A Problem Squared - 081 = Pi Prize and Square Stairs

Episode Date: March 11, 2024

In this episode… 🥧 Can Pi ever repeat itself?  🪜 Is it impossible for cube-based stairs to be identical? 📈 And some Business of an Any Other Nature.  If you want to see the biggest hand calculation in HISTORY, you can! Just head over to Matt’s YouTube channel on the 14th of March: https://www.youtube.com/@standupmaths An Evening of Unnecessary Detail is BACK IN NEW YORK! On the 14th of April, at The Bell House in Brooklyn from 7:30pm!  You can get tickets for that here: https://www.eventbrite.com/e/matt-parker-an-evening-of-unnecessary-detail-tickets-847453715877?aff=oddtdtcreator Ever wanted to know the answer to whether ALL Japanese snacks scale with the packaging imagery? Well now you can, on Bec’s YouTube: https://www.youtube.com/watch?v=C-roi6FdISY&ab_channel=BecHill GO SEE BEC! She’s playing in Reading and Louth in April. You can find the dates for that, here: https://www.bechillcomedian.com/tour-gigs YOU CAN PRE-ORDER MATT’S NEW BOOK ‘Love Triangle’ NOW!  For a signed, but full priced copy with a special dust jacket follow this link: https://mathsgear.co.uk/products/love-triangle-by-matt-parker-signed Or for 21% off your pre-order, follow this link to Waterstones: https://tinyurl.com/2fdjnwwz As always, please send your problems and solutions to the Problem Solving Page, here: www.aproblemsquared.com. If you want more from A Problem Squared, you can also find us on Twitter, Instagram, Discord and on Patreon.

Hello and welcome to A Problem Squared, the problem-solving podcast which is a lot like Pi in that we have a habit of going on forever. So true. But unlike Pi, we also have a habit of repeating ourselves. So true. Your hosts are myself, Bec Hill, and Matt Parker. That's me. And Matt, you are also like pie oh yeah because you are maths related correct and have a very large but niche fan base

yeah and i think it overlaps a lot with pies fan base yeah i think i think there is a and you're in the venn diagram is essentially a circle. Appropriate. And I am also like Pi because I'm full of chunks of meat and sometimes a little crusty. So close. A little crusty. A little crusty in the mornings. Fair enough. I have no follow on questions. In this episode.

I work out if Pi can almost repeat itself. I'll be staring at some stairs. And we got any other business about me doing unnecessary detail in New York on the 14th of April. Ha! I got it in the menu as well. Damn it. You didn't want to talk about pie day? Oh, and that, and that. Hey, Matt.

Bec. Well, I mean, now that I've mentioned it in the menu. Pi Day. I'm guessing because this episode comes out just before Pi Day. Just before Pi Day. Monday before Pi Day. Yes. Very exciting.

Yeah, I've noticed that here we celebrate Pi Day based on the American dating system. We do. Only because we haven't got a better suggestion. Right. If we had a 31st of April, which we would write as 3-1-4, we would have a superior Pi Day. But we don't. That date doesn't exist. Yeah, that's true. All we get is the 14th of March, which you have to write the wrong way around to get 3-1-4. But I'm not going to pass up an opportunity to celebrate Pi.

What about the 3rd of February, which is the 14th month if you count rollover? No, I like what you're doing. That's Valentine's Day. Valentine's Day is Pi Day. About what Pi Day is. No, I'm... All right, everyone. I want everyone to understand that I am now...

I'm starting a rival Pi Day on Valentine's Day. You've all got 11 months to get ready for it. Yes. Because what is a love heart if not a circle with a point? That's what they say. Two points technically. I think it was you, Glenda, who said that. Yeah. So it's Pi Day. I celebrate, well, I have now for almost a decade celebrated every Pi Day by calculating Pi a ridiculous way. Yes. Which I kind of started doing by accident. And it just so happened the first time I did it. Sorry, how did you do it by accident? Well, I always liked calculating Pi silly ways.

And then when I first started doing my YouTube channel, one of the first videos I made, because I started at kind of the beginning of the year and then Pi Day came up. This was my original mini-oit. I was like, oh, I should do a thing about pi. And so I calculated pi by weighing a circle to work out the value of pi. Oh, yes. Yeah. Which I thought was very funny. And I didn't think anything else of it until the next Pi Day rolled around a year later, still doing YouTube. And I'm

like, oh, what's another ridiculous way? And I quite like there are some infinite series where you're adding or subtracting like infinitely many terms. But the more you do, the closer you get to pi. So you get more digits as you do more of them. And there's a terrible one that I did. And so I thought it would be kind of funny to do. Worked it out on a big chalkboard. Vaguely pi-ish.

Job done. What I didn't realize is I'd now set myself up into a rhythm. Every year I calculate pi, every odd year I do some kind of ridiculous experiment to physically calculate pi. And on the in-between even years, I try and work it out by hand. And my working it out by hand gradually escalated. The next year or two years later, I then did a better calculation, but it took me all day to try and get more digits. And then two years after that, I got five people in a room to help. And then two years after that, I had 20 to 30 people in a room for a weekend for two days to help.

And then two years after that, which brings us up to this year, I had 200 people in a room. Oh, my goodness. For six days, calculating pie by hand. Oh, my God. It's kind of got out of hand in that regard yeah yeah we went all out oh my goodness i'm just i'm nervous thinking about 2026 oh yes we'll get to that so our goal was so the current world record if i may yeah you're using quotation marks because no one's ever contested this record. Because the record was set before we had electronic computers to calculate pi. Oh, okay.

We're not 100% sure if the person who did this, William Shanks, who calculated pi on and off between about 1850 and the 1870s, if they had any kind of mechanical calculator to help them. We don't know. So for now, we're assuming they did it all by hand until we have any evidence otherwise. They thought they had 707 digits correct. Nobody double checked. In the 1940s, someone did double check now with a mechanical calculator.

