A Problem Squared - 012 = Series of Teeth and Queries of Cheese

Episode Date: October 31, 2020

To celebrate 12 episodes we're collecting some data! If you can, fill in the listener survey: http://thatsurvey.ilikeit.aproblemsquared.com Does this cheese look like it has 41% less packaging? https://www.dropbox.com/s/kuj16lmkgrx8w7i/41cheese.jpeg?dl=0 Here's the wikipedia page we used to make sense of shark teeth: https://en.wikipedia.org/wiki/Shark_tooth#/media/File:How_to_count_shark_teeth.png The Weaire-Phelan structure: https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure What a truncated octahedron looks like: https://en.wikipedia.org/wiki/Truncated_octahedron And a spinning one!  https://upload.wikimedia.org/wikipedia/commons/7/7c/Truncatedoctahedron.gif MATT'S CALCULATIONS A 140mm × 60mm × 46mm block of cheese has a volume of 386.4 cm^3 and a surface area of 352 cm^2. To give a 41% reduction the original needs to have the same volume of cheese but an area of 596.6 cm^2. Matt assumed there are 4,200,000 current infections and each person has 400,000 virus particles per mL across 2L of fluid. If the SARS-CoV-2 particle has a diameter of 150nm and stacks like spheres with 74% efficiency: those 3,360,000,000,000,000 current virus particles would fill only 8mL, aka "about a teaspoon".

Welcome to A Problem Squared, a podcast dedicated to solving problems you didn't know you had. My name is Bec Hill, although seeing as this episode will be coming out around Halloween, well, let's just say I put the chill in be chill. And I am joined as ever by the co-host, the man who puts the R into Matt Parker. It's Matt Parker. Oh, thank goodness. I certainly was. I hadn't planned a spooky name version of my name, but you were there. You're taking your intro duty seriously. That's excellent work. I know.

Look at me sounding like a pro. Yeah, nailed it. All right, bye. In today's episode, we will be solving problems such as how many shark's teeth can you pull? What is the total volume of coronavirus? Does wearing a jacket in hot weather make you cooler at any point? What's the most efficient way to package cheese? And what should we put on our Christmas cards?

It's going to be a pun. Spoiler. Puns are spoiling. Yeah. Matt, it's been a month since our last episode. What have you been up to? Well, continuing to be locked down, which has been great fun. That sounds like a line from a bad sitcom where they're talking about how's your marriage going? Yeah, the old ball in lockdown.

Yeah, well, no, I said that and then immediately thought, well, actually, we are still frequenting the Step and Conquer, which is what we've named the distant end of our backyard in an attempt to feel like we're going to the pub when we just walk to the other side of the garden with a drink. And we've converted it into winter mode. So like a lot of pubs, we bought the cheapest possible kind of, I think it's advertised as a marquee, but it's basically just a kind of shade frame. Yeah. Which is about 2.4 meters in each direction and comes with the least reassuring ropes to tie it down. It's like that very plasticky, nylon-y, but almost like thread. So I'm amazed it's still there. I think it's going to get blown away at a moment's notice. So we've installed that so we can still get outside even when it's raining.

And we made a fire pit. Oh, I mean, arguably anyone can make a fire pit, but yours sounds official. Well, no, we took the legs off the barbecue. Yeah, but it's not like you dug a hole and put some fire in it. No, that's true. That is true. That is true.

We didn't just dig a pit. See, the problem is I get caught. I've always did a fire pit for a long time, but I get caught in this loop where I go, go oh i'll just get some kind of quick and easy fire pit and i'm like oh actually you can get some cool like geometric looking ones and they're not that expensive because they're like you know like hexagons and stuff joined together yeah and hexagons are universally known as the cheapest shapes they are they are they are the bargain basement of shapes yeah that's why 50ps are so like is that is a 50p hexagon i don't actually know i think it's a heptagon yeah i think you

might have to um yeah they see exactly they're splashed out on that they did and the pound coin that's like 12 sides or one or one um but so think, well, what if I could get some pentagons in there, maybe some triangles, squares, and I get really excited. First of all, I think about more expensive geometric fireplaces I can get off the shelf and then off my pit. And then I think, wait a minute, I know geometry. I could design one. I could find a bespoke fire pit welding person yeah to make and i could pick like an

archimedean solid and which is which is a whole family of shakes pick a nice one of those yeah just to have the bottom bit and then i go oh that's gonna take forever i know i'll just get a cheap fire pit in the meantime and then the loop continues and i just endlessly go around this cycle yeah so to shortcut that loop i just took the legs off the barbecue and declared it a fire and buried that's is it low or is it just sitting on the ground it's low but not touching the ground so the way the legs detach it's like half an oil drum what was? It's just hovering. It's just hovering. The legs are touching the ground. So it's like a couple centimetres off the ground.

Yeah. So it's not destroying the grass. And we took the grill out and just fill it with wood. And there you are. This is fascinating. It's this sort of content that our listeners keep returning for. Well, what have you been up to, Bec? How's your...

I'm very aware that I've been talking about making do in the garden, but I'm also very privileged to have a garden that I can step outside and make do in. I'm very excited about your crappy fire pit because I have a bunch of stuff. You sound it. Well, I have old receipts and things that I keep meaning to shred like or just like old files and stuff because obviously I did loads of cleaning during lockdown this year so I've got like bags and bags of stuff that I keep meaning to shred

but I don't want to buy a shredder because my flat's so small I'd have like nowhere I could really easily store it and I've got one of those handheld ones. It would be a decent percentage of the volume of your flat. Yeah, yeah. So I keep saying to myself that I want to take it somewhere and burn it. So this is perfect. When the restriction is a little easier, I'm going to come around with all of my files and burn them in your backyard.

