A Problem Squared - 131 = Microwave Conventions and Pythagoral Dimensions

Episode Date: March 30, 2026

👋 When microwaving food, should you place it in the centre, or off to the side?📐 Does the longest diagonal in a 1m sided cube in 4D square equal root 4? 🐨 And there’s some Bonus Other Business.Head to our socials to see Bec’s Photographic Exhibition of Australian Wildlife, Cubie the Moon Rover, Bec’s Poppadom Experiment,  Matt’s Pythagorean Whiteboard Action, and the first glimpse of the most confusing game of Connect 4 ever to be played. A surprisingly visual audio podcast… so it’s all happening over on BlueSky / Twitter / InstagramJoin us on Patreon for early releases and our monthly bonus podcast I’m A Wizard!If you’re already on Patreon and have a creative Wizard offer to give Bec and Matt, please comment on our pinned post!  If you want to (we’re not forcing anyone) please do leave us a review, share the podcast with a friend, or give us a rating! Please do that. It really helps. Finally, if you want even more from A Problem Squared you can connect with us and other listeners on BlueSky, Twitter, Instagram, and on Discord.

Hello and welcome to Some Problems Squared, a problem-solving podcast that has the acronym SPS, which is palindromic. And this is episode 131, which is also palindromic. I'm joined by Beck Hill, whose name is not palindromic. But if it was, it would be beckheb, bekeb, bekeb. As a performer, she tells lots of gags. That's the palindrome. brings a lot of wow to the podcast and always has something interesting on her radar. Also a palindrome.

There you are. And I'm always saying, ha! Yep, that checks out. Whereas I'm Matt Parker, mathematician who brings the stats. And I'm sometimes described as a rotator. Did you know there are surprisingly few words that are palindromic? Matt is a bit of a race car if you're a. ask me.

There you go. If I'm ha, then you're a-ha. Aha. Excellent work. And on this episode, micro, I'm looking at microwaves. I'm not going to do a palindrome.

I can't pull that out. Wow. You didn't give me any time. Short notice. I'm taking Pythagoras to higher dimensions. And there'll be some, any other. business or I don't know.

I hate today's theme. I was going to go with bonus other business just to make the acronym. Oh yeah, that makes way more sense. So Beck, how have you been? Well, right now, I don't like that you changed the name of the show to fit some problem squared. No, that's not correct.

Aw. You've really tapped into that side of me. Like one of my earliest childhood memories is. is standing on a table at kindergarten and yelling, frogs don't go lardie da-di-da-da, they go ribet. And I think that this is very much the same as when you were like, welcome to some problem squared. And I was like, no, actually, Matt.

I'm good. I'm in Adelaide. I'm in Australia. It's nice to be here. Do you remember how last time I was in Oz, I was like, everybody thinks that Australia's just full of all these animals and they're everywhere. I'm not that stereotypical.

And then at my auntie's place, then a kangaroo just let itself into the house. And so I don't know why, but I'm always amazed at just how many animals are here. Like the first day I arrived, Mom found like a massive blue-tong lizard under a bin outside, like a wheelie bin. They're huge. Well, like, you know, in terms of compared to British lizards, they're pretty big. And then I've seen two redbacks, one in the house, which mum removed, which I was grateful for. There's been cucobaras, coales, possums, loads of cockatoos, just like... Now, for the record, you're not living, like, in the middle of the bush.

I'm in the burbs, I'm in the suburbs. No, no, and like, pretty metropolitan suburbs. Like, I'm not, like, out in the sticks. I'm not on the fringes of town, so to speak. Yeah. A lot of these animals have to commute in. No, yeah. What I started doing was like actually sending, taking photos or videos and sending them to some of my friends back in the UK. And it blows their minds. And I haven't, I've just been sitting on this like, oh yeah, this is just what life is.

And not realizing that there are people who are out there who were just amazed at the amount. So I will send you both all of the photos of videos that I've been taking. You can put that up on. the socials, very cute sleepy possums and koalas in trees or loud cockatoos. I've got my favorite photo was a photo I took of a cockatoo that flew away as I was taking it. And you can really tell. What I find interesting is it's difficult trying to convince people in Australia there's more wildlife than they expect. Because everyone thinks you get there and it's like, here's a problem. People think in Australia you're going to be instantly killed by some bit of wildlife.