And they realized they made a mistake in the 528th digit. Oh. So they put the effort in for 707 and 527 were correct. I really want to beat that. I mean, you don't want to cross. I want to have the record. William, because I've heard that William Shanks. William Shanks.

Oh, Billy Shanks. I know you've all made, yeah, you've made all the jokes. It's great. So yeah, so that's the record to beat. That's the record to beat. And it might be less than that. But hey, why not have a benchmark? If we beat that, we would unambiguously have the world record for most digits.

And is that what you were going for? We started with the capacity. If everything went perfectly, we were in with a vague shot. Can you explain to me how you calculate pi by hand? Great question. So, there are a bunch of different equations that give you pi as the answer. But some of them are quite complicated. They contain lots of summation.

What's summation? Where you're adding lots of things together. Okay. Some of them involve multiplying things together. Some of them involve factorials, which is those are the numbers where you're multiplying by every number smaller than them. And the reason that they're not good is they get very big very quickly. Okay. Some of them involve square roots or other kind of roots, which are very difficult to do by hand. Some involve raising things to weird powers, like non-integer powers, which are difficult to do by hand. And some of them involve dividing very big numbers by each other which is

hard to do by hand and the reason we have all these complicated equations is because most of the time mathematicians only care about is it a cool equation that gives you pi and the the actual practical ones are optimized for one definition of practical are optimized for computers to do they're not optimized for humans to do so i had to go through all these different equations and try and find one which could be done first of all could be done by humans by hand secondly could be split up between a lot of humans to all do it once yeah because that's where my like where i sort of get stuck because i'm like how do you yeah how do you divide up the sum? Yeah.

So, first of all, we picked an equation. I say we, me and the council of calculating pi by hand. We picked an equation, which is actually seven different equations. And at the end, you've just got to add their results together. Okay. So, each one can be done separately. So, already now, we've got seven different things that can be done independent of each other. Within each one, it's actually split into a whole bunch of things you've got to add together. And you can kind of,

once you've started, like, let's say you've calculated the first one, you've done the first 20 digits of it. We were doing them in 20 digit chunks. So, the first person does the long division to get the first 20 digits. They can then hand that on to the next person who uses it to get the first 20 digits of the next term, who then hands it on to the next person who uses it to get the first 20 digits of the next term who then hands it on to the next person to get the first 20 digits of the next term okay but meanwhile the people who did the first 20 digits of the first term they've now done 20 to 40 digits then they've done 40 to 60 and they're working their way down so you've got some people who are doing digits further and further into one of the many many different things we need to know.

And other people are taking what's already been done and starting on the subsequent things we have to do. Yeah, my brain is sort of imagining like a family tree, like a sort of flow chart. Very much like a family tree. Yeah. But it's weird because depending on what you're doing. So we call each sheet is a chunk. So the goal was in terms of the end user, we had, oh goodness, somewhere between three and 400 unique humans came and helped out at some point in time. For each human calculator, all they have to do is walk up to one of the handing out tables we had, get a sheet, and all the sheets are long division. They'll

have a big number at the top. They'll have a smaller number. They have to divide into it. And they go away and do that long division, accurate to 20 digits. And then they bring and hand it back in again. Yeah. So from the end user's point of view, you're just doing lots of long division. From our point of view, running the machine, each one, like each sheet that's done, the things like the answer has to go to two different places, to two different sheets. But some of the sheets require two different bits of information to come in before they can be sent out to be divided.

And the logistics behind tracking all of these sheets and making sure the right things are copied into the right place with no mistake. Because we cut a single mistake and we're sunk. I was going to say, is there like a quality control element? Is there anyone that checks someone else's sheet after they've done it? So we sent out each sheet three times independently. Waited until two came back that matched. That was the first verification. That was done by the verification station. They would then pass it on to the mod squad who would do a modular arithmetic check,

which is where you do, like you can't use it to get the answer in the first place, but you can redo the calculation. They were doing it, for people who are familiar with these things, modular 9 and modular 11. And I mean, mod 10, you kind of get for free. And then they would compare the results from those to make sure it lined up with the answer that was on the sheet. So it's a way of doing a real shortcut version of the calculation, but it will indicate if the one on the page is probably correct or not. By doing that, we found on one of the long divisions, two different people doing the same division, both made the same mistake at exactly the same place.

And that meant it passed the verification check of comparing their answers. Their answers were identical, but they just accidentally made the same mistake at the same time. And if you're doing, we were doing thousands of sheets. Yeah, I was going to say that's like human errors. Yeah. But the mod squad caught it because it didn't pass the mod test. We estimate the mod squad would catch 90% of mistakes that got to it.

Wow. Which is good, but... But still leaves 10%. Still leaves 10% chance. And if we were to do this again for even more digits... So, the video came out today, on the 11th of March, when this podcast goes out. Okay. So...

So, the listeners know something that I don't. Potentially. But I don't want to ruin the surprise if you haven't seen the video yet. So, I'm going to give away most of the result. Okay. We, at the end of six days with a lot of people, had calculated 140 digits of pi. Calculated, not checked. Okay.

We then had to check that against the real value of pi. Last time we did this, we got 11. 11 digits correct. Oh. We want more than that. Wait, hang on. When you say 11 digits correct oh we want more than that wait hang on when you say 11 digits correct yeah do you mean like in the places they're supposed to be in the place where they're supposed to be yes yes yes what like the first 11 the first yeah at the beginning we got 3.1415926 and so on

for the first 11 digits and then we had a mistake and then the rest of it is the rest is wrong so you can so that we only had like 30 people for two days then. Yeah, but you had 30 people for two days and you calculated by... Stop it, stop it. I know, I know. Look, you're late to the party. A lot of people have already found that very funny. I know.

