A Problem Squared - 041 = Table Surface Braced and Toothbrush Versus Paste

Episode Date: August 29, 2022

In this episode... * How do you stop a wobbly table from wobbling?  * Which is more effective for your dental hygiene: a toothbrush without toothpaste or toothpaste without a toothbrush?  * A-to-the-O-to-the-B.  * Plus! Some exciting merch news.  If you want to see more of Matt's wobbly tables on ridiculous maths surfaces, you can do that here: https://arxiv.org/abs/math/0511490. For the best system for teeth brushing and more tips on dental hygiene have a listen to ‘APS 03 - Teeth Tips and Defining Decades’ https://aproblemsquared.libsyn.com/aps-003. And, you can now buy APS merch! If you want a t-shirt of your own, head on over to: https://a-problem-squared.teemill.com. As always, if you've got a problem or a solution, hit us up on our website aproblemsquared.com. And if you want want even more from A Problem Squared, find us on Twitter and Instagram.

it is a period of civil war podcast presenters hosting from a hidden base have won their victory against the evil problematic empire during the battle the podcasters managed to steal the secret plans to the Empire's ultimate weapon. The analogy. Was that it? I gave up at this point. Wow. I was like, I am too tired to try.

Can I just say? Yes. First of all, strong start. Thanks. That's it. End of list. First of all. Yes. Yeah.

Yeah. Yeah. Hello. Welcome to A Problem Squared. If you are new to the podcast, I am Beck Hill, a comedian, presenter, and author. And I am joined by mathematician, YouTuber, celebrity, and friend. Wow. Matt Parker.

And in this podcast, we will try to solve your problems. Our listeners send us problems. We try to solve them. And that's basically it. That's it. Welcome new listeners. And welcome back old. On this episode, I will de-wobble tables using maths.

I'll solve a little question about teeth brushing. We've got some light AOB, including we check back in on our review situation. Ooh. Stay tuned. So, Matt, it's been, I think last time we saw each other was Blue Dot, wasn't it? Yeah, we were in a field. Yeah.

And actually, you left before I did. Yes. That's going to be left before I did. Yes. That's going to be important in a moment. Okay. So Lucy and I stayed until the very end of Blue Dot. We saw Bjork, which was hilarious. Okay. Did you see any online chat about the Bjork performance?

No. So do you remember when we saw Groove Armada? Yes. And there were like lasers and lights and they were projecting on the radio telescope at Jodrell Bank. Yeah, it looked awesome. And it was just all going off. Bjork did none of that. A tiny, tiny Icelandic lady came out with a full orchestra and she would stand stationary in the middle of the stage.

No one could see her and sing a song over an orchestra, which is not the kind of music that really pierces through a huge crowd of people standing in a field. So don't get me wrong, I had a great time. Number one, I thought it was very hilarious how ill-fielded this was for a festival environment. Yep. And second, I'm a big fan of Bjork. Lucy's a mega fan, right?

So we had a good time in that regard, but it wasn't a spectacle. It didn't fill the space. And it was the same thing every time. I love that you're like this megast star major singer came out with a full orchestra and yet it didn't fill enough space and didn't feel it would have been great in an indoor tiered seating venue yes yeah would have been spectacular but in a massive field where no one can see and the default setting is standing around chatting. Yeah.

It just felt like quiet background music in a pub because you couldn't see. It wasn't that loud and everyone was chatting. And I had a good time. I caught up with some mates. They didn't project anything on the dish. They just didn't utilize anything that's part of the Blue Dot Festival.

I think that's my arm. But something amazing did happen during that performance. And actually I have posted you something back because we are recording separately today. Do you have the delivery that arrived at your house? I do have the delivery. Now, I need to warn you, it's a two-stage delivery. So we'll just get the first stage out of the way. So if you pop it open.

All right. First of all, the merch showed up. Okay. We now have our Problem Squared merch. First of all, the merch showed up. Okay. We now have our Problem Squared merch. So I divided it into two piles and put one pile in that envelope.

So you can open it up and you can grab it out. Okay. Yeah, so I've got this. Don't open the sub plastic bag. Oh, okay. That's inside there. So I have for listeners at home a large, you can probably hear it, a large brown posting sack. Yeah, posting sack seems accurate. I reused the posting sack that the merch was sent to me.

I was going to say, it feels like it's quite a nice one, actually. If anyone wants their own paper posting sack sent to them in the post. If they went to problem squared with hyphens between the words dot tmail dot com. Yeah, we can pop that in the show notes. I feel like we probably should link that somewhere. So for people who were just catching up on the podcast, we ordered these t-shirts to be delivered to us at the festival so we could launch them into the crowd using a trapeze.

And they arrived after we had left. And the hotel, in fact, refused delivery. Becca's holding up a t-shirt with the word ding in giant letters across the front. Amazing. The hotel refused delivery they went back to t-mill t-mill great customer service got in touch said at no extra cost where do you want them sent instead i was like send them to my house and they sent them over t-mill

they're the best environmentally friendly print on demand t-shirt one oh and you've got the holding t-shirt do you want to just read us out the text on the holding t-shirt i know some people will have heard this in the blue dot episode, but I feel like it bears repeating. Yeah. When we created the shop, we just popped some holding place merch on there until we had graphics and stuff. So this one, we mentioned it in episode 039, the Blue Dot one. Yep. And it says, name another podcast who launches their merchandise via Trebuchet.

I'll wait. And then a problem square dot com. And wait as in a mass. W-E-I-G-H-T. In a gravitational field. Yeah. That's very funny. Although we ended up launching my own personal T-shirts into the crowd because they didn't

arrive in time. But now you have them. Yes. So first of all, that's just distribution of T-shirts. So you've got some. Now I got some. We can wear our own merch. Yes.