And you're going to try and convince people, yes, there is a lot of wildlife that you'll see, but also it's perfectly safe. Like you're not going to, you know, unless you trip over a wombat, for the most part, you're fine. And even then, you're just going to like graze it. Unless you trip over a wombat into, just sprinkled some broken glass and then some, yeah. Yeah, it's very, very rare. I know we've spoken about it before, but it is, it does. surprised me as well at just the amount. Yeah, it's just a lot of wildlife around. How about you, Matt?

I've been good. Big update is we know how we're calculating pie on the moon and we've announced the method. Hang on. You did a Kickstarter where you were raising enough money to pay the fees that are necessary in order to put a program on a moon probe. On a rover. You sold this. I think there was like 5,000 people or so. Yeah, 5,000 people. And you still hadn't actually decided on how you're going to do. No, no, hey, look. You're one of those people who's like, who goes on Dragonsden and it's like, I've got this idea. It's like, you're one of those kick starters. It's a magical box.

That says, I've invented a hat that's two meters wide and it will protect you and your loved ones from the sun. I just need this much investment. And then you get the investment. You're like, I haven't actually worked out how to make the brim stay up. Yeah, it's like one of those things where, oh, I invented a box. and you just put food waste into it and out comes perfectly edible snacks. Everyone's like, how does that work? You're like, I don't know, I'll work out the details inside the box later.

My point is I'm the visionary. I've imagined the thing. Yeah, you're that blood lady. You're the lady with the... I'm the blood lady. Yeah, the drop of blood lady. I'm like, you put a little bit of blood in the box and all the information comes out. We'll worry about the details later.

Give me money. Yeah. That's literally what I did. It's a good point. I was like, we'll put a box on the moon and we'll put data into it and then pie will come out. Don't worry about the detail. Now, there are different types of things that haven't been solved or fixed yet. There's ones where you can work out the actual solution.

And there's other ones where you can just show a solution must exist. And so I was very confident when I did my pitch for everyone to join in that I'd proven a solution must exist and we'll just find it later. Let's not get hung up on the exact details. It's doable. Because the method we wanted to use, we're not allowed to drive the rover around, which is obviously very upsetting.

But I never thought that that would be the case, but I love that you've had to check. Oh, they were very clear. I didn't have to ask that the first meeting, they said two things. One, you can't drive the rover. Two, you're not allowed to use Python. How?

We did it in C++. And that second one is specific to you. Oh yeah, they know me. I mean, the reason I was getting involved is because they know me. So they came in swinging. Now, because we're not allowed to drive the rover, but I still want the calculation of pi to depend on the moon. Otherwise, I'm just running, I've just got a server on the moon,

which is very cool, but I want the moon to impact the calculation. So we're using the sensors as a source of randomness because they've got so many decimal places. After a while, it's just noise. And so we're using the noise as such from the sensors as effectively rolling dice to give us random numbers. And I know there are a lot of ways

you can calculate pi using random numbers. And so I was like, oh, it's fine. We'll use the random numbers and we'll get pi. And the example I gave early on was you could use the random numbers to pick a random point in a square. If one random number is like your X horizontal coordinate and the other random number gives you your vertical coordinate, you're picking random points in a square and you can calculate pi by working out if those random points fit within a circle, like a dartboard in the middle of the square or not. And that will give you pi if you do enough of them.

So I was like, oh, there are ways to do it. But then a friend of mine, a guy called Colin Beveridge, was like, hey, did you know that if you pick a random point on a sphere and you compare how far away it is from an axis through that sphere, the average distance is pi on four? And I said, I don't know that. And Colin then said, well, that means you could use your random numbers to move, you know, a virtual point around. a sphere and keep track of how far it is from an axis. And if you do that for enough points on the sphere and multiply the average distance by four, you'll get pi. And I was like, that's great.

Are you telling me that you're going to create a moon rover simulator that will play on the rover that you can drive? Yes. Yeah. Yeah. Found a loophole. We're running a lunar rover. simulator driving a virtual rover wherever we want around a virtual moon and the rover sim is running

on an actual lunar rover on the surface of the moon and that's how we're going to calculate by yeah i can't see how that could possibly go wrong i'm so it's genius uh so whenever the rover has downtime because we're the lowest priority part of the mission whenever there's downtime and it's not doing anything else. Effectively when it's asleep, it dreams of being able to roam free on a virtual moon.