But I think you deserve your moment to appreciate it. Thank you. Yeah, it's very funny. Your response is valid. But I would also say that uh i mean you know me i'm always about going the long fun way even if it's wrong oh yeah because it's more fun yes so i i applaud and appreciate that effort so it would have been very embarrassing to get hundreds of people together for a week only 11 correct so i can reveal for our listeners that we made over 100 digits okay that's impressive i is

astounding even if it's 101 i'm not gonna tell you how many over we got it's still impressive we locked in over 100 digits correct i'll be honest believe it with the because i don't understand what what like you've even though you've explained to me what goes into calculating it oh it's a whole thing i i'm i'm still like i thought you would have gotten way over 500 and whatever you've got to see the chaos that led to i also forget like what if i was to write out like as if I was tallying for 500 and something. That in itself is like long, never mind doing the maths to. For me, it was a real experience.

Like you think you know pi, but watching that chaos. And yet you get these exact digits out the other side. Like why? Why on earth did that ridiculous calculation? I was very emotional. It's a pretty special beginning to end. Awesome. And how have you been, Bec?

I'm good. I'm good. Speaking of YouTube videos, you finally watched mine. I did. Yeah. I did, of my own free will. Yes. It's not like you came around to my house and sat me down and said, watch the video.

I was like, Matt, it's six minutes long. Oh, that did happen. And you're in it. I mean, I consume a lot of YouTube, big YouTube fan, as well as making it. You've done what a lot of people on YouTube don't do. And you've put thought and structure into the whole video as a whole. Well, you could say that I was inspired. You work as an actual comedy writer. It a good starting place i figure oh yeah yeah excellent start to you eight the year

of youtube thank you i was mainly just wanted to make sure that you were happy with your cameo it's good i do anything i regret is i should have given you some b-roll of me buying the snacks in japan oh yeah well a bit nice it's good it's short no youtube videos ever finished at some point you is I should have given you some B-roll of me buying the snacks in Japan. Oh, yeah. That would have been nice. It's good. It's short. No YouTube videos ever finished. At some point, you just got to draw a line under it and put it out. I think it was Da Vinci that once said that.

I think it was. Yeah. Yeah. And I do have a little bit of other stuff to say because this comes out in the 11th. Other than everyone go watch the video. Yes. I know that Pi Day is coming up for you. But last night, well, I guess both UK and US time, was the Oscars.

Maybe I'm just announcing this after it's already happened and there's nothing anyone can do about it. But I'm going to be doing a little spot on. Jonathan Ross is going to be doing a little pre-Oscars companion show before they start streaming live from the States. And so I'm going to go on with some flip chart and do some. Movie flip chart. Moving flip chart stuff chart stuff yeah so it should be fun but there were some suggestions that i had uh like one of them was uh i wanted let's see if you can guess what this would have been okay if i'd drawn it it's a picture of uh nicholas cage yep and he's got a very big breast on him just the one yeah right the breast is then is put on on onto him like as like a pin the tail on donkey type thing okay so you're gonna guess what

guess what the film is yeah and i'm gonna say it's more of a say what you see don't get thrown by the fact that it's it's an actor uh no but you're kind of like you're so close to it already tit on nick what titanic titanic titanic yeah yeah the execs didn't like that yeah how about that vetoed how have you made me side with tv execs well thanks everyone for sticking with us because this was quite a long catch up. That's a lot of catch up. I think very fascinating. Let's hope so.

It was Pi Day. This first problem is for you, Matt. Yep. It's from No Name. They didn't enter a name. They went to the problem closing page at problemsquared.com. Yep. Skip it right over that.

I mean, no text fields are compulsory. No, that's it. They don't have to get in there. That's how we live. You choose what you want to fill in. Maybe you left us this problem. It could be me. No, I meant the listener.

No, the listener. You. Yeah, you. You. I mean, I wouldn't put it past you, Matt, because surprise, surprise. It's pretty mouthsy. It's about pi. I'm Brian.

So can numbers like pi almost repeat themselves? Like could pi go 3.14... Stuff, stuff, stuff. 1, 2, 3. Followed by 3.14... Almost again. 1, 2, 4. My instincts are telling me no, but I don't really have the math knowledge to back that instinct up.

My instincts are telling me no, but I don't really have the math knowledge to back that instinct up. This came up because I had an idea for a short story where a supercomputer calculating the digits to pi has an existential crisis because pi seemingly starts repeating. I like that. Yeah. Yeah. I would read a short story about a supercomputer having an existential crisis. I don't think anything I'm going to say is not going to devalue that. Yeah. Great story idea.

Yeah. And we can all take it now because you have enough to say. Yeah, steal it. Everyone, go. No, no one steal it. But if you do write this short story, please let us know. But in the meantime. Can pi almost repeat?

I mean, if it's technically infinite, then yes. Well, I mean, do they want it to start repeating from the beginning? Or do they just want like a patch that repeats partway through? They're subtly different questions so you know what you know i'm going to give you one of my one of my favorite facts about pi there's a thing called a pi prime which is where you start from the beginning of pi with the three one four one five nine a prime a prime and you keep going and you see how many digits. If you stop there, everything so far is a prime number. So it works for one digit because three is prime.

Okay. It works for two digits because 31 is a prime number. Oh, okay. Yep. Three, one. It doesn't work for three because 314 is not a prime number. It's even for a star. And so the first couple are 314159 is a prime number. Okay.

And it works for the first 38 digits. That's a big prime number. And you think, oh, I wonder when the next one's going to be. The one after that is 16,208 digits. That's a big old leap. That's a big one, yeah. And the one after that is 47,000 something digits. Okay.

But that's a less of a leap than. That's true. Yeah. Yeah. There's a big, there's just a big gap. Because instantly my brain's like, ooh, any patterns? Because my brain likes to do that. But I don't immediately see any patterns.