There's quite a lot here and we made them big because we wanted to make sure whoever caught them would some are big some are small I just gave you a whole spectrum of different sized t-shirts

yes secondly there's another bag in there because as you remember we were very excited that we were on the poster for Blue Dot

because you know we're just a podcast we do because it's fun and they didn't put our names on the poster they put a problem square yep which meant the first five minutes of us doing the a podcast we do because it's fun. And they didn't put our names on the poster. They put a problem square. Yeah. Which meant the first five minutes of us doing the performance,

people realising, oh, it's those people. It turns out, as I was just looking around, a bit distracted during the Bjork concert, I noticed someone nearby wearing the official festival hoodie. And only a very small subset of all acts get put on the hoodie. Oh. And would you believe, beck we made the hoodie oh no way so i ran to the merch store this is last thing on the sunday they had sold out of

everything apart from large white hoodies so i got us both a large white hoodie i've got my mine's here somewhere hang on do you want to grab yours out yeah oh man this is gonna get so much spaghetti on it. It's a real lounging around the house size hoodie. So on the front, it just says Blue Dot. The look on the back. Good quality. There we are.

Oh, my goodness. I mean, second row from the bottom. But still, not many people make the hoodie. No, there aren't that many. Look at that. To put it into context for people listening the top of the listing is like bjerk groove armada mogwai matronomy tim peak that guy hannah peel and then a ton of other like we've got stewart lee adam buxton adam buxton yeah a problem squared that's very nice so that means that if anyone listening went to blue dot festival and bought

a t-shirt or hoodie or something with the lineup on it, they now have that. They've now got a problem squared. Amazing. Isn't that something? Well. Doesn't say it's a podcast or anything. Who knows what people think that is? This makes up for when we showed up at the security and they couldn't find us under a problem squared.

They couldn't find us on the list. You said it might be under Matt Parker. It wasn't under that. And so then you were like, oh, well, you know, we'll radio someone. The guy walked off. And I was like, well, it's not under my name. It's not under our names, right? Yeah, it turns out it was under Beck Hill.

It was under Beck Hill. It's alphabetical. It was under Beck Hill's A Problem Squared with some guy. That's right, yeah. So there you are. Thank you so much. Do you know what? I started putting it on

and then realized I can't put this on. I'm in a cupboard on a hot day with multiple lights around me. So I'm not going to wear it right now, but I will put it on. Yeah, I'm in my study. I am already sweating a lot in my study. I'm not putting on the hoodie.

So there you are. Oh, thank you. That's made my day, that has. There you go. That's nothing. We made the hoodie. Yeah, thank you. So apart from your recent hoodie acquisition, how There you go. That's nothing. We made the hoodie. Yeah, thank you. So apart from your recent hoodie acquisition, how are you doing?

I'm all right. I'm all right. I did Deer Shed Festival the weekend after I did Blue Dot Festival. But at the festival, the night before we filmed, I did a reading from one of the Horror Heights books. Are you a book to do this? Yes. Yeah, yeah. No, I just gathered all the small children. Guerrilla marketing yeah yeah no so i did a reading and um sold some books and did some signings it was lovely i met all these great kids i did a q a after the reading and one of the girls in the audience popped up her hand when i was doing the q a and just said who looks after pudding when you're away and i was like oh then I had to explain to everyone,

like pudding's the hamster. And right now my husband's looking after him. Afterwards, when I was doing the signings, turns out that their dad listens to A Problem Squared. I think that's how they knew about pudding. And then all the kids had questions about pudding because, you know, as soon as you bring up that, they've got no more questions about the book. In fact, at one stage I said something about how pudding's nocturnal

and sometimes he gets a bit bitey, like a vampire. And then a kid put up their hand and said, will there be a vampire hamster in the next Horror Heights book? I was like, well, maybe there needs to be now. But that leads me on to some sad news because a week after that, last weekend, unfortunately, Pudding has left us for the big wheel in the sky. It was very sad. It was very sad. And he was too, which, as we know from previous episodes, is sort of their average lifespan. He was very happy. No suspicious circumstances. Great two years.

I don't think anyone did it in the dark of night. Didn't have any enemies. Wasn't many of Pudding's many, many nemesis. I think he just got super excited and it was a bit much for him. But, you know, in a nice way. That's a good way to go. Yeah. So the reason I say this, I wouldn't normally say this

because it sort of brings the tone down a bit. But I'm not sure whether to continue with the Pudding Squared and still do hamster-related things or whether we shelve that for a while. I might get another hamster in the future, but we're going to leave some time so that it doesn't feel disrespectful. I feel like at a minimum, out of respect for Pudding,

we should retire the segment until we decide otherwise. Yeah, so he's gone past the edge of the solar system now. He's off to other solar systems. Moving forward through the solar system. Yeah, so I just wanted to thank everyone because I've had so many questions about him from people i have just met and it's really sweet so um yeah so thank you thank you everyone i don't think he will ever know just how incredibly loved he was and also my love and heart goes out to anyone else who's lost a pet because it really sucks but hey we're on a hoodie.

Unless we did that first, right? Because we can't go from in memory of Pudding to, hey, here's a hoodie. Yeah. Pudding. We'll live on through being such a big influence on the podcast. Yeah. So I'm going to dedicate this episode of Problem Squared to Pudding, the unofficial mascot of the Pudcast.