Well, whenever it is allowed, you can come and destroy whatever. All that evidence. Paperwork, evidence, etc. Anything incriminating or you just want to get rid of it comprehensively, bring it around. But apart from evidence disposal, how has your month been?

It's been good. I've got a pet now. Oh, yes. I've got a little pet hamster and he's so, so cute. You sent me a photo. Yeah. His name is Pudding. He's only a couple months old.

And yeah, he's real sweet. I'm a bit obsessed with him. Been taking loads of photos. So if anyone follows my personal socials, you can check them out. I'm Beck Hill Comedian. Has Pudding learnt any tricks yet? No, although I did put a tiny easel in his playpen with a little, like a tiny notepad, so it looked like he was doing a flip chart gig,

which was very, very cute. Yeah. So I'd like to do more things like that if he's not annoyed at me. But it did lead me to think of something because I was looking at, you know, how much space his cage needs and things that, you know, I want to be a good, responsible pet owner. And I also know that a lot of people who listen to this have children who are interested in problem solving of many kinds. So I wanted to propose a very small segment that I would like to occasionally add into future episodes, including this one, which I'm going to entitle A Pudding Squared.

A Pudding Squared? Yeah, where I would like to pose simple hamster problems for people's children to solve. I love it. It's a great segment. Yeah. So the problem for this one is going to be the fact that hamsters can run up to around nine kilometers a night in the wild. They're pretty energetic creatures.

And Pudding's little is uh 14 centimeters in diameter so i would like to know how many rotations pudding has to run of his little wheel to get the amount of exercise he would normally get if he was running nine kilometers a day How many revolutions to get nine kilometers of exercise on a 14 centimeter diameter wheel? Yeah. That's a great pudding problem. Yeah. So if anyone's got any answers for a pudding squared, hit us up on social media. We are at a problem squared on Twitter and Instagram. And yeah, I guess the first answer that we get, or if there's any extra details that kids want to add, I will give them a shout out on the next episode. Excellent.

Our first problem was sent in on Twitter by Sam Miller, and they say, sharks constantly grow new teeth. Just a statement so far. What is the fastest rate you can pull their teeth out? Wow, that got aggressive fast. So that they always have

at least one tooth. And then they've clarified assuming one tooth at a time. I assume that's the removal process. So, Beck, you looked into Sam's problem of how fast can they harvest shark teeth? Yeah, and I took this upon myself to answer because as you and some of our listeners know, I have a slight obsession with teeth. I just think they're really... Yes, you have an ongoing tooth fairy project. Yeah. Well,

I might be half remembering because I'm no biologist, but isn't it the case that a lot of animals, once you run out of teeth, because they wear down, they get, you know, go rotten or whatever. It's just like, well, I'm out of teeth. That's it. They just lie down and then just. I'm dead. I'm dead.

Yeah. It's like, that's the limiting factor For a lot of creatures, I believe Is once your teeth run out You're like, well, that's the end of my run Otherwise they just have to hope Hope that they find some naturally occurring soup Yeah, exactly I could only eat rotten fruit

Oh well, you know I bet they wish they could access all those plastic straws now, eh? Yeah, exactly Toothless animal deaths have skyrocketed Since we access all those plastic straws now, eh? Yeah, exactly. Toothless animal deaths have skyrocketed since we got rid of plastic straws. So I'm amazed more creatures haven't evolved to regenerate teeth, but the shark is definitely the poster animal for doing that. Yeah, and so I wanted to solve this one.

And you, in all honesty, I actually did have to ask for your help because I got confused because they have some fun shark facts. First of all, the number of teeth a shark has and the rate at which they grow their teeth changes depending not just on the species, but on the individual animals. So there isn't really any definitive answer to it. The most sort of average answer I could find across my research was that a great white shark has about 300 teeth and about 48, let's say 50. Let's call it 50. Well, they said around, and if you've got 300, it's just easier if you've got multiple rows. So about 50 teeth per row. And because they were saying that they can have like around five rows, but I figure if they got 300, then let's say that it's six. See, this is what I mean.

Like the information out there is so already this problem actually became far more complicated than I thought it was because not even marine biologists are entirely 100% sure what the answer is in terms of exactly how many teeth they have. Yeah, when Sam sent this problem in, I thought, okay, it looks like it's got all the dressing of a nice, simple, open and shut mathematical problem. But I know as soon as there's biology it's a mess no one can agree this is somewhere between this and a much different value and so i was very pleased when you took this one on yeah just because i was too naive to understand what i was taking on what i

offered to the another fun fact is that the way that the the teeth sort of come in is that you've got the front ones and they're often referred to as a series. So what we would call a row, they would say a series of teeth. So I have one series of teeth in my mouth. Yeah, whereas sharks get six series in a movie. Actually, several movies if they're Jaws. Yeah, but the first one was superior. Yeah. So the front row, the front series,

what will happen is as teeth break or fall out, the next row will kind of rotate on their sort of, I guess you would say the root. They don't actually have roots, which is why they fall out so quickly. But they'll sort of rotate and then, so they'll come up sideways and then go and then like clock in and come in so that they're in that straight line that we're used to seeing in pictures and stuff um and then yeah they just sort of come in again and again like a conveyor belt

um if you can i would recommend you google um shark teeth and uh or you know the way that the series or the rose works because there's some really fascinating pictures of their jaw lines and the way that they would come in oh it's disgusting and amazing so i realized that this isn't as simple as like it would be with us where it's like, oh, the tooth comes out and it comes back in from beneath because these babies kind of like roll on in like on a conveyor belt. So there's several waves of them coming in one after the other. Yeah, yeah, exactly. And so you and I actually had a discussion about it to see the best way we could answer this problem. agreed that in order for it to be as clear as possible the best way to do this would be to only take out teeth from the front series yeah so that it would be replaced by the teeth behind them

because the shark can partially chew with the other series yes all the the reserve teeth on the way out are already pulling their weight. Yeah. But there's no point preemptively harvesting them until they're ripe, until they're ready. Yeah, ripe. I don't know. There's so many questions about shark husbandry I don't want to get into here.