It's just because each next digit becomes less likely to make a prime. And so once you're kind of escaped from the small numbers where there are a bunch of primes, once your numbers got any kind of length to it, it now becomes really unlikely that each subsequent digit is going to make a prime. Yeah. Because there's just a smaller percentage of numbers of prime at that size. Of course, yeah. And so that once you've hit that escape velocity,

once you've escaped the small numbers, it's very unlikely it'll stop. But for the case of primes, likely enough that, you know, once you're in the tens of thousands, hundreds of thousands of digits, you've had so many chances for it to be prime, eventually it will stop and be prime. How do we apply this now to repeating with the same digits repeat at any point? But first I thought i'd just look for repeating sections inside pi so i thought i wonder i've got a document on my laptop yep which has the first 1 million digits of pi is this gonna at some point involve some terrible python code oh whoa can i smell that it's coming it's terrible python code i don't know if you noticed. Can I smell that? It's coming. It's terrible Python code. I don't know if you noticed this morning, after I got back from my bike ride.

Okay, yeah. I then sat down, because you were working at the dining room table. Yes. And Lucy was having a call in the front room. I then sat down at the dining room table while the coffee was being made. Yep. And I knocked together some terrible Python code. Fantastic.

Because I was like, I've already got a million digits of pi here. Yep. For people curious, it's from when we filmed a video years ago for Numberphile, where we printed out the first one million digits of pi. And I did a walking tour of pi. And I'm like, oh, you can see this number here and this bit's over here. And so I wrote some code that would take the million digits and I could search for different things.

So I just dusted off that code and rewrote a bunch of it to search for repeated strings in Pi. Yeah. So the question is, what do you think is the longest repeated section? So you have a run of digits and then immediately followed by the same run of digits again. How long a run do you think you'll get in a million digits of Pi? I'm going to bring up the results while you're having a look. As in like, you know, the same five digits are the same or six okay yeah the longest run in a million in a million and if

there's one thing i've learned it's that a million is bigger than i think it is having walked we printed them out tiny little digits and it was a mile to walk no what is so a million such a big paper it was it was a ridiculously long piece of paper. Was it done on like one of those old-fashioned ticker machines? What are they called? No, like the old stock market ticker. Yes. No, like what Mr Burns has. No.

Yeah. Brady, who got a printer for Numberphile, found a printer in, I think it was, I want to say Belgium, somewhere like that. And what they used was you can buy very very long rolls of like packaging brown paper so it's brown paper but it's not plastic but it's slightly waterproof and a bit more robust than you'd expect yeah i know this almost like wax paper yeah almost like wax paper and what they did was they just cut the end 20 centimeters off a massive roll because

when you buy them from the factory to go to the packaging plant it's like huge rolls of the stuff and they just sliced the 20 centimeters off the end of a roll and so now he had a one mile long piece of paper like 20 or whatever it was centimeters white printing on that was a non-trivial did you please tell me you did a video just about the paper yes brady did a whole second video about how it was printed okay good it was that's my kind of it had to go through the printer twice because i requested labels i said we're going to get so lost in pi like i wrote a bit of terrible python code we could type we could look at the pi in front of us because we put it on a runway at an airport great we could look at the pi in front of us and type in a few digits and hit go. And it would tell, oh, you're at this point in pie.

Okay. Yeah. But I was like, it'd be way easier if we could just label every 10,000 digits or something like that. All right. Not like you didn't label it like a ruler. No. Like every hundred meters. No. It just flags up.

I think we might have done every thousand digits or something. But it was done by running it through the printer twice. So, one print run put all the digits on the bit of paper. And the second it through the printer twice so one print run put all the digits on the bit of paper and the second print time through the printer put all the locations through oh man i'm such a sucker for that stuff oh it was a logistical it was a whole thing at the end of the day everyone kind of packed up into the cars because it's like support cars driving us along pie yeah and everyone drove back to the beginning and i was like no no i want to walk and so everyone else drove back and i was by myself and it wasn't raining but it was like, no, no, no, I want to walk. And so everyone else drove back. And I was by myself.

It wasn't raining, but it was like that kind of weather. And I could just see pi to the horizon. And so I just started walking. And I spent like 20 minutes, whatever it is, walking by myself along pi. It was a religious experience. I was going to say, that's quite the mental image. Yeah. And just like walking for so long with these tiny digits one after the other.

They're all so certain. It's similar to, you know, when we calculated 100 and they matched. Yeah, I was going to say it feels like another emotional moment. A million digits that are all so seemingly random, but so defined. Ah, incredible. Yeah. Anyway, so that was a lot of fun. A million is a big number. It's a lot of digits of pi.

Yes. Yes. That was a lot of fun. A million is a big number. It's a lot of digits of pi. Yes. Yes. And your question to me was, how long do I think a patch of numbers would be that matches another patch of numbers?

Yeah. And now that I know how long that would be if it was just written out, I don't know. Ten. Good guess. Five. Ah, damn. Do you know what? I was going to say six.

Yeah. It's a tough one because there's no intuition to go off you did say five and i was like oh i was avoiding saying a number yeah so the first batch of five which occurs at position so this is digit 107 472 ish this is terrible python code yep there's probably an off by one error in there it's around there within a couple couple. The old TPC. The old terrible Python code. 64015.

That's my phone number. That repeats twice. So pi, once you're 107,472 digits in, goes 64015, 64015. So it repeats for five digits. 64015. Yeah repeats for five digits yeah in a million and i don't think it gets more likely like i know we know pi now to tens of trillions of digits i suspect we're not getting much bigger it would be bigger than that but maybe it might only be 10 maybe a bit more it's not going to be long so pi is we think what's called a normal number.

And normal in this context in maths means any group of digits are equally likely. So, if you pick five digits at random, that's as likely to appear in pi as any other group of five digits. Yeah. And we've not proven that, but as far as we can tell it's true for pi it's a very hard thing to prove for numbers imagine that we got to a point where we like calculated pi to like quadrillion yeah whatever and then and then just afterwards and then afterwards it repeats from the beginning from the beginning but like obviously like the final number has to be wrong.