Oh, I can't believe we never thought of podcast. The podcast. I mean, it feels weird going straight to the agenda now. The first problem of this episode comes from James Tunstall, who says, I, along with what I can only assume is everyone, dislike tables which wobble. Most of the time you just shove a bit of anything under the leg and you're all set. But what if you don't have a bit of anything? Given a four-legged table, can you always find a spot on a surface where the table will sit without wobbling? Or put another way, does there exist a surface on which a table will always wobble? So Matt, I mean, I think you could find a surface that will always wobble if you just cut

one of the legs short. Ah, yes. And in fact, you've got straight to an important point here, which is why is the table wobbly? Is it wobbling because the legs are different lengths or is it wobbling because the surface is uneven? Or because the legs aren't attached properly. Or because legs aren't attached properly. Correct. Yeah. And at that point, you know what? If it's a manufacturing defect in the table, there's not much that can be done. If you're thinking this sounds familiar, this is a concept in maths, the wobbly table problem.

Yeah. And it's been around for quite a while. Yeah, because I'm sure in those like life hack listicle things that you get online, I've definitely seen the advice that if you have a wobbly table, you should just rotate it a little bit by a little bit and eventually it will get to a point where it doesn't wobble. I'm pretty sure that's worked for me before.

Yeah, and that's why I thought it was quite interesting about this because it's been a mass thing for a long time. Look, I remember at university this being a mass thing. It's been around since the, oh goodness, late 20th century, I think. thing it's been around since the oh goodness late 20th century i think but the kind of rigorous proof behind it has been lacking okay and so this is an interesting one where even in math circles it's kind of this oh there's a uh the concept that if you're on a wobbly table if you rotate the table you don't have to move it somewhere else but the table already is you rotate the table at some point it will not wobble and actually practically that has always worked

for me and there's always a point where it stops wobbling but proving that rigorously proving it will always work and if it doesn't work what are the conditions under which we know if it will or won't work is super vague. And so actually I brought up the most recent bit of mathematical research into this. It's called Mathematical Table Turning Revisited. Oh, man. From 2018. I bet there's so many DJs out there who were disappointed when they opened this up.

Yeah, they were just searching on archive for math papers about turntables. Yeah. They're like, oh, Mathematical Turntabling Revisited. So the first paragraph, we investigate under which conditions a rectangular table can be placed papers about turntables. Yeah. They're like, ah, mathematical turntabling revisited. And so the first paragraph, we investigate under which conditions a rectangular table can be placed with all four feet on a ground described by a function.

And then they got the math symbols for mapping R2 to R, which is already leaping into the proof. But first of all, for people who aren't familiar with the concept, let's do the hand wavy solution to this problem. Okay. The solution technically only exists

if the legs are all of the same length. So if the legs are different lengths, you're in trouble. It doesn't work anymore. Because you can think of a perverse example. Let's say you've got two very long legs, actually. The diagonally opposite legs are super long, and the ones on the other corners are super short yeah yeah that's just gonna fall over or you've got to kind of lay it down on just three legs and we don't accept just three legs as a

solution because maybe it might be stable but in theory an annoying table is one that rocks between the two different three leg arrangements where it's stable like it leans on one set of three and then it rocks the other rocks backwards and forwards so we don't accept just being on three legs as a solution to this problem it's got to be all four cannot budge no matter what you do but if your rectangular table it used to just be square tables we now know this is true for rectangular tables does have four legs that are all the same length and it wobbles, we know for absolute certain if you get that table and you rotate it on the spot, there will be a point at which it does not rock, assuming the surface the table is on is continuous, which I'll explain in a moment, and there are no angles in the surface greater

than 35.26 degrees. Oh, wow. That's so specific. And also, just to clarify, when you say square or rectangular tables, you're meaning the configuration of the legs rather than the shape on top. Because you could have a square table that only has, say, three legs. Or you can have... Yeah, a disco rectangle. Yeah. Yeah, our old sausage body. Yep. That's a reference to something we found out.

Wow, that was a while ago. Yeah, yeah. Or you could have like a square configuration of legs, but with a round table on top, which actually I think is really common. And actually, it's an interesting point saying if you're on a table with three legs, that's always stable. So the power of triangles. Three legs. Yeah. Can't go wrong. Issue is, of course, there, if you put too much weight outside of those three legs, you can tip the table over. Yeah. Yeah. They've got to be wide legs. So the advantage of a leg in each corner is there's no tabletop outside the legs so you can't tip it over by putting too much force on the table there is a argument for yeah three legs would work but it

you can tip it over if you like you sit on it or something whereas four legs that won't happen so strictly speaking everything is in terms of these four points which are the bottom of the legs and we assume they're all quote unquote in the same plane which basically means they're all the same length and we assume they're in a rectangular arrangement that that's the situation for which we know it definitely works and practically i've never come across a wobbly table that i couldn't fix by rotating it unless it was you know obviously legs were super short in fact yeah a lot of the time even if it's the legs problem it'll still be fixed by rotating

it because it will still match somewhere on an uneven surface well i was going to say for the sake of clarity as well when we say rotating the table we're meaning like clockwise anti-clockwise in as if you were looking down at the table not rotating it not flipping it side to side yeah yeah yeah putting it on its upside down around the z-axis i would uh yes it'd be fair to upside down maybe it'll stop wobbling it's a solution yeah the hand wavy argument for this uses something called the intermediate value theorem which i'm a big fan of great theorem and what's interesting about the intermediate value theorem to my mind is it's surprisingly easy to state and sort of intuitive to understand but yet very powerful

when you apply it to different situations now we'll get into the weeds in a second because actually using it rigorously is super difficult but for an intuitive way of is there a solution to a problem it's it's actually handy. So the intermediate value theorem basically says if something changes continuously and it starts at one value and ends up at another value, it must have gone through every value in between along the way. Okay.