You've got to squish them to see if they're firm enough. Yeah. I don't want to get into here. You've got to squish them to see if they're firm enough. But we're only going to take from the outermost layer of teeth. Yeah, yeah. So the other thing that marine biologists don't seem to be able to fully agree on is how long it takes for a tooth to be replaced because they move at different rates in different sharks.

It's probably really difficult to routinely check inside a shark's mouth to work out how Do you know what? They just don't check their schedules enough. They never show up for their dentist appointments. Like, where do you send the letter to remind them? Just throw it into the ocean. Yeah, just throw it. It's what all those bottles are doing.

They're just dentist checkup reminders just floating about. doing they're just they're just dentists check up reminders just floating about um and so the best answer i could find was that it can take anywhere between a day and several weeks that's a big range yeah it's a big range biologists sort your stuff out so i thought okay if it takes a week for that tooth behind to come in and replace the one that you've taken out, then you want to work out how many teeth you can pull in a week from that front row. Ah, yep. Yeah. I think you might want to have gone for 49 in a series over.

Ah, yeah. Well, you know what? What I read was 48 and then I took it up to 50 to fit in with 300. But I'm happy to say 49. Yeah, let's split that difference. Yeah, let's do that. Let's do that. So you're going 49 divided by seven days in a week,

which means that you can then pull seven teeth a day. Yep. But to make it even more succinct for lovely Sam, I'm going to say so you've got 24 hours in a day divided by seven so that gives you 3.4 so you know you can every sort of roughly every three hours and a bit you could pull the two every two hours during working hours maybe yeah if you want to do it what what are business hours it's uh eight hours isn't it sharking hour it's it starts just before sunset yeah i don't know yeah well because if you're

going to do business hours that's like eight hours for a working day you know if what you could do is you could pull a tooth an hour and have a yeah take lunch off there you go there's your rate job done a tooth every hour every hour. One an hour. Nine to five an hour for lunch. But you've got to work weekends. Sharks, like they can't stop moving, they can't

stop working. You don't get a weekend off. Sharks are always on the go. Yeah, you're right. You sink. If you take a weekend off as a shark, you die. Yeah, exactly. Fact. So pretty sure I answered that you die. Yeah, exactly. Fact. So pretty sure I answered that, yeah? Yeah. So you can harvest a maximum of about one tooth an hour during working hours around the clock,

but you should probably take lunchtime off. Yeah. Perfect. I mean, I don't want to speak for Sam, but I think that is a problem solved. Ding. Okay, Matt, we've got a problem for you. Yep. This one came in via email.

It says, hi, love your podcast, which is lovely. Thank you. I have a question. A lot of people are infected with COVID, but a virus is also very small. That made me wonder how many litres of the virus are there? I guess it can't be that much. Kind regards, Menno Duck or Duke. I'm not quite sure of the pronunciation, but I know that that is a fantastic problem.

Matt, you had a look at it. That is an excellent problem. And I, because I was kind of aware that a virus is very small. But when I double checked, it's incredibly small like nanometers across so like a human we're on kind of a um we're about a meter two meters give or take yeah and our cells are about give or take 10 000 times smaller than we are. They're one 10,000th of us, which is why we can't really see individual cells. But then bacteria are about 1,000 times smaller than our cells,

but a virus is 1,000 times smaller than bacteria. Like they're just tiny, absolutely, just vanishingly small. However, because it's biologists, when I looked it up, they didn't have a very good answer. It's like, oh, between, probably between 60 nanometers and maybe 150 nanometers. So the estimations are all over the place. Starting to sound like shark teeth. It really are. Biologists, sort yourself out.

Come on. Just pick an answer. Pick an answer. Biologists, sort yourself out. Come on. Just pick an answer. Pick an answer. I want that on a t-shirt. Biologists, pick an answer. Just pick one. So, you know, you see these graphics of the coronavirus. It's like a big sphere with little pointy bits sticking out.

that is only that whole spherical blob i have estimated to be about 150 nanometers across which is like around the wavelength of light like that's insane wow super super small that is small i was like okay so meno has got a good point they are small, but there's also lots of them. So now I had to estimate how many coronavirus particles there are in the world. Yeah, because if you catch it, you might have more than one particle in your system. Hey, it's not going to be just one. No, it won't just be one, you know, wrecking the joint. And they replicate. It's not like the tom cruise of viruses it's not like gonna sneak in and like dual spy havoc it's like an army yeah yeah it's like it's

like tom cruise in oblivion there's gonna be a lot of them and so and it's about the same size it's about the same size they're smaller than you expect so i looked up research into how many particles of coronavirus you will have if you're infected because also you can't get infected just from like one or two like there's a certain viral load at which the infection will take root and then you've got your viral load of how many you end up with and so i found a study in the lancet which is like this um the lancet is like a medical research publication place and obviously as soon as the pandemic started research kicked in to try and understand it they took their samples from people who had been admitted to hospital with coronavirus and i believe the severity of the covid symptoms does depend on

your infection level so because of the selection bias by only sampling people who had to be hospitalized this is probably an overestimate but it's fine they estimate that there are and they measure it in a log scale 10 to the power of 5.6 particles per milliliter on average which is roughly if you turn it into normal numbers 400 000 viruses per milliliter of fluid wow now i couldn't get a straight answer on what that fluid is i assume it's like initially this whatever they swab that's the density they're seeing and it's not like the coronavirus is evenly distributed throughout your body it's like a lung infection and so it's in a certain type of lung stuff like saliva or something yeah and well there's certain types of lung stuff. Like saliva or something. Yeah. And well, there's certain types of cells in your lung. Yeah, exactly.