I mean, it could. There's nothing stopping it from repeating from the beginning. But it won't definitely just keep doing that over and over. Like it could do it once. Yeah. And then other stuff. We know that. We've proven that. We're very confident in that.

Now, the issue with getting repeating patches is if you've got a run of five digits, which is one of a hundred thousand options the chance of the next five being the same is one in a hundred thousand yeah which is why i'm not surprised the first one happens at 107 000 digits in like that's about right yeah yeah so each time you want to have one more digit it's 10 times less likely to occur. Okay. And will probably happen 10 times on average further into pi. Yeah, I got it.

So, if you get, if you know 10 times as much pi, the string of repeating digits you're likely to find goes up by one. Right, yeah. So, it's real unlikely. And that's before we get to if you want it to start at the beginning. Yeah. Because we're going to have that same problem where once you're a few in, like once you're 100 digits in and we know it doesn't repeat in the first 100 digits, the chance of the next one happening is one in 10 to the power of 100, like one in a Google. Got it.

Yeah. in 10 to the power of 100 like one in a google got it yeah so if we don't have a repeat of three one four something near the something whatever then yeah what you're saying is we're getting further and further away from the start until that three one four has to happen again in which case yeah it's going to be uh i imagine the probability of the whole next block matching is one over ten to how many digits we've done so far. Yeah. Because this unnamed person, they haven't necessarily said how long it would need to be

to be something that feels like it's repeating itself. Oh, that's very true. And they did- Because you could say it's just two digits. They did say almost. So there is some flexibility. Like, oh, maybe like one digit in the middle is wrong. But that's not going to-

It'll gain you a bit, but not a lot. And it's even worse than the primes because each time you go one longer, because you're thinking, well, sure, it's very unlikely, but pi never stops. However, the probability is not like converging on some kind of non-zero value. My hand wavy argument, and people can maybe put this on more rigorous footing than me, is it converges to zero. So, the probability tends to zero. It doesn't matter if you've got infinitely many digits, it's a zero probability chance of getting that repeat, because the chance of it happening gets small so fast.

So, some people would argue that if something is infinite, then it increases the possibility of something happening. Yes. But however, something being infinite also means that it increases the impossibility of something happening. Yeah, because what we want to happen depends on how long, how far we are into quote unquote infinity so far. Yeah. And the getting difficultness as you go infinitely far gets bigger, faster. Yeah.

Like if it was happeningness. One out of three, something will happen. Two out of three, something won't happen. Yeah. And if we just would go, okay, let's like expand that. Let's just like make it big. Then you've still got more chance of it not happening than you do of it happening yes but more complicated than that more complicated yes yes yes

but this is this is the first step my brain is taking to sort of vaguely understand what you're saying great great zero step but the problem is both probability and things tending to infinity are famously woolly and complicated things in mathematics. But to a very simple, and I'm hand-waving it. To a very simple-minded person such as yourself. Such as me. No, I'm just saying very, very hand-wavy. Yeah. This is the vague sense of what's going on.

Yeah. Yeah. Yeah. Like how emojis don't really tell you exactly how you're feeling. Yeah. But they're a zero step. They're better than the null message. Yeah.

I think you've absolutely answered that. Thank you. And I think that this no-name person has, I mean, it's a great concept for a story because I've almost had an existential crisis. It's a good story. While trying to wrap my head around this. I love it. Write that story.

Yeah. That's a great answer. Thanks, Matt. My pleasure. I'm going to give that a ding, ding, ding, ding, ding, ding because it's repeating itself. Ah, repeating dings. Next problem. Put into the problem posing page.

And they did enter a name. Jelly. Gel. Next problem. Put into the problem posing page. And they did enter a name. Jelly. Gel. Or yell. Or yell. Or yelly. Let's go with yell.

So, this is yell speaking. Or hell. Hell. Depending on where they're from. There's a J. It starts with a J. It starts with a J. J-E-L-L-E says that they have a solid one meter by one meter by one meter cube unknown substance and they want to cut it okay so this i think is is a confusing way to put it

they want to cut it around the diagonal of the xy plane let's not worry about that for now okay they want to cut the cube that's what we're taking away from this. Right. To produce two sets of stairs. Or in other words, this is Jill speaking again, they want to build two sets of stairs to use in a stage play. So I guess like a fancy prop such that one stair, I guess I'll set a stairs, can be put on the other one to form a perfect cube. Also, the stairs should be walkable on by an adult human. Well, that depends what the cube's made of, Jill.

Yes. I guess. But they want to have two sets of stairs, such that if you stack one on top of the other, these two identical sets of stairs form a complete cube. Yep. Okay. We got it. They say that after a few paper experiments, it seems to be impossible

to produce two identical stairs. One always seems to have more stairs. This is a good problem. It's deemed impossible. That's where we come in. They want to know, is it really impossible for stairs to be identical? If so, what shape of stairs could be close to identical? And they then go on, blah, blah, blah.

Thanks for awesome podcast. If people want to have a go at solving this themselves, now is a good time to pause. Give it a go. Because, Beck, have you got a solution for us yes so uh i love this because this is like a puzzle to me and i love puzzles and so my you know your first instinct is to draw a square that represents a cube and then you sort of draw some steps. And when you're drawing the steps, you know, if you're sort of doing them like cube-shaped or whatever, you do find that if you go from like one step up and then across and then up again, another step and across,

and if you're doing that in sort of even cubes, even squares, then what happens is you come out the diagonal top of your square on the same side of that diagonal. Does this make sense? Only because I did get a bit of paper and try and do this. Yeah. If you're able to, you should do it right. You should try it right now.

But essentially what it means is you end up with a smaller side. Yes. You've got one side that's got, so let's say I've got a three by three square. Yep. That first square is your first step. First step. Next two squares.