Which is to say, if you go from one place to another place, you've got to go everywhere in between to get there. Sure. So for example, let's say you're currently a meter and a half tall. I don't know how tall you are. That's to the nearest half meter. Is that about right?

Yeah, a little bit more. A little bit more. Okay. And when you were a child, at some point you were about half a meter tall. Yeah. So at some point in your life, you grew from half a meter to one and a half meters tall, which means if you pick any height in between there, at some point in your life, you were definitely that

height yeah because you had to go through all of them yeah that's the intermediate value theorem now we can't say what age you were when that happened we can't say anything about the situation where that occurred we can just say it exists so this is an existence theorem this this thing exists and the same deal with like if you wanted to cut i don't know like a pancake or a pie or a sandwich in half with a straight line that you want to you got a knife you wanted to cut i don't know like a pancake or a pie or a sandwich in half with a straight line that you want to you got a knife you want to do a single cut but it's some crazy shape and you're like that's annoying how do i cut this in half but if you think about it if you were to cut it all the way to one extreme side it would be unfair because the piece on the

left would be really big and the piece on the right would be tiny and if you cut it on the other extreme side it'd be the reverse situation and so one would be bigger big and the piece on the right would be tiny. And if you cut it on the other extreme side, it'd be the reverse situation. And so one would be bigger than the other. But actually, if you move your knife continuously between those two positions, it must go through every possible ratio of the two halves in between. And so somewhere in the middle, there must be a point at which it switches from the right being bigger than the left, the left being bigger than the right. And that's the exact moment at which it would be perfectly fair to cut it at that point. I don't know why you've chosen that as your analogy, because it's a seesaw. It's like you're using balancing scales, where you put a little bit on one side, a little bit on the other, until it all balances out. If you can only

add or subtract the mass continuously, because seesaw, you put a whole kid on at once or something and so you're suddenly abruptly changing the amount of mass on one end or the other the seesaw will tip at some point it can't go from one to the other without having gone through all the other degrees uh oh yes but doesn't mean it will balance it just means it'll move through the middle point but that's the same as your pancake so yes but the point of the pancake thing is there's a perfect equilibrium point in the middle where you cut whereas a seesaw you're just gonna the reason i'm being hesitant about the seesaw oh what i'm saying is you have to put someone else on the other side yeah to try and balance it out and then you just

keep adding kids until you get it balanced the reason i'm being hesitant is you're adding a whole child at a time if you are gradually pouring water onto one end yeah well you're the one that introduced kids i know i'm just saying that's why i'm being hesitant because one might have a backpack yeah yeah yeah yeah i'm only being careful because the intermediate value theorem only works if something has to change continuously yeah and so if there's any any ability for an abrupt change it no longer applies so when you're cutting though so you were saying are you saying that you never finish cutting no no you move the knife continuously from one side to the other as you're cutting let me go back each time you cut you're doing the same as doing a whole child you're not no no okay so so you've got a sandwich

in front of you i feel like you're choosing this because you know how much i like food maybe trying to distract you with with the food so i'm talking about all the places you could cut it so you're hovering your knife above this sandwich and you're like oh i can totally cut this sandwich right here but if i cut it right here the left side's bigger than the right side that would be unfair and so you move your knife over to the other side and you're like i cut it here but if i cut it here the right side's definitely bigger than the left side yeah but at some point as you are moving the knife from one position to the other because it starts being unfair in one direction and ends up

being unfair in the other direction and you've continuously moved your knife as you are moving it across at some point it must have flipped from one side being bigger to the other side being bigger. And that point where, and you don't know where, this is the problem. This tells you nothing about where to cut. It just shows you that in the middle, at some point, unbeknownst to you at the time,

you would have gone through the perfect cutting position without realizing. And so all this theorem time, you would have gone through the perfect cutting position without realizing. And so all this theorem says is there would have been a perfect point where if you'd cut it then, you would have cut the sandwich perfectly in half. That's such a dumb theory. It's great. It's the same thing. Do you know what? It is the theory version of Morecambe and Wise.

I was playing all the right notes, just not necessarily in the right order. It's not far off that. I will accept. Yeah. playing all the right notes just not necessarily in the right order it's not far off that i will accept yeah and it like it belongs to this class of theories that are called existence theories it proves it exists but is in no way helpful yeah yeah exactly other than you know that there is a solution yeah like i feel like inherently we all know that there is a point we could have got something right and didn't and i feel that we are all used to doing that constantly. So this is just a theory that says,

do they use the pancake in the theory or did that come from you? So they use the pancake in the 2D case. They use sandwiches in the 3D case. Amazing. That's the traditional way of... That's the traditional way of testing theories with pancakes and sandwiches. Yeah. And actually the way the theory goes is you can actually, with a single cut,

cut through two pancakes and cut them both perfectly in half with one cut by using the same theory. But this time you're also changing the orientation of the knife continuously. And because you can move it backwards and forwards and move it on the angle, you can prove it's unfair one way, unfair the other way. And in the middle, it must have been fair to cut through both pancakes equally through the center. Okay, so if you had two pancakes stacked on top of each other but it looked like a Venn diagram. Yeah or even even next to each other on the counter but you're going to cut in one continuous slicing action all the way across your countertop going through both pancakes in the same line you can cut them both fairly in half with a single cut. That's very interesting. You can also do it in the Venn diagram arrangement that you have mentioned yeah and then the ham sandwich one gets a bit

ridiculous you can use the same idea of the intermediate value theorem that if you had two slices of bread and a piece of ham which are separated anywhere in the solar system on earth so the community can totally deconstructed sandwich you can still there's a direction where it can be a totally deconstructed sandwich. You can still, there's a direction where you can make a single cut in one plane that would cut all three perfectly

in half without having to change the direction you were cutting in. So another super useful bit of maths. You've just blown my mind. Wow. That means anywhere

there is always a ham sandwich that exists. It's just that all the pieces might be in separate places. Exactly. That's deep. I mean, that wasn't the intention of that theory, but that's what I took away from it. Good. They're suggesting that the hams, it's a sandwich when they're nowhere near each other.