So, and how much of that have you got? I tried every sorts of weird Google search and I could not find out the total amount. I'd love to see your search history. The amount of bodily fluid someone has. It's not, it's not pretty. No. someone has it's not it's not pretty no so i ended up estimating two liters as a kind of upper bound i mean you've only got five or six liters of blood so two liters probably too much but i thought i would go big and then we know that's probably the maximum amount of

coronavirus in the world now i need to know how many people were infected. There's on the order of 300,000 new cases per day worldwide. Again, big old estimate. The average infection length, who knows? But I've guessed about two weeks on average from the research I saw, knows but i've guessed about two weeks on average from the research i saw which means at any point in time there's just over four million people currently infected and i'm ignoring any reservoir of the virus in animals i'm just looking at human coronavirus sars-coV-2, whatever it is, that's currently in humans. And so I then multiplied the 4.2 million current people by their two litres of fluid each times 400,000-ish viruses per millilitre.

liter and i got the total number of current virus cells which is hang on let me look um thousands millions billions trillions which is roughly 3.3 quadrillion virus cells so 3 times 10 to the 15 cells at this point i had a choice am i just going to work out their total volume based on their spherical size? Or am I going to worry about how you'd arrange those spherical cells in the container? Is it a Ferrero Rocher problem? So, I went with Ferrero Rocher I'd assume they were stacked as tiny spheres. Oh, Matt, you're spoiling us. Exactly. In which case, as previously discussed,

the best you can do stacking spheres is to fill up 74% of the space. And so I factored that in, and then I turned the handle on all the calculations, factored that in and then i i turned the handle on all the calculations and i can tell you how many liters of virus cells are currently in humans at the moment and it's less than a liter wow it's roughly eight milliliters oh my gosh that's so little insane yeah isn't that like a teaspoon about a teaspoon and even if i tweak the numbers because I was trying to go for an upper estimate, but if I make the cell... So I've already got the maximum possible size cell for big volume. I've taken a biased higher value for the number per milliliter in a human because it's the hospital measurement.

I've upper estimated how much fluid per human. And I could maybe be a bit more generous with the number of cases but no matter how you tweak the numbers you'll never get more than a shot glass worth of coronavirus and by my estimations it's it's about a teaspoon i've just looked it up a teaspoon is six is it roughly six mils so yeah it's the opposite of a spoonful of medicine so all the all the chaos in the world the pandemic destroying lives ruining livelihoods is down to a teaspoon's worth of trouble like that like you could hold a shot glass with all the coronavirus i reckon you can chuck in the animal coronavirus and other types of coronavirus as well and all of it would fit in a shot glass so like so many life problems it starts with a shot

shot of the shock so meno duke um there you go it's causing chaos but you're right a virus particle is incredibly small because it borrows the machinery for reproduction from other cells. It doesn't have any of that. All a virus is, is like pure data to get other things to do its work. It's literally like a computer virus. It's just the code to wreck and commandeer other cells. And so all that code is just DNA or RNA. It doesn't need much space.

Hey, Mel, there you go. It's like a bad idea. It is. It's just a bad idea. An infectious bad idea. Matt, I'm very impressed, if not horrified. But I'm happy with saying that that is definitely a problem solved. Excellent.

One footnote I'd like to clarify, though, is the virus particles are incredibly super, super small, but it's still worth wearing a mask, even though a mask, in theory, might struggle to stop things that small. The medical ones, the N95s, will probably stop individual virus particles on that scale, I think. Virus particles on that scale, I think. However, when the particles are in the air, they're on water droplets from someone coughing or breathing or singing or whatever. And so the droplets that carry the particles are substantially bigger.

So any kind of mask is still useful, even though it wouldn't stop lone virus particles. That's not how they travel. They travel in teams, in droplets, and that will be stopped. They move in herds. They move in herds. This problem comes in from Mike Berman, one of our Patreon supporters. They've got the usual, Matt and Bec, wonderful podcast, blah, blah, blah.

Here we go. I used to think of big fluffy jackets as being inherently warm, but of course they're really just insulation against the cold. Oh, I guess they're not, yeah, they're not innately warm either. They're just insulating you, right? Okay. How hot does it have to be outside before wearing a jacket will help you stay cool does it depend on how long one spends in the heat wow okay mike's question which i'm going to paraphrase as we're going to need a bigger coat okay beck what have you got first of all i want to tell everyone that when they say they like our podcast i don't go blah blah blah okay i love them all yep yep yeah that is true emailing us and be like i don't care for your podcast straight to business no yeah

okay that's right no i should flag up that's a very good point beck is in charge of appreciating that so thank you beck um yeah i offered to take this one because, yet again, I thought, yeah, that's a pretty simple and obvious answer. A doi, I'll do it. It was a good run this month, a good run. I retract my doi because the way that my reasoning was, was that, yeah, basically as soon as, because your core body temperature is around 37 degrees. Celsius. Some people are slightly hotter or colder than others generally. I come in at like 36 point something.