Yep. That's your next step up. And then the next three squares. Yep. You know, going upwards. Three steps. That's your final step.

But then the other bit left over, you've only got three tiny bits. You've only got like one step two step up and then yeah so then embarrassing how embarrassing but i was like that it's weird i guarantee the the first time you try this if you don't put any thought into it in advance you will draw asymmetric stairs yes yeah and i i'd had a i realized i'd had a similar problem to this because in a flip chart that I recently made and filmed. Actually, this one is on YouTube. This one's on YouTube.

Yeah. Yeah. It's called This Page is the First and Also the Last. On one of the pages, there's like a broken heart. And when I was drawing the broken heart, I was trying to do like the break in the middle. Yeah. And I was trying to do it so that they had like an even number of. Pointy bits.

Yeah, pointy bits. But that was really hard. Same problem. Same problem. And so I was like, oh, I've come across this before. I was sitting with you when this problem came in and I was like, this doesn't seem right. I feel like there is definitely an answer. And we had very different ways of.

Yes. Of doing it. And I, my way was hands on. I like your way, it's good. So I was like, do you have any cube, like anything cube shaped? You had a bunch of die. I went and got the dice box from upstairs. Yes.

I was like, wait here. And I ran up, I came back with a box of dice. Yes. Most of which are not cubes, but enough were. No. And then I instinctively went to do a cube of three by three by three. Yep. I don't know why, that's just always my instinct. Three by three. It's a classic. Yeah. You of three by three by three. Yep. I don't know why.

That's just always my instinct. Three by three. It's a classic. Yeah. You're obsessed with the Rubik's Cube. That's true. Do you know what? I was playing with the Rubik's Cube.

There you go. So that's definitely where my brain went. And I was like, oh, this is impossible. And then I realized that three by three by three. Yeah. It's 27. Yep. It's an odd number.

It's an odd number. So I can't get an even number of cubes. Can't divide that by two. Then you were like, well, just do an even number. a number so i can't get an even number of cubes can't divide that by two then you were like well just do an even number of that so i was like of course i did a four by four i don't think i use that tone of voice no you didn't that was me that was me being like silly with myself i told you as we were doing this last night i said tomorrow when we're in the office recording the episode i both have a lot of dice in the office. Yes. I can see three different containers or bags that are full of dice. Including a jar that I'll try and guess later. You're not allowed near.

I've also got like cubes that link together. Would you like a bunch of cubes that link together? Yes, please. Wait there. They're under my desk. While Matt gets those, I'll explain what I did. So I realized that 4x4 by four by four is 64, which means that then I need to make half a cube that looks like stairs that consists of 32 cubes. If I could build up something that

looks like two sets of stairs that uses 32 of each and they're identical, but then match together, then I've solved this for the listeners at home these are all the cubes that i used i did a video about how you can find all the 3d nets of the 4d hypercube and then i showed how they can tile to fill space and so that's basically got big chunks of those that were left over that she's going to pull some cubes off i sat down and i realized that if you think about it like from a square point of view is fine because you're basically going to cut one side all the way through and you end up with a prism of whatever your staircase looks like so i realized that the issue is that you're you instinctively want to put one of the

corners of the stairs right in the center but that doesn't work and so i thought well if that doesn't work and hypothetically if there is a way to do this i need to do it the other way as i realized the other way was to put the center right in the middle of one of the steps like halfway across the flat and so i drew a square i drew i put a dot in the middle for the center. And I'm going to resketch this now so we can all see it. And I then drew the middle flat first, a third of the way across. And then I just kind of joined up the rest. So I filled that in like that. And I filled that in like that.

And that, there you are. That's two identical halves. So you have drawn a picture of the first step starts about, about a quarter of a way up. Yeah. And then, so that's sort of, if that first bit was a cube, like that first square, that first step could be a cube from the floor up. Yes. But then I would say the next one is like two cubes across.

They're each a quarter height and they're a third of the length of the cube across. So each of these runs is a third and each of these heights is up as a quarter. My brain is just calculating. Tell you what, let me do another sketch. Let me know if this is better. Because I would say that isn't that essentially one and then that's two and then that's one?

If you were to draw this as a... Oh yeah, but I've just drawn that badly. They can all be the same. They can all be a third. Yeah, yeah, yeah. It's just longer steps. Yeah, it looks like an Ikea manual. Anyway.

It does. Get a friend and make that out of a cube. Yeah. So the short answer is yes. Yes. And this method would generalize. You could do this for more steps. So if you wanted more and smaller steps, you'd split it in one direction up into an odd number of pieces and the other

direction into an even number. And then you would zigzag your way up. Yeah. Yeah. If you want tiny steps for tiny feet. Yeah. And if you want mine, which I'm still building. Oh, it's coming along nicely. I'll finish that afterwards and then we'll take a picture and we'll put it on socials. Yeah. But my version, to be fair, in all honesty, I would go with yours, Matt. They look more like traditional stairs. They'd be much easier to use, a lot less chance of falling around.

But if you want to get all ranchy, my version sort of goes from a corner. Yours are more theatrical, if I may. It is more theatrical and you could put them together to make almost like a half a pyramid type. Yeah. Type set of stairs if you wanted to sit them up against each other.

Well, I want you to finish yours. I think you should do both halves. I want to see them. I'm going to. I'm going to. But I'm going to wait till we take a break so that people can go. Ah, right.

Okay. Yeah, we'll have a break you can do that yeah yeah but in the magical listening land it's done oh my gosh if you want to see a photo of it go to a problem squared on instagram or on twitter so i'm prepared to say that yours are way more theatrical i love them thank you mine are deeply boring and functional but i but i love them like so yeah but Mine are deeply boring and functional. But I love them. So, yeah. But between us.