No, it's a conceptual sandwich. It's a generalized sandwich. But to be fair, what you've taken away from that theorem is just as practical as the actual intention of the theorem so you know that's true yeah okay i'll let you get back to solving the problem now when it comes to the table now it's actually practical because as you rotate the table you're testing every intermediate position and you just keep rotating until it stops wobbling so in that case the existence proof is enough you just start rotating and you wait until it stops and the kind of hand wavy version is if you think about it when you set the table it's going to rock in a direction so let's say

it rocks backwards and forwards in one direction and it's because of the ground not the table table's fine if you then rotate the table around 90 degrees it's going to rock in a different direction yes relative to the table but the same direction relative to the table, but the same direction relative to the unearthing. Yeah. So from the table's point of view, it starts from rocking this way and ends up rocking the other way. And there must have been a point where it swaps from rocking this way to rocking that

way. And the point where it transitions from one to the other is where it's perfectly stable. Got it. So you're kind of drawing a line between two legs. Yeah. And then it's at what point does that line start to be between the other two legs? Yes. Perfect.

Except mathematicians don't like that kind of hand wavy, this sort of works. And it turns out turning that intuition, which actually works all the time, into a rigorous proof is super difficult and is not yet finished. So we have not finished proving that this definitely works, which is why this group of four mathematicians from Australia, they're Monash mathematicians, are still chipping away at the problem. And I imagine a non-zero number of people who listen to this podcast also watch maths videos on YouTube because that's our primary way of promoting the podcast.

There is a fantastic channel called Mathologer, which a lot of people may have come across. And the main presenter on Mathologer, a guy called Burkhard, Burkhard Polster. Oh my gosh, that's the coolest name ever. Burkhard Polster. Yeah, German mathematician, lives in Melbourne. Great guy. Why isn't that your name? Because Burkhard got there first. Oh, fair. Yeah, otherwise, of course. Burkhard Polster. Yeah, German mathematician, lives in Melbourne. Great guy. Why isn't that your name? Because Burkhard got there first.

Oh, fair. Yeah. Otherwise, of course. Burkhard Polster. Why would I have Matt Parker? That's a crowded search term. And Marty Ross. That's the other person who works on the channel, Marty Ross.

Oh, the back hill. Wow. Yeah. You're the Marty Ross, I guess, if I get to be Burkhard Polster. And then I have a few other people who also help out with working on the scripts and testing things and all that kind of jazz. But they're the Marty Ross, I guess, if I get to be Burkhard Polster. And then I have a few other people who also help out with working on the scripts and testing things and all that kind of jazz. But they're the main two.

They're also two of the authors on this paper. Wow. So a lot of people will know Burkhard from Mathologer. They're also one of the leading world experts on the wobbly table problem. Wow. I think that's amazing. So there you go.

I mean, I have a channel, but I'm just a recreational mathematician, whereas Burkhardt's doing proper math research. Well, I say proper math research on wobbly tables at Monash. And so they did a bunch of research. And that angle, that 36 and a half degrees is actually the cube angle. So if you get a cube, I'm holding a Rubik's Cube for reference, and you balance it on one of its corners perfectly upright.

So Matt is currently holding the Rubik's Cube so that a corner is on his thumb. If you imagine someone spinning a basketball, a bit like that, but with a Rubik's Cube. Yeah, exactly like that. I'm spinning it very slowly now. Yeah, he's spinning it with his other hand. And so if you do that, and then you look at, so there's the top corner. And now he's put his pants on his head. He's dancing around.

Nothing to do with the maths. So you've got the top corner up here. Yeah. Then there's three edges that meet at that corner. Yes. If this is perfectly balanced, those edges are exactly 35.26 degrees. Ah. And they used the notion of a cube like this as part of their proof to generalize all rectangular tables.

I won't go into the details. there's some great diagrams in the paper in fact we could tweet the diagram let me just check to make sure it doesn't look terrible oh they got some great diagrams of wobbly tables on ridiculous math surfaces oh we're tweeting that yay so we will tweet the crazy tables on the surfaces because that is awesome and then there is a diagram there it is oh it's not very exciting just shows a cube on its corner. And there's a great sentence. I mean, it's a great sentence from a maths point of view. Now, if I read it out in isolation, notice that every table is similar to one of the gray rectangles shown in the right triangle created by moving the point A prime from P to B prime. Amazing, amazing stuff. Real. Some great stuff.

We'll do the tables on the weird surface. Sounds like a haiku. So they've managed to prove on a continuous surface with no angles greater than this cube angle if you balance it on a corner, the theorem definitely holds all tables can be de-wobbled

by rotating them around. So what happens if the angle is greater than that on the surface oh it it almost definitely probably still works we just haven't managed to prove it definitely definitely well that's just the limit of the proof but not necessarily a practical limit because we're pretty sure it works anyway like my argument be like what i was thinking is if it's over that degrees does that just mean that the table will slide? You're on like a hill and so you can't turn around because you're just going to slide down the hill.