Oh, it's dreaming. It's so good. Yeah. It dreams about pie. So that's what we're doing. Okay, that's adorable. We'll be dreaming of pie whenever they've got some downtime. Is that its name? Kuby. I call it Kuby. It's called

Cube Rover. But let's go with Cuby. Cubie is adorable Cubie is adorable it's a cute rover yeah I would buy that plushy let me give you a picture I'm sure I've shared pictures of Cubie before I would like to see a picture of cubie

that is not a moon rover that is that is a Victorian pram you're not far off that is a Victorian pram that a child made at school yep yep Look, I'm not on the big rover.

No, it's very cute. It's very cute. And when does this Macano set come out? So we don't know when we're going to launch. It will hopefully be sometime this year, but space schedules are, you know, all over the place. I won't find out until pretty much fairly close to the date once everything's locked in. I'm hoping to go and visit QB before. they're put into the lander that will then be launched to the moon.

You can give it a little... I do think QB is very cute. And I like laying into it, but actually it's very impressive. Yeah, it's very cool. And I would still, I would genuinely buy a kit where I could make it. Like a little cuby. Yeah. Now, I know I've talked a lot about Moon Pie.

I'm very excited, calculating pie on the Moon. I mentioned the patches last time. And this is why the patches have the Moon inside the Moon, the virtual moon, being simulated. on the real moon. Ah. Very pleasing all round. It's such a great mission.

I promise to not give another moon pie update for a while. But that's where we're at. Calculating pie on the moon. It's moon all the way down to the core. First problem was sent in by Jethro, who went to the problem posing page at a problem square.com and said, hi, Matt and Beck. LTP, LTF, BBB, two-thirds of which are palindromic.

So, a long-time listener and blah, blah, palindromes, love the pod. That could be prefer the podcast. Yeah, PTP, we'll go with that. So, Jethro asks, if I'm heating something in a microwave with a turntable, is it better to put the food right in the middle or off to one side? If it is off to the side, it feels like the heating should be more consistent. Question mark? So I guess Jethro's preferring off to the side because they feel like it should be more consistent,

but they're uncertain about that. Additional details here, the food is lasagna, veggie lasagna, in a round bowl in case that matters. Jethro, all the details matter around here. So, in order to answer this question, I realized I have not. idea how microwaves work. Like absolutely none. You're perfectly suited to solve this problem. Well, I was like, what better time than now to learn? And so I'm going to start by explaining how microwaves work, knowing that you will probably fully be aware of this based on your brain and personality. Look, I learned how microarrows worked like 25 years ago.

and haven't thought about it since. Yeah. So I'm going to be rusty. But I figured there's got to be some people listening who didn't. Like I only recently realized that spaghetti westerns are called spaghetti westerns because they were made by Italians. Oh, did you know as well that tea light candles were created to keep teapots warm? No.

Like a tiny cauldron. Yeah, I used to have like a coffee pot from the 70s that would sit on a base. You put a tea light in. Yeah, that's why it's called a tea light. That's nuts. Blew my mind. Well, Jethro, I hope that's answered your question. So I'm going to do my best to explain my understanding of how microwaves work.

Go, and I'm going to do my best to not be Googling it at the same time. See, well, you're going to ask me questions, and I definitely won't know the answers to them. So in a nutshell, the microwave is a Faraday cage, which, I mean, that part I know, it's a metal box so that these microwaves, these microwaves, these microwaves, can't get out. I know that they're created by something called a magnetron, which sounds awesome. The magotron's very cool, yes. But the waves go in and they bounce all around and they're all like like bouncing off the sides and stuff. The waves start to sort of sink up in a way so then you end up with the positives at one side and the negative sides at another and then they keep jumping really, really, really fast. So you're getting a positive and negative side where alternates really,

really fast, those microwaves interact with the largely water molecules. They do also interact with the fat and other molecules in there, but mostly with the water. It excites all the water molecules. And basically because they keep trying to switch themselves so they're facing like with the positive up and the negative down or the other way around and then they keep switching like, they're turning around so much that they're moving real fast, that creates energy, which creates heat, which means that anything with, with, water in it, which is most things, that starts to heat and then it cooks, cooks everything.