So I actually run a little bit cold. But yeah, on average 37 degrees Celsius is the core body temperature. Well, yeah, your body temperature. Celsius is the core body temperature. Well, yeah, your body temperature. And I figured, okay, so yeah, as soon as it gets to over 37 degrees, technically your coat should, you know, act as an insulator against that heat and you should remain at 37 degrees. You see, Bec, I had exactly the same thought.

I was like, it's going to insulate you. And then I was like, wait a minute, this involves biology. And you know what I think the biologists haven't done? Sorted out a simple answer, that's why. It's almost like biology is like life in that there's no definable terms and we should just accept that. Biology is literally like life. So the thing is, because the body doesn't, like when you've got a thermos, that acts as an insulator. You know those hot and cold thermoses.

Sometimes they keep the contents hot. Sometimes they keep the contents cold. But the reason that is, is because, you know, the inside sort of acts like a vacuum. But the reason that is, is because, you know, the inside sort of acts like a vacuum. So it's stopping heat from escaping or stopping coolness from escaping. Well, it's a barrier to stop heat flow. So if heat can't flow out, heat can't flow in. Yeah, if you want to put it in tentacle terms.

If you want to say it correctly, then yeah, I guess. And so the problem with that, of course, is that the human body is not a vacuum. We generate heat. So the heat is constantly coming from within us and it disperses off of our skin. And depending on what conditions our bodies are in, they're actually quite good at regulating and making sure that our core temperature stays the same so that everything can function.

And the way it does that, of course, is by obviously with our body hair and the way that our pores act with goosebumps and stuff, we can do that to try and capture and trap heat. And then when we're overheated, we sweat. And then when we're overheated, we sweat. And because water acts as a better conductor than just air, it means that we can disperse the heat from our bodies much more quickly and cool down. That means as soon as you put a layer on top of your body when you're hot, you're stopping the system your body has in place to get rid of that heat. Ah, so as well as we're constantly producing heat, that's a problem.

Just by wrapping us in an insulative layer, we're altering how the body regulates its temperature. Correct. That's going to be messy. Yeah, which is why sometimes you know you'll notice uh in winter you'll be outside it'll be freezing outside but you'll still feel really sweaty and that's because your body doesn't realize that it's really really cold outside it just thinks oh my face is cold but the rest of me is really hot so it starts sweating because it thinks it's overheated so we're sort of starting to to mess with that it does mean that you're making yourself hotter by putting on a jacket that said then i realized of course if it's hotter outside

then does it really matter if your body can sweat or not and then of course it depends on the type of heat as well if it's a humid heat your sweat's not going to do as much. Yeah, that's when it started to get really tricky. So then I started to think, oh, yeah, so wearing a jacket wouldn't help. It would only serve to make you hotter. It's never going to make you, you know, you're constantly going to get hotter. You're never going to be cooler. And then I started to confuse myself because I thought about firefighters, right?

Because firefighters wear those massive jackets and they don't get burnt. And then I was like, it must keep you cooler at some point. Otherwise, firefighters would do their job naked. But then that would also explain why they have those calendars. So then I got really, really confused as to how it all works and why. But really, really confused as to how it all works and why. So I reached out to Rowan Francis, who is Medlife Crisis. Great YouTube channel if people want to check it out.

Yeah. Yeah. They're fantastic. And also, as you know, it is a very difficult time to get hold of anyone. Yeah, that's what I told Lucy that you were investigating this and you were going to contact some of our medical buddies. And Lucy's like, yeah, it's a real good time to approach medical people with some kind of frivolous. Like, I know you're saving lives, but I've got a question about firefighters and their jackets. Yeah.

But Ron was fantastic and replied back really, really quickly. Yeah. And so basically they said um it is a complicated question they're not entirely sure if they have the skill set because it's not just a simple medical question we're also talking temperatures and physics and the way that all that works because i suspect it's a function of how long you're in the heat as well like it's not going to be simple at all correct yeah it's not it's not easy at all so they actually came back and said that they asked a firefighter and um and the firefighter

said that the suit allows them to literally walk through flames due to the material it's made from and avoid burns but only for a short while and they have they have to retreat to cool slash avoid smoke inhalation etc so it a jacket made of the right material could provide some barrier from direct heat radiation but obviously it depends on the materials and the length of time you're there and the type of atmosphere that you're in as well. So then I realized this is far more complicated than I ever considered. So I've decided that in order to answer this question, we're going to have to set some rules. I think firstly, in the original question, they talked about wearing a nice warm jacket. So I think I'm going to have to choose a very typical winter jacket, the sort that we wear in UK around this time of year.

They specify fluffy, big fluffy jacket. Well, I do actually have a big fluffy jacket. You've seen it, Matt. Not surprised at all. Yeah. Yeah. So it's sort of a faux fur. So I have access to a fluffy jacket. Also, I've got a friend who has a small sauna.

It is about the size of a cupboard and they're not using it at the moment. So it's technically been quarantined because no one's used it for a while. That's great. So I could use the sauna and my fluffy jacket to check this, but we need to work out how we're going to put this experiment together in order to come up with exactly what question we're trying to answer and whether I'm testing correctly. It's not a hot stone sauna.

