

• 31:13Michael Stevens?

  Have you used it for anything, even like a game?

• 31:16Hannah Fry?

  No, I haven't. I have used, because now you can get computer, if you're like programming, so you know, for my PhD and in my younger years, I used to build a lot of mathematical models. And there are pseudo-random number generators that you can, that you can get that sort of do this job for you. They're called pseudo-random because they're not totally perfectly random, like nothing is apart from physical processes. You can't sort of generate randomness by doing like, you know, deterministic processes on a, on a machine. But I, so I've used lots of random numbers. I've never used the actual book. I'm cheating, really.

• 31:56Michael Stevens?

  All right, well, you showed off a book. I've got a book of digits that are not random.

• 32:03Michael Stevens?

  You can easily calculate them. One million digits of pi. This book contains pi in order, right? So it starts with three, but it's got a million digits. Here's some in the middle there. Oh, man. Amazing. And I cannot tell you how it ends because this is just part one. Oh, of infinite number. So this is beautiful. And related is a book that is quite controversial. But I own it because I'm not afraid. This is the square root of four to a million places. Hang on. Square root of four?

• 32:48Michael Stevens?

  Yeah. It's just two. The first million digits of the square root of four, right here. Show it to me. OK, so square root of four. It also begins in the way that we would all expect. Do you see the two? Two point zero. Is it just zeros? Yeah. And then it's just a million zeros. The whole way through. A million zeros. Michael, how many trees had to die for the purposes of that joke? Well, when I had it made, I said, just one tree, but make it painful. And so that tree suffered. No, here's, here's, here's the way I feel about it. I think that this is much less than one tree.

• 33:33Michael Stevens?

  The trees are replanted twofold when they're cut down. And I've used this so many times to get people excited about math that I think that that tree died with honor. I think that's great. And if you, if you think there's a problem with printing a million zeros and you get mad at me for it, I get it. But you are playing big pollution's game. They want you to get mad at the little guy who's trying to teach math. Well, they, well, they just deforest for fun. So look. I take it back. But that's also what's beautiful about the book. It's, it's a, it's a funny joke. It's a cool mathematical teaching tool, but it's also a philosophical lesson about the rights of plants.

• 34:16Michael Stevens?

  It's also an incomplete series, right? You need an infinite number more, more copies in order to fully complete the square root of four. The script of two. Yeah. You, you, I don't have enough precision. No. A million zeros and then what?

• 34:33Hannah Fry?

  How does it end? Tell us, how does it end?

• 34:36Michael Stevens?

  I don't know.

• 34:38Hannah Fry?

  Do you remember when there was a friend of ours, Brady Haran and Matt Parker, the YouTubers, where they took a million digits of pie and they printed it out on a roll of paper?

• 34:49Michael Stevens?

  They printed it by hand, though, right?

• 34:51Hannah Fry?

  No, no, they printed it. They constructed it by hand, you know, they cheated slightly. They went in and they annotated it. I think it still took them four hours.

• 34:59Michael Stevens?

  Oh.

• 34:59Hannah Fry?

  They had to go to a, to an airport runway and unroll it. Took them ages. Something fun to do in an afternoon, roll out a million digits of pie. It was actually a really interesting video if you want to go and watch it. But I do remember Brady telling me that once they rolled up that sheet of paper, he got a message from someone on the internet asking to buy it. And he sold it to them and used the money to buy himself a very nice watch. I mean, that doesn't really, that doesn't really say very much, I think, because I've got a nice watch that's like 120 quid.

• 35:31Michael Stevens?

  But how nice of a watch, I guess.

• 35:33Hannah Fry?

  Yeah, how nice of a watch. But somewhere, somewhere out there, there is, there is a person who owns a mile of pie on brown paper. I also like the idea that this is a person who is, who's carrying their mile of pie around with them at all times, just in case they bump into someone like Vsauce in a museum and can use it as an opportunity to make a friend.

• 35:54Michael Stevens?

  Well, yeah, so that person might be listening to this podcast. I'd love to know what they're doing with it. Do they have it shoved in the back of a closet? Is it on display? If you own the mile of pie, reach out because I, I want to meet you and I might even have an offer to make you.

• 36:12Michael Stevens?

  A mile of pie annotated by Matt Parker? I mean, that's Smithsonian stuff right there.

• 36:17Hannah Fry?

  That's, that's, that's the big dog. So there we are. That was an episode of Field Notes where every week Michael and I are gonna bring something to show and tell the other. It might be a little object, it might be a riddle, it might be a thought experiment, it might be a question or a story even.

• 36:34Michael Stevens?

  Yeah, a life story.

• 36:36Hannah Fry?

  Hey, why not? I want to hear about where you've been. A confessional even, indeed. If you have something that you would like to contribute to Field Notes, then you can write in to us, [email protected] with your ideas, with your thoughts, with your stories, with your questions.

• 36:52Michael Stevens?

  Yes, please do. I cannot wait to read those and experience what you guys send over. And until next time, see you later.

• 37:13Speaker 05

  There we are getting the statement from number 10. On that flight, with 350 passengers on board.

• 37:22Speaker 06

  The Great Game: Strategic Struggle Between Britain and Russia.

• 37:30Speaker 07

  We assess, Prime Minister, with high confidence it was Russian.

• 37:36Speaker 08

  Holding the Russians accountable is World War III.

• 37:38Speaker 09

  They took down a British passenger jet containing 350 people.

• 37:41Speaker 06

  Man in the Kremlin is playing, Alex. Got to ask yourself, do you want to play or not? You're the Prime Minister. You've done nothing!

• 37:54Speaker 10

  We would need to understand the policy context.

• 37:57Speaker 06

  All the relevant context will be structured as a framework today, disseminated into an operation order, or FRAGO.

• 38:05Speaker 11

  A FRAGO that says to kill the President of Russia?

• 38:10Speaker 06

  Yes.

• 38:12Speaker 08

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• 38:31Speaker 08

  Where Britain plays its strongest arm. He's the best. Against its strongest foe. Don't do it, Duffy! Bear Hunt. Available now wherever you get your podcasts.