So when people talk about it like cooking stuff from the inside, it's basically saying like the water molecules start moving so much that everything around it cooks. It is difficult because thicker bits of food, the microwaves find it a bit harder to get closer to the centre, like they get slowed down so you don't necessarily get stuff in the very middle, sometimes if it's a big, thick piece of something. Also, when stuff is frozen, it's not enough to excite it because it's too locked in, which is why, like, you have a defrost setting on the microwave, which sort of does things like more slowly, because it's basically trying to heat up the stuff on the outside enough to start heating up the water

molecules next to them. Basically, it starts defrosting itself because then you get some, the bits that are a little less frozen start to heat up. And then they heat up the frozen bits next to them and it thaws. And once that water thaws, then those water molecules can start moving around and then so on and so on and so on. It's also why they tell you to leave your stuff to rest for a moment because then the heat disperses and becomes more even. Does that sound correct? I feel like you hit on some good points in order. The Faraday cage.

So it turns out you can block radiation, as in like, lots. light and radio waves and microwaves. Not like nuclear radiation kind of atomic stuff, just like regular radiation. And I'm aware of the overlap. By, you don't necessarily need a solid bit of metal. As long as the holes in the metal are sufficiently smaller than the wavelength of your waves,

you can still block it. And so your Faraday cage around a microwave, the reason you can still see in through the screen is you've got loads of tiny holes in the metal in the mesh. Yeah. So a mesh is sufficient. to block radiation is the short version of that. And so if you put like your phone in a microwave,

you can't ring it because the radiation won't make it in there. Huh. Don't put the microwave on at the same time. It's also really hard to ring my phone when my phone is not with me. Yes, you can't ring it because you've no longer got your phone. That's how it works. Yeah, it's in a microwave. Yeah.

So you also can't make lunch at the same time. Now, I think what I'm being cautious about, is I believe people mix up needing the waves to jostle around your dielectric molecules with them forming standing waves within the microwave. Because if you're bouncing or just injecting a lot of electromagnetic radiation into a Faraday cage, it's going to bounce around a bunch inside and that will form standing waves. And the standing waves means you've got points where all the radiation's kind of cancelling out, so it's a low energy spot, and you've got points where

the waves are combining constructively, so you're getting high energy spots. At the top level, microwaves preferentially can transfer their energy into liquid water. And fun side fact, that's why human bodies are very good at blocking Wi-Fi, because I think if you check the frequency microwaves run at, it's about the two. 2.4 megahertz that Wi-Fi runs at. And I think that's why we put Wi-Fi there because we're not using it for anything else. And so we put it on the old microwave band. And because that's specifically the frequency at which energy can be transferred to water easily,

humans full of water are very good at blocking Wi-Fi signal. That's the least concise summary of that fun fact. I think that has ever been put to podcast. All right. Well, luckily, the specifics of that don't actually apply to Jeff Rose problem. Part of me was just checking to see if I'd done my research correct. And it's good to know that I've done enough research to have fallen into a pit that many other people have of not entirely understanding how microwaves work on a much deeper level. I don't know how big that pit is, but I got to talk about Wi-Fi and that makes you happen. Well, to get back to the problem, one I realized is that because, the radio waves sort of start sinking up and going like the way I imagined it is if you were to sort of

draw out a wavelength and imagine that sort of bouncing up and down in a microwave and so that there's points where it's hitting the bottom which is where your food would be and those points are where your food's going to get those microwaves the most it's going to start hitting up in those points the most you will have waves combining and you'll often see this drawn in one dimension like a one-way wave and there's like high points and low points in reality this is happening in three dimensions in a microwave and the moral the story is you will end up with hot spots and cold spots based on if the waves are combining constructively or destructively if they're adding or subtracting from each other you've just said i was right but it's in three dimensions instead of

two it's in three but it's not that it's able to like reach the food in those spots it's just the up and downness in the diagram is just using up and down to show intensity. It's not actually that kind of up and downness and the microwave. Yeah, yeah. Right. Okay, okay, okay. I mean, they are waves. Is it more like if you had a swarm of bees that are flying everywhere?

I think I can confidently say no, carry on. You had a swarm of bees and you're measuring. how loud the bees are. It's not actually got anything to do with how they, like how they look where they're flying is like not the point. We're actually thinking of like where they are the loudest. Beck, that's actually really good. That's top notch, top notch analogy.

Because if I did, let's say the bees are changing their noise over time, And I did a plot with a wave going up and down showing the noise level. The bees aren't necessarily flying in a wave shape. They're still going straightforward, but their level of noise is pulsing. And that's a much better analogy for electromagnetic radiation. It goes straight, but it has got a wavelength because it's pulsing how much is electro and how much is magnetic. Great.

So in conclusion, microwaves work with bees. Big whole box of bees. Can you put a lot of bees in a box? They get angry and heat up. It's like that. And it's loud. That's what makes the noise in a microwave. Okay.