Fun fact, nothing we're talking about here says anything about the angle the surface of the table ends up on, just that it won't wobble. It can be at some crazy angle, like all your drinks will slide off, but not because the table's wobbling. That's what I was thinking, because it doesn't mean that the surface of the table will be flat. It just means it won't wobble. Absolutely not. No.

It just means it won't wobble. That's all we were asked about. Which kind of brings us to the second half of the problem. Is there a surface where they won't? Well, it just needs to be a discontinuous surface, and then it won't. And they point out in the paper, considering our example of a discontinuous ground, it should be clear that a wobbling

table on a tiled floor may defy our table turning efforts. Oh, because of the grouting. So if the floor is tiled, you can have sudden jumps. Yeah, and I guess it can be at different, some might be at an angle slightly. Yeah, and they're all different angles and it doesn't smoothly go from one tile to the next. It abruptly jumps yeah and i guess yeah it can be at different some might be at an angle slightly and yeah and

they're all different angles and it doesn't smoothly go from one tile to the next it abruptly changes and so it doesn't mean that you can't do this and in fact in my humble practical experience it still works but technically if you've got a discontinuous ground this will not necessarily work so they're they're the exact parameters around when you can and cannot de-wobble the table. And I think it's interesting that of all the things in maths that have, number one, pervaded most of maths, like

anyone involved in maths or recreational maths has heard more often than not of the wobbly table problem. That's one of the better known problems. If you haven't, I hope you've enjoyed hearing about it for the first time. It's an amazing theorem. And even general public because it makes it into those life hacky kind of fun fact lists that you can rotate a table stop it from wobbling but yet it's not a mathematically proven result yet we are still chipping away to make sure the math is definitely correct

and that's that's the end That's my slightly wobbly argument, which if you rotate enough, shows you that you can de-wobble tables under some circumstances. Okay, right. So to bring it back to James's problem. Oh yeah. So James said, given a four-legged table,

can you always find a spot on a surface where the table will sit without wobbling? The answer is yes. In fact, you don't have to move it. You can just rotate it. Assuming the four legs form a rectangle, they're all the same length. There's not a slope greater than 35.26 degrees. And the surface is nice and smooth. Like there's no sudden jumps in the surface. It's like a smooth, continuous thing. But within those conditions, which in my experience, most pubs conform to, then yes. Okay. And the second part of James's problem said, or put another way, does there exist a surface on which a table will always wobble? Well, yes, sort of. Yes, if the legs are weird lengths, you can't fix that by moving it around.

And almost definitely if you include discontinuous surfaces that like tiled surfaces with a like or if you're dining on a staircase situations like that then there very well is not a solution to stop the table from wobbling got it so a continuous surface is a flat surface but can go in any angle no it can have bumps and things but as long as there's not a sudden jump to a different level it just bows up and down. It's like a warped surface. Now I see why this is so hard, because at what point does a curved surface become a, like a continuous become a discontinuous? As soon as the derivative, first derivative decides to disappear. That's the short answer.

I don't even know what that means. There you go. But you know what? That's not the problem that James sent us. Yeah, that's not the problem. I mean, I'm annoyed at James because James definitely gave me more problems than i had before but in relation to james's problem i feel like you've answered both of those thank you so there was uh two parts of that problem so i'm going to give you a ding ding ding ding all right thank you and i've learned a bit more about that wobbly table thing i didn't realize it was so complicated and about conceptual

sandwiches i want a conceptual sandwich people try it try it. Try it in a pub. Let us know if it works. Beck, someone titled Mandy from Mafra, went to our problem posing page at a problemsquared.com and they put in a problem that reads as follows. If you had to choose

which would be more effective for your dental hygiene? A toothbrush without toothpaste or toothpaste without a toothbrush? And they have come to us because, as it turns out, we have a history of discussing optimal toothbrushing. Yeah. Well, first of all, the way this question is phrased,

you could suggest that you're saying if you had to choose, what would you make the most effective way of doing it? Like if you had to choose what would you make the most effective way of doing it like if you had to choose if you had to choose oh so also if you were the tooth hygiene deity yeah and you get to i could choose which one works best but i'm pretty sure that's not what mandy meant i'm just being pedantic so i thought i would answer this as a little uh dinglet little wing ding which we still haven't decided which one we prefer, so we've decided to go with both. So in episode 003, Teeth Tips and Defining Decades, I think it was called. It was early, really early. We answered a different question about the optimal way to brush teeth. And I got some advice from a professional dentist, wonderful Sophie from Adelaide, where I hail from, and also one of the honeymooners who came to visit me on set of Make Way, Take Way.

Look at that. We've come full circle. Pretty sure I talked about that in the main pod and not on the Patreon bonus podcast. I think that was the main pod. But, you know, if no one's sure, you should just sign up as a Patreon and listen to our bonus show just in case. It's the safest way to go about it. So I was pretty sure that I already knew the answer to this,

but I did go back to Sophie and check and she has confirmed this. So one of the things that we learned from that episode, I'm not spoiling anything, but one of the things we found out was that saliva is like a wonder substance. It's totally underrated and it has amazing antibacterial properties. Basically, saliva is almost like our mouth's own toothpaste. Nature's toothpaste, they call it. It is nature's toothpaste. That's why if I ever have a dry mouth they ask someone to spit in my mouth. I don't do that. I never do that. I wouldn't recommend it

especially in this. For other reasons. Yeah. The great thing about toothpaste is that often it has fluoride in it and that helps add a protective layer to our teeth. Toothpaste have different qualities, but mainly it's the properties that it has to help create a protective layer assisting our saliva. That's more of a long-term game, right? Actually, it only sort of lasts for about 24 hours. It's not like you get a buildup of toothpaste. It's like it wears off. The opposite. It is the opposite. So, you know, your teeth are most protected probably from about half an hour after you brush your teeth and then get less and less. That's why we found out in that episode why you shouldn't rinse after you finish brushing your teeth. You should only spit. People should go back and listen to it because it's really fascinating. You learn loads