It's an infrared sauna. Oh, okay. But you can set the temperature, I guess, so you know. Yeah. Because I suspect, and this is just me guessing, if you bake all the biological complications together, there'll be some combination of the temperature difference between body temperature and the ambient temperature

and a critical point when they're the same and if ambient's higher if you look at the insulating properties of the fluffy jacket there'll be some over time delay or impact of keeping you cooler than you would have been if you weren't wearing the jacket but you're going to be fighting a losing battle pretty quickly because your body's going to start you know overheating or doing something yeah i don't know you're right it's complicated i can't even i can't even say why i think it's complicated in a well also we need to agree i mean i'm assuming it's a case of putting the jacket on when you're in a sort of average room

temperature and then entering into a hotter environment. Because if you were to put on the jacket while you're in the hot environment and your skin is already hot, then surely that would make a difference. So what we're going to do, because I'm not going to answer this problem this month round, is that if anyone out there can definitely help or has a suggestion of how we should do this experiment in order to test whatever theories we've got going, please let us know. We're at A Problem Squared on Twitter or Instagram, or you can send it to me via email. It's beck at aproblemsquared.com. If you've got a solid idea of how we should do this experiment, let me know. And if I get a chance, I will try and not only do the experiment and carry it out so that we could solve this problem in the next episode,

but I might even try and turn it into a little YouTube video because, hey, watching me sit in a sauna with a fluffy jacket on video might be pretty entertaining. So for now, that is a problem unsolved. No, you don't get any percentage of a ding until you come back with some results. Mm-hmm. This problem comes in via Instagram from Adam L., who says, Hello, Matt and Beck. I recently bought cheese and Tesco's in a more unconventional shape. Tesco claims that this squarer packaging uses 40% less packaging than its previous

rectangular block. They said, unfortunately, they did forget to measure a whole block of previous cheese for comparisons, but they have sent measurements of this new block of cheese. Then they go on to say firstly is tesco correct in saying there's 40 reduction in plastic what would be the most efficient packing solution that minimizes wasted space and cuts down on plastic and which products would have the greatest improvement in reducing packaging by just changing its shape. Thanks for your help. Sorry, Matt, I figured that this was a nice complex mathsy one for you. What have you got? Yeah, I'm glad Adam sent this in

because they also included a photograph of the new block of cheese. And I zoomed in on it, and we'll put this out on our various social medias and link it below if you want to check it out. They claim it's a 41% reduction in packaging because of the new shape. And so I figured I would work out the area of the new packaging,

which Adam sent the photo in, which I calculated. So they said it's 140 millimeters by 60 by 46. I then calculated that surface area. It's 352 square centimeters of packaging plastic. And because they say it's a 41% reduction on the old packaging, I was able to work out that the old packaging must have been about 597. We could potentially call it 600 square centimeters and then I thought I'd try and work out what that must have looked like

given Adam didn't save their old packet and I don't think it's possible I am completely unconvinced that the old packet had so much surface area that they could cut it down by 41% to this new one I did run some numbers

you could potentially do it if the old packet that they could cut it down by 41% to this new one. I did run some numbers. You could potentially do it if the old packet was like a square, which is 158 millimeters in each direction. So about 16 centimeters, but then would only be 15 millimeters thick. So it would be like a massive flat square. And that's not a normal cheese. Just a big slice of cheese. No, exactly. It's like a single oversized cheese slice i would love that why doesn't cheese come in sheets well because

too much packaging as we've now discovered and so i was like well what if it's more rectangular so i tried to get it in the same kind of aspect ratio as the new packet, which would be like about 24 by 10 centimeters. And then it would be 16 mil thick. So it's still too thin. Like to make it bad enough that you could get a 41% reduction, it's going to be so poorly packaged. It's basically a pancake because the flatter it gets, the more packaging you're going to need. Right. So if the cheese came as like an A3 book.

Yeah, that would be the worst. The flatter. Pages of cheese. The worst it gets. So I think they must have reduced the thickness of the packaging. There's something else going on. of the packaging there's something else going on they cannot have achieved that just by changing the surface area without changing the volume of cheese so just out of the gates i question their claim but if we take that on board and adam has now asked what would be the best way to do the

packaging well the answer is cheese balls the answer is always a ball and when we were discussing this when the question came in from adam uh and you were like oh this could be interesting i'm like it's going to be a ball because a sphere is the way to minimize the total area for any amount of volume and so you could just make a cheese ball, which is nine centimeters across, and that would minimize your surface area and give you the same amount of cheese as the block that Adam sent the photo in for. But how would you wrap it? Because mozzarella balls tend to come in rectangular packets. Well, you've hit the nail on the head that it's not practical to wrap a ball. You've hit the nail on the head that it's not practical to wrap a ball. And one aspect is it's just like, how do you automate that process? Because it's a ridiculous shape and it's wasteful.

Like, because you can't take a flat bit of packaging and then wrap it around a ball without a bunch of waste. Yeah. As you would have seen from Frere Rocher wrappers. Bingo. And speaking of Frere Rocher, what if you want to stack a bunch of them on the shelf or in transport in a box spheres don't stack well been through this many times it's a bit like a teaspoon of coronavirus it's not well packed and the old ferrero rocher problem and so i thought well hang on what you want to do is you want a shape which minimizes the surface area for a certain volume of cheese,

but yet stacks without any gaps. And that is actually something called the Calvin problem. And Lord Calvin... As in Calvin and Hobbes? As in Calvin and Hobbes. No, as in the person after which we named the temperature scale Calvin. So Lord Calvin or William Thompson, who became Baron Calvin later in life. So actually, William, Bill, Bill Thompson is their actual name.