Yours makes sense to me. We have shown it is possible. So, the problem technically is, is it really impossible for the stairs to be identical? We've managed to show, no, it's not impossible. And it's proof by doing one version each. Yeah. Proof by sticking cubes together. Love it. The each. Yeah. Proof by sticking keys together. Love it.

The end. Yeah. And I want to see a photo of these stairs on stage. Oh, yeah. Yeah. Jelly, let us know what you end up doing. Yeah, yeah. If you end up making him, take a photo of it during the play.

No rush. Sometime in the next couple of years, we want a photo. Yeah. Upload it. Send us a link on the problem posing page. In the next couple of years, we want a photo. Yeah. Upload it.

Send us a link on the problem posing page. And just like Pi, we're at Any Other Business. There's always more business than Pi. Because it starts with three letters, A-O-B. And has a point, which is one, four. No, I'm trying real hard. I like what you're doing. Thanks. Yeah, I'm getting there.

But let's just get into it. What's our any other business? You pulled out a package and now I'm excited. I've pulled out a package. Yeah. Not a euphemism. Would you like to, would you like to open this? Oh, you know what?

I'm going to open it and I'm going to reveal the content. Oh, that's so mean. I know. I dangled it in front of you and then I took it away. I know. I just. You're like, do you want to get the button? Oh, I'm going to get it instead.

You're the guy in the lift no one likes ready yeah oh okay describe for the listeners what i have just removed from the bag matt has just pulled out uh his book that he has not finished writing yet correct it raises a lot of questions yes but you have pulled out what i imagine the book is exactly going to look like once it is available you you've correctly worked out this is not the book for start the back is completely blank okay i hadn't seen this before but this seems to be like a cover yep and the inside is this a proof copy of your book no it's so the inside is someone else's book it's the trading game by gary stevenson amazing also published by alan lane so the same imprint at penguin as me does this mean that someone else is getting your book i guess so do i do i pay it forward

so my um people at penguin got in touch and said look we've made a mock-up version of your book for photography reasons because they're doing promo stuff. And they're like, do you want it? And I'm like, yeah. Of course, yeah. So, there it is. So, they posted me someone else's book with a mock-up of what the cover of my book would look like on it. Yes.

I love it. It's a great, it's a very eye-catching. I quite like it. It's a good cover. This is a UK cover for people there's a different cover in the us so your your third book's gonna be called love triangle love triangle perfect it's third book pre-sales have just started which is very exciting yes people want to uh pre-order the book it'll be out in the uk on the 20th of june and out in the us on the 20th of

august this year 2024 and i can see that the subtitle of your book is the life-changing magic of trigonometry. I've got to say, I'm very sad that they didn't go with your original choice for a subtitle. Getting triggy with it. Yeah. Yeah. Yeah. Yeah. You can only make so many great suggestions.

I know. Before you just got to go with what they got. So I'm very excited about this and we'll put some links in the description. So if you pre-order it, that really helps me. But if you can't afford links in the description so if you pre-order it that really helps me but if you can't afford the hardback or if you pre-order it through my website it's full price but i sign it and you will probably still get a limited edition dust jacket if you pre-order it somewhere else it's cheaper particularly if you're in north america because postage for me to send it to you from the UK is quite expensive. So we will link to a variety of places where you can pre-order it.

Yes. Including Waterstones, who they got in touch with. My publisher went to them and said, hey, could you offer a discount for Matt's listeners and viewers and fans? Yeah. And they're like, oh, yeah, sure. We can do up to 20% off. And I said, well, 21 is a triangle number.

So if you made it 21% off. You're the guy where someone at the other end is getting the emails going. So true. This guy. This guy. They're like, fine, 21% off. What code would you like? You now realize there's going to be other people like, oh, but Matt Parker got 21%.

I know, I know. But I had a good math reason for it yeah and they're like what code you want like you know like a triangle discount or you know stand up maths whatever and i'm like if i'm allowed plus and equal signs and i am the code is one plus two plus three plus four plus five plus six equals 21 there's a triangle number yes so so if you pre-order on Waterstones UK edition, 21% off. If you pre-order through me, 0% off. But assigned.

But I will sign it. You'll have a signed first edition with a limited edition dust jacket that I will give you for free. So the choice is yours. Or wait for it to come out on paperback. It'll be much cheaper then. Yes.

That's my Any Other Business. Any Other Literal Business. I'm doing business. You are. Yeah. That's my Any Other Business. Any Other Literal Business. I'm doing business. You are. Yeah, closing deals, making sales. Nice one. Thanks, everyone. Well, let's bring you down a notch.

Uh-oh. Because we got a response to the poll that we did for our Californian episode. I haven't seen it yet, but I suspect it's going to remind me why I hate democracy so much. So, we did an LA episode and our question in that was about the Hollywood stars. Yes. And if you were to follow that in alphabetical order. Alphabetical order. Then what distance would you cover sort of going in between all the stars?

And you went, okay, I'll go for the first name because some things weren't named. Some things are like TV shows. I went by whatever string of characters was on the star. Yes. And some, I think it was three, three of them started with punctuation. Yeah, like a asterisk for NSYNC. Yeah. Like a quote mark for Weird Al.

Yes. Yeah. Like a quote mark for Weird Al. Yes. Yeah. And I included, if they put the punctuation on the star, I used it as part of the text string that I then sorted. You were like, how does Excel sort it? Yep.

Because I was like, how do you know if an asterisk comes before a quotation mark? And you were like, that's how Excel does it. Yep. I thought maybe it's to do with a QWERTY keyboard. That's a whole other question. Different question. My argument was, thatTY keyboard. That's a whole other question. Different question. My argument was, that's dumb. That was your whole argument.

That's a good point. I remember that. If I say Weird Al, whenever I say Weird Al Yankovic, I'm not doing quotation marks with my fingers every time I say it. Like, it is just a given. And so, I was like, Weird Al Yankovic should just be listed. Under W. Under W for Weird Al. I'm having a look.