more about teeth. But to answer this question, essentially marketing is the reason that we think the toothpaste is so important versus a toothbrush. Basically basically the brush is what gets rid of the plaque and the build-up of food around and on your teeth it's the brush that does the majority of the work the toothpaste is sort of there to age your saliva and something that it already does and to make your breath smell nice you know and again that's a relatively new thing that humans feel that we need this minty freshness to make us feel clean. Minty fresh, yeah. It's like in the same way that I take peppermint oil tablets for tummy bugs. So even when I burp, I feel quite clean.

But I'm not. I'm just burping peppermint. Anyway, I'm a bit all over the shop today. Mainly because I'm aware of how short this problem is if I don't go off on tangents. But the answer is toothbrush. Toothbrush is loads is toothbrush. Toothbrush is loads more effective. Toothbrush. So if you could only fit one item into your toiletry bag and you had to choose between a toothbrush or toothpaste, take a toothbrush because in terms of dental hygiene, you will do far better brushing your teeth without toothpaste than you would just

moving toothpaste around your teeth with your finger. I wish I'd known when I was a teenager running late for school because I used to just put the toothpaste in my mouth. Yeah, I hear. Yeah. I wouldn't even use my finger to rub around my teeth. Oh, really? I would just squirt some toothpaste directly into my mouth. Into your fresh time. So if you're packing for a trip under a really restrictive weight limit, or you need to run out the door now, you've only got time to grab either the toothbrush or the toothpaste grab the toothbrush yeah i'm not saying don't use toothpaste it still has good qualities but just saying that your toothbrush does does more than um heavy lifting then you're probably giving it credit for i would very much recommend anyone who hasn't listened to episode 003 because we do a deep dive into more of that stuff sophie's given the thumbs

up so that's absolutely correct. Toothbrush is more effective. So there you go. Given you went to our resident dental expert, who turns out we have on call, and it's one-sided. It's not like it's all this, if, that. It's no. Toothbrush.

Toothbrush all the way. I'm going to give you a ding. My only extra question would be be if you haven't got your toothbrush what's the optimal strategy then eat an apple or apples are quite acidic so i'm gonna say cheese eating cheese is calcium okay so that's my guess why do we keep creating more problems when we're solving them poor sophie she's gonna get some really random questions now we've just created a problem i guess this will be my Any Other Business, not next episode because- In the distant future.

In the distant future, I will come back to you. Do you know what? Let's put it out to you guys, the listeners. Yeah. We've spent so much time- We're doing all the work here. Solving your problems. Yeah.

Why don't you tell us? I want expert opinions or something with research because that's what we do. I want guesses. Wild guesses. You want guesses. Wild guesses. You want guesses. Okay. Oh, yeah. How about if you have a guess, send it directly to Matt on Twitter.

No, no, no, no, no, no. Send it to both of us. But just give us a measurement where you are on the guess expert spectrum so we know. Because it makes me wonder where gum comes in. Oh, chewing gum. You know how they say like, oh, gum helps fight decay. The reason it helps fight decay is because chewing means that your saliva glands start producing more saliva. Like gum takes credit for all the stuff that your mouth is capable of already.

But I do wonder if the gum helps remove the stuff between the teeth. Random guesses from the internet. Let us know. Yeah. Or a problem posing page at aproblemsquared.com. You can put your solution there. Select solution in the drop down should we get into any other business so a lot of people enjoyed your summary of the different wedding anniversary gifts yay but we have what i have deemed the best suggestion that came in from French guy,

CH like a French guy living in Switzerland, I guess at French guy. CH says, hello, a problem squared regarding the wedding anniversary. I think they should be named after the periodic table of elements. That would make things a lot easier to remember. They addressed me directly and say,

so happy oxygen anniversary. And oxygen, yeah, eight protons. It's the eighth element of the periodic table. Great system. The problem with that. No problem with that. How do we work out what the negative anniversary is? Because they've started on the first element. It's antimatter. You get antimatter for the first one. Okay, right, yeah. So you get antihydrogen, which I think you can get from cosmic ray collisions. So you get a

cloud chamber for your negative first birthday. Heavier elements are harder, I think, to get the antimatter version of. You just had a wedding anniversary, did you not? Yeah, in fact, the episode that that came out on was on my wedding anniversary. Serendipitous. On my iron and or candy anniversary. And what number is that? Six. Carbon.

So I would say diamond. That's your diamond wedding anniversary. Or graphite. Or graphite. Or a pencil. Yeah, you've got options, as previously discussed. The downside to this suggestion is, of course, your gold wedding anniversary isn't until 79 years. Your platinum isn't until years but you get them both at once that's kind of nice 10 10's your 50th that's nice yeah the jewelers association won't be happy with that but then again you can get a beryllium thing

on your fourth wedding anniversary that's pretty cool neon neon on your 10th that's a party come on that's nice that's good i'm on board that's in two years can we do a neon wedding anniversary for you and lucy absolutely and we'll have a big party that would be great 90s themed i'm so excited and we did get an email from the original couple who were due to celebrate their negative one wedding anniversary and they have confirmed they say absolutely 100 earned a ding on your solution. So they emailed in to give us total sign off. They were very, very impressed that we sent them a gift and they have given us the negative one thing. Yay.