They, in 1887, said the best shape, not specifically for wrapping cheese, just if you want to minimize area, but it still stacks perfectly with no gaps. They claimed it was a truncated octahedron. That was their solution to the problem. They couldn't prove it. They just said, pretty sure it's this shape. Prove me wrong. No one could prove them wrong. What is a truncated octahedron? What would that look like? It's a shape. It's got 14 faces. And it looks a little bit like a classic football.

Okay. You know classic football is like hexagons and pentagons. It's the same deal, but only 14 panels, and it's hexagons and squares. Okay. So it's six squares and eight hexagons turn into a ball. Okay. And it's an amazing shape because it stacks perfectly with itself. Very clever shape. So Lord Calvin in the late 19th century said, it's this shape.

Prove me wrong. No one could until 1993. Two researchers in Dublin came up with a new structure. It's called the Weary Felon Structure. It uses two different shapes. One of them is very similar. It's got 14 faces, slightly different shape. And one's got 12 faces. It's like a distorted dodecahedron. And combined, they use 0.3% less packaging or surface area for the same volume and it's still unknown if there's an even better solution out there so it's an open problem what's the best way to wrap cheese well apparently tesco have found it well they've gone for the cuboid

so this actually brings us back to kind of what you were saying in terms of practicalities all these other shapes are terrible for stacking inside a cuboid box which is how we transport things so actually the correct answer is gonna be pretty much what tesco came up with because you want to be able to stack them into a standard size box that fits on a standard pallet which potentially fits in a standard lorry or however you're shipping it around the country because it's not just the packaging that you see on the shelf there's the entire supply chain i actually spoke to i don't think i'm allowed to name the company or industry but i was talking to an engineer who designed major electrical devices and the engineers who work on the actual electrical

device also do the packaging i thought that'd be a whole separate department but it's the same engineers do the packaging because they want to make sure that the packaging fits perfectly a certain number of times inside a shipping container. Because then it's more efficient to ship around the world from the factory in China. You can fit more of them or you're not wasting space when you put them in the giant shipping container. And so that even impacts the design of the product. So people will have these products around and their actual dimensions. Matt, stop being so coy.

We all know that you're talking about electric cheese. Sorry, electric cheese, cheese graters. So I just don't want to get them in. Electric cheese. I have a lot of chats with people I know in the industry and I couldn't get in touch with them to verify if I'm allowed to say where they work. So I'm not going to.

But the same thing applies to any products around the house. If you look at things that have been designed, they've not just been designed for your current use, be it cheese, be it an electrical device. They've also been designed to change the shape of their packaging, to change the way that they're shipped. have been designed to change the shape of their packaging to change the way that they're shipped and actually if it's environmental concerns you could probably do more for the environment by having less carbon to ship them because of the shape of the packaging than just reducing the amount of plastic in the packaging and so it's actually just saying we've changed the shape to reduce the packaging may not be net better for the environment. I suspect that the block of cheese is actually shaped that way because of logistical

reasons, but that's just hard to explain on a block of cheese in Tesco. So they've gone with the far more easy, there's less plastic. But I think actually that's not the biggest deal. So do you think that instead of it being individual packaging that has lost that 40% could it be that they've changed the shape of the cheese to make it easier to ship so that they're actually saving that packaging in terms of transport because they can fit more of the blocks of cheese

into each shipping container? You're right. That could be it. Maybe the 41% claim factors in the whole supply chain. So I can't say either way, but I think you're probably correct. At a minimum, I would say with all these things, the easy to explain marketable version is often not the biggest factor. And in this case, i think it's the behind

the scenes aspects be it packaging be it carbon emissions and in transport all these things we probably have a bigger impact on the environment from the cheese you know what the one thing we didn't do oh to try and solve this problem buy any cheese contact tesco oh that feels like the obvious step. You know, I know some people in the cheese biz as well. I should drop them a line. Right. So email Tesco.

You know what? I might just go in and measure a bunch of cheese. Measure. I'm making a list. Cheese. Get some baby bells in. Yeah, you're right. What are they?

Cylinders? Oblate spheroids? Who knows? Wheels. I should find out. I think we should all ask Tesco. Everyone. Yeah.

Make them back up their 41%, right? Because they're going to ignore me. But if a whole bunch of people take them to task on, hey, your packaging, what do you mean by 41%? Yeah. We'll see if they get back to any of us. Matt, we need a hashtag. Cheese cover up. Oh, that's good because it's a cover get back to any of us. Matt, we need a hashtag. Cheese cover-up. Oh, that's good because it's a cover. It's a cover-up.

It's what's covering it. It's covering the cheese. Hashtag cheese cover-up. That's good. So we want to know what's the cover-up of the covering up of the cheese. Just thought the more I explain it, the funnier that's going to get. So everyone on Twitter, let's put some pressure on Tesco. Even if you don't live in the UK or somewhere where there are no Tesco's, it's a big supermarket,

join in. It'd be fun. We're going to get to the bottom of where that 41, you can't just throw a percentage on a block of cheese and not back it up. I want to see that working. Yeah. So if everyone listening to this, who's on Twitter goes and tweets Tesco, it's at tesco on twitter uh feel free to copy us in a problem squared so that we can see if they reply and ask them to explain where that 41% packaging has gone why is that why is it less 40 if you if you near the cheese take a photo of it i'll definitely visit a tesco and get a disgruntled photo holding the block of cheese like like a kind of photo from a uh community newspaper someone looking disgruntled holding a product it's gold yes hashtag cheese cover-up i have one last problem for you beck

which i need some help with but I might get your actual solution off podcast for reasons that will become obvious in a moment I so this podcast is funded by our fantastic patreon supporters huge thanks yeah who supports us on patreon and we hit our target so this is episode 012 the 12th episode we are 11 months into this and we originally said if we have a hundred or more patreon supporters by the end of the first year we'll keep making a problem squared yeah which has been achieved yay thank you so much everyone who's joined in we got 136 now amazing At the time of recording. So we hugely appreciate that. And much of my life is funded by people thinking there should be more maths videos in the world. There should be more podcasts like this.