And we did a poll oh dear that is decisive yeah you know what i don't know if this is better or worse because if it was like 98 against me and two percent four then i'm very clearly in the wrong yeah totally totally misguided the results if i may reveal them are against me 64.8% with me 35.2%. So I'm wrong, according to the people, but totally wrong. One in three people agree. One in three of you is an idiot. Yeah.

Though I can't believe two-thirds of you would disrespect Weird Al like this. I think you're disrespecting Weird Al. Why, by using every character and their name? By suggesting that the first letter of Weird Al's name is a quotation mark. Okay, okay, okay. I mean, I can't argue. The results are in. Yeah.

Wow. Which does bring me on to some other, any other business about the Hollywood stars. Yep. Alan wrote into the problem posing page. Yep. Selected solution in the drop down. Oh, good on him. A possible explanation for the incongruence between the, I just had to make sure I was

pronouncing that right, between the experimental Hollywood star distance and theoretical distance is that certain names would be disproportionately more common within certain generations who would all age up and get stars at similar times. Okay. Which is similar to our, I believe it was the last episode where we talked about names and trends with names and everything. So it's funny how these sort of linked. They said, for example, a bunch of Michaels born in 1970

might all grow up to be famous during a certain prime become famous enough for a star window. So they're all put in together. Exactly. Therefore, a small but significant amount of Michaels would all get stars at a similar time and be nearer to each other. Interesting. Which. It might be interesting to look at the distribution of dates they were installed and where. Yeah.

And if the distances between. If they clump. Some of them clump. So I thought that was really interesting. Another thing I noticed pop up on the Problem Squared Reddit. Oh, yeah. Which I occasionally have a little. Oh, good on you.

A little lurking. Yeah. Was that Chad. Chad. Got a shout out in the episode. Said I'm so excited. Now I need to look up Charles Butterworth. Because we mentioned that in the Hollywood Stars. Yeah. Everyone out in the episode. Said, I'm so excited. Now I need to look up Charles Butterworth. Because we mentioned that in the Hollywood stars.

Everyone's favorite person. Butterworth. We were like, oh yeah, he probably looks like a butler or something like that. Chad did a little deep dive into Charles Butterworth. Said, Reddit never disappoints. And then has linked to another Reddit thread where it quotes, he's literally the inspiration for the beloved naval serial mascot, Captain Crunch.

Wow. Good work, Chad. If you're on Reddit. And the whole subreddit crew. Yeah. My last bit of any other business. Oh, yes. More business.

More US. More US. Putting the US into business. There is no US in business. Yeah. B-U-S. Oh, it's right at the top. It's right at the top.

I knew it. Brain. I doubted. Brain. I doubted it again. So I'm going to the US. But the reason I'm going is there's a big solar eclipse and my wife is a solar physicist. So off we go. And while I'm there, we were going to drop by New York, see some friends. And I decided to put on an evening of unnecessary detail on April 14th. It's a Sunday. I'll be there. Grant Sanderson from 3Blue1Brown. Yes.

I'll be back. Emma Haruka-Awao, who is the current world record holder at the time of record for the number of digits of pi ever calculated by any means. She commandeered a lot of Google Cloud server time because she works there to calculate, oh goodness, I think we're up on 100 trillion digits now. Wow. Phenomenal. But Emma's going to come along and I've seen her talk about how she calculated pi

to hundreds of trillions of digits before and it's so fascinating. And we've got a bunch of other incredible guests. So we're at the Bell House in Brooklynoklyn on uh sunday the 14th of april it'd be great to see a bunch of uh dinglets there dingers that's what we call you all yeah dinguses wasn't it dingus ding guy be great to see all you ding guy yeah i won't be there but if anyone's in la i'll be you'll be around i'm gonna be in LA from mid-March until early May. Yeah.

So I don't have anything booked yet, but keep an eye out on the socials. I'm Beckill Comedian and I'll announce. Yeah. And before we finish up, we'd like to say a few thanks, some gratefulnesses, appreciation of our Patreon supporters who are the reason that we can continue making this podcast and the reason it's accessible to everyone. So in order to do that, we choose three Patreon supporters at random.

Yep. Using Matt's very... Spreadsheet of randomness. Yes. And then thank them them and in this episode those three patreon supporters are yule ian dream and our next one is i'm a cat there you go ema cat ema cat denis la cita yes i pronounced it dinas

la cita i can see your dinas i'd also like to thank my co-host matt parker that's me and our fantastic producer, who is also a lot like Pi, in that without it, stuff wouldn't work. Yeah. Yeah. Everything would be all over the place. Yeah, madness. No circles.

No circles. Lauren Armstrong Carter. Thank you very much, Lauren. And thank you to everyone else for listening, especially if you've told other people to listen to this podcast. You're extra special. Yeah. When you share it, when you, I see you on the socials tagging us and recommending us to people.

It always lightens my mood. The podcast is like a pyramid scheme. You get half the credit of the people you recommend to listening. I don't think it works that way at all. I think you get the same amount of joy when you share it. No matter how many steps it came down to get to you. So in a way, we're the ultimate get rich screen. I don't know how it works.

I don't know where I was going with that. Bye. It's not quick. No. So Bec, I just had to look up and remind myself. This is why I'm on my phone while you were talking at the end there. To remind myself how many dice are in that jar. Yes. And I've got the correct number in front of me.

Although I haven't double checked for a while. Maybe people have been helping themselves to the dice. I'll have to check. So, you recently picked 469. Yeah. To which I would have said lower. Oh, which you would have said lower. Yes. And the lowest one below that is 453, to which I would have said higher.

So, that's the window we're in now. It's not a big window. It's not. I could get it. This could be it. This could be it. This could be it. I'm going to say. Yep.

Four hundred. Mm-hmm. And sixty-one. Lower. Oof. Can't be long now. Not long now. Before the hundredth episode.

Bring it in, baby.