That makes me happy. Although what I will say is, I mean, that episode has not been out for long. No. But I've just gone on Wikipedia and I can see that no one has added the negative and zeroth anniversary suggestions. So, just putting it out there. It's been what, 24 hours? By the time you listen to this, if it's still not up there.

Get onto it. This episode comes out two weeks after that one. So, surely in two weeks. Yeah. We also have some Any Other Business regarding the Spotify ratings of this podcast. Oh, yeah. Because we worked out in the last episode that if you don't have Apple products

or anything like that, actually the majority of people listening to this show listen on Spotify. So they've introduced a rating system where it's very easy to give a star rating to any podcast that you listen to on there. We heavily implied people should do that. And we had somewhere,

I think at the low 300s, we had sort of a low 300 amount of ratings and five stars which is great five star podcast and then we looked at a podcast of unnecessary detail just to benchmark which is your other podcast with helen arnie and steve mold yeah and they had more reviews sort of in the late 300s but they had 0.1 less of a star so they're on 4.9 stars and then we also looked up no such thing as a fish which had like five stars and over 7 000 reviews or something yeah one thing at a time we put it out there that because i think it was like 22 or somewhere around the 20 of our listeners listen on spotify it was quite a high amount So we worked out that if everyone who listens to

this on Spotify gave us a five-star rating, we could beat No Such Thing as a Fish. We could beat fish, right? Yeah. The episode went out 24 hours ago. It's not been long. It's only been out for a day. We've gone up from below 300s to, I checked just before we started recording our Any Other Business and we were on 515. Yes! Come on! We've gone up by almost 200. That's great. In the last day.

So we're beating Podcast of Unnecessary Detail. And we still have five stars. Take that, me. Yeah. That'll show me. But what I thought was interesting is that a Podcast of Unnecessary Detail has jumped up in the number of views as well. Not as much as us.

And they're still on 4.9 stars. There's been no more episodes. Huh. They've gone from like mid to late 300s to 464 ratings. Which means I think about 100 people listening to this. Interesting. Went and gave a podcast of unnecessary detail a rating after hearing this. And they're like, you can't tell us what to do.

Yeah. We will five star whatever podcast we want. So. And rightfully so. No, exactly. Exactly. And I, by all means, do not want them to bring down the rating for any podcast. I just want ours to beat them.

Are we closing in on fish? That's the important thing. Well, I mean, look, fish have gone up, but obviously not as intensely as we have. So fish is like 7.3. We're gaining on them. 7.3 thousand they have with five stars. Oh, okay. We are gaining.

I don't know why i said 7.3 thousand what a weird way to say that great way to put it that's what i would have said i'm very tempted to look at our listener statistics for each episode to work out within the first 24 hours how many do we get how fewer do you get oh and how fast it tapers off yeah and then take that and apply it to how many people gave us five stars in the first 24 hours so we could maybe guess at what point will we overtake if such things are fish yeah i would love to do that but i only just thought of it now i never thought i'd be the one to say this but instead of doing the maths i'm just gonna wait let's just see what happens all right

or you know everyone listening can go do that and we'll be fine. And then it will be two weeks. That's when it will be. Yeah, come on. Spotify, get on there. So there we go. There's a little update on that. Now, each episode, we like to thank our Patreon supporters who allow this podcast to exist through their financial support.

And so that anyone out there who isn't able to financially support us can also enjoy it as well. So in order to do that, we randomly have three names picked out by a system that Matt has in place. I think it's a spreadsheet. It's a spreadsheet. So, yeah, if you're a Patreon supporter, you may get thanked personally by us at the end of the episode. So jumping straight in today, we would like to thank Mike Bell, Elias Sodonis, Blake the

Pattern, all one word. Thank you very much. You are stone cold legends and may the force be with you or I don't know, something to tie it back to the beginning, Matt. Oh, wow. Well closed. Nice. Now it looks like I was planning it all along.

This podcast was brought to you by myself, Bec Hill, my co-host, Matt Parker, and our producer, Lauren Armstrong Carter. You can find pictures related to things we talked about in this podcast or other stuff in the show notes of this episode or on our social media accounts, which are at a problem squared on Instagram and Twitter. So Matt, as you know, after the last episode. Yes. I'm trying to slowly poison us with. My office still smells of awful artificial cheese that's been aged. See, I wasn't sure if that was your office or the fact that it never left my nostrils.

Which of these stale waterlogged crisps would you like to try? I picked the first one you made. Do you want the one with what looks like a monkey skateboarding? Yeah. Okay. Here you go. All right. So these ones are called Chi.

They're called Chi.toes, T-O-Z. So C-H-E-E-T-O-Z. And they're made in Iran. I believe these were given to me by a member of Unnecessary Detail. I think some of the audience. Not a member, but some of the audience brought them along to the next one. Everyone who comes along is a member of an evening of Unnecessary Detail. So, I mean, all the ingredients are in Arabic.

I think that is. I don't think any of us are fluent. Okay. They look better than the other ones did. The tradition we've established is whoever opens smells, other person tastes. Okay. I mean, they smell quite stale.

They smell like packing foam. They look like twisties. They're more like the Cheeto ones from the States. Bright orange. So stale. Good crunch sound, though. What's the flavor going for a snack. Bright orange. So stale. Good crunch sound, though. What's the flavor like?

Yeah, it's like styrofoam. They taste good, though. They taste good. Let me do a snap in the mic so you can... Ready? That's not bad. I'm not eating another one. Considering they've been in there since I did that

an evening of unnecessary detail about 2018. 2018? These are four years old. When did they go off? Let's have a look. When's their use by date? I mean, according to tradition, we should have bought them a clock. Call back.