And that's the only reason I can do what I do. And so every year for the previous four years, I've sent Christmas cards to my Patreon supporters. And I know people live in different cultures and different backgrounds. Everyone celebrates the same Christmas. But I send some kind of festive card, often Christmas themed, to everyone. And I email it to some supporters. I post physical ones that I've signed to others just to really say thanks for what they've done. And it's always some kind of mathsy pun.

Oh, yeah. And so this is my problem for you to help me with back i don't want you to give the answer right now because the card's a bit of a surprise and so i don't want everyone to know what you know mathsy christmas card they're going to get if they're one of my patreon supporters but i'm going to show you the last four years worth okay i'm i'm dumping them all into uh whatsapp chat there they are okay so i've whatsapped oh they've appeared in chronological order too that's my file naming convention and so they're all mass punish so the first one i made a christmas tree based on a Sapinski triangle. And it's got another fractal for the box. It's a manger gasket, I think. Yeah. And so it's a fractal Christmas tree I've made.

And at the bottom it says, may your Christmas be merry on every scale. Yeah. Because fractals are the same on every scale. Fantastic. It's great. And then the next one that looks pretty bad, that's I've made a 4D cube Christmas present.

And it says, happy hyper holidays. Yep. I mean, it's a bit of a reach, but yeah, sure. Okay. Good. Okay. That sentence could have ended much worse. The next one's got an infinite spiral of like a wreath you would have.

Only it goes on forever. Yeah. May your festivities never end. See what I've done there. That's good. Then I did another fractal. I did like a Christmas fractal, but now I feel like I've been doing a lot of fractal-based puns.

I was scraping the barrel last year with, I hope you enjoy the season's fractivities. And it's got a picture of a fractal, like an ice fractal. Yeah, it looks a little bit like a snowflake. I don't think you're laughing directly at the puns, just more... I mean, no, no, I am laughing directly at the puns. I don't think that clarification puts my mind at ease. Actually, there's two things here. First of all, if you can help me out off podcast,

I'm going to have to come up with my Christmas card for my Patreon supporters this year. Okay. Yep. Also, I thought maybe we could do the same tradition with our supporters. Yeah. Because with the primary reward of getting to 100 was we're going to keep making the podcast. But I feel like that was more of a threat that would stop, I guess. So something more of a gift we could we

should make a christmas card and we'll send it out oh yeah so we can do like an email one for the main supporters and yep maybe for our wizard supporters we'll send one in the post oh yeah we've only got two tiers that's easy yeah and actually we need to set our target for next year and so the consensus seemed to be that we should go for pi hundred supporters after the second end of the second year okay so in in layman's terms that means 314 and a bit supporters okay um or just 314 314 supporters in a year we will continue to continue to make the podcast um because then then i think we'll actually um it'll be part of our real our real work my goodness i think that sounds like a great

idea i also think if we hit 314 in the next 12 episodes we should offer more than just continuing to do this let's do another special little thing if we can hit 314 in total in the next 12 episodes let's do a live recording for everyone oh yeah oh goodness we would normally do a live live show like in a theater yeah as our jobs used to entail but i mean who knows in 12 months though but we'll do something online for everyone because i think like the supporters, they support the podcast above and beyond so that everyone gets to enjoy it. So I feel like that perk will make available to everyone. Yeah, because everyone deserves to listen

regardless of whether they can support financially or not. Just supporting with it is enough. So thank you to everyone listening and to everyone who's told their friends about it too. Yeah, and thank you so much for everyone who's been with us through 12 episodes and 11 months. We hugely appreciate everyone who listens to this.

Before we finish this episode, this is a call out for more problems, specifically more problems for me. It's really cool and exciting how much everyone listening loves maths-based stuff. But as you know, Matt, my specialty is in creative solutions in terms of... Yeah, what's a Beck problem? Yeah. Something more creative.

We start filming my CITV show next week, Makeaway Takeaway, where I get to solve solve kids problems using arts and crafts which is really exciting but i'd love to be able to extend that to our wider audience with this show as well so if anyone has any problems that they would like solved that doesn't fall under the remit of maths um yeah i mean don't stop those ones either no no do send them matt loves them he even the ones that we don't get around to reading out, he reads in his spare time. He smiles. I do, I do. How many X's in a Y?

How big can a Z be? Oh, they're great. Great questions. Yeah, he's nuts for it. But yeah, anything like that, please do send them over at A Problem Squared on Twitter or Instagram, or indeed you can pop it in an email to beckataproblemsquared.com. mattataproblemsquared.com is also available.

This podcast would be nothing without the listeners providing all of their problems. And we're curious to know, after 12 episodes, what were your favorite problems? Which bits of the show do you enjoy the most or the least? So we're going to do a survey. enjoy the most or the least. So we're going to do a survey. If you don't mind filling it out, you can go to thatsurvey.ilikeit.aproblemsquared.com. We'll link to it in the show notes and you can tell us what you enjoy the most, what you'd like to change, answer as many or as few of the questions as you want. We just want to get to know you, the listeners, so we can solve your problems better.