It's great. I'm glad we're on the same truck. So there's been like high school experiments. This is not one that I did. So I was very excited. But I found lots of different experiments where people cover a plate. with something that would sort of show the heat reaction. I did see versions where one in the States was like,

someone was saying that when they were in school, they did it by doing like a tray of marshmallows. And so like it, you could see where it had like cooked. I was limited to what was in my mum's pantry, which was some popadums. She had all these little popadums that thankfully had already gone off. So I wasn't wasting food.

She hadn't realized that they'd expired until she, to check them and they were about seven years old. But what do you know? Still good for the microwave. So I covered the rotating tray with pop-at-arms and then filmed where it started. And also then I ended up with a sort of like pattern

because obviously it's rotating, all the food was right, so it was like a line. Yes. through the, because my thinking was, this way I can see if I was to put it in the center, would I get more? And I actually did try a version where I put a plate in the center, a small plate in the center covered in small poplaroms, and then compared it to how well it cooked if I put the small

plate on the side. And what I discovered is my mom has a really good microwave. Yeah, not all microwaves are created evil. I've just sent you a video of when I covered the whole rotating plate with with popper-dums, you can also hear the fact that my bum's microwave makes a sound that is very adorable at the end. Okay, here we go. I love the fact you've also made it a circle packing problem.

I'm trying to arrange a disc on a disc. Oh, look at it go. Yeah, it's definitely forming a circle. It's playing a whole song at the end. The full song goes do-do-do-do-d-d-d-d-d-d-d-d-d-and then it sounds a lot like the beginning of Glenn Miller's in the mood. So every time the microwave finishes, mom and I then continue going, da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da.

Every time. Now, here's a side fact. I don't have a microwave. One of those guys. No, it's not deliberate. We used to have a microwave, but I don't believe any electrical device in my house should be allowed to make noise unless it's been explicitly authorized to do so. I'm okay with smoke detectors and whatnot.

So I dismantle all my devices and take out the little piezo electric speakers. So my kettle used to beep, taking that out. The microwave had the most piercing beeps. So I dismantled the microwave and took out the little speaker. And it was perfectly silent and worked great for another good three or four years and then mysteriously stopped working. Unrelated, I believe, to my alteration. But whatever the case may be, I modded my microwave and then a couple years later it stopped working.

Yeah, I'm pretty sure you're not supposed to deconstruct those. For everyone listening, don't open up a microwave. It's very dangerous. Even if it's unplugged, they hold a phenomenal amount of charge. Do not open a microwave, even if you think it's not connected to the power, it could still have a lot of juice in there to cause some very serious problems. Even if you want to take out the beepy thing. Even if you want to take out the beepy thing. I'm an official dofus.

Don't do what I did. So what are you saying? You want my mum's microwave that makes an adorable song instead? That I'd kind of be okay with. So you can do the same experiment with the plate not spinning. You know, obviously you're focused on Jeff Thro's problem here. If you were just interested in the waves, if you have it not spinning, you'll get discrete hotspots. And if you measure the distance between those hotspots, you will get half the wavelength as a distance.

So you can actually calculate the wavelength of the microwaves in your microwave by doing the experiment without the spin and measuring the distance. And then if you know the frequency of the microwaves, you can actually calculate the speed of light. So it's a famous experiment where you can calculate the speed of light at home. This whole thing's big up about calculate the speed of light at home from first principles. Or you get the wavelength. And then you get to the part of the experiment, which is calculate the frequency. And the step is look at the back of the microwave and read the frequency because it's written on a sticker on the back of the microwave. No, technically, yes, you're using the microwave.

But I feel like it being on the technical information sticker is a mild cheat. But I still support that as a fun experiment. I say measure how high your popodoms get. Exactly. That's the amplitude. Right. So your popodoms here have popped. Yes.

And there is a definite pattern. It's interesting because each time they sort of, all start in the same spots. Obviously, there's a hot spot that's actually slightly off center. And I realized that that'll be why that there's a rotating plate in microwave so that the food always goes through that point. With my mom's microwave in particular, the only parts that sort of weren't affected were like the very center.

There's a very small bit in the center and the very edges. Pretty much. Everything else is fairly consistent. It depends how much how large you're, lasagna is that you're cooking. But if you're putting that in the very center, that center of your food is going to be surrounded by stuff that's getting very hot and will probably start heating up towards the center anyway, just by being close to the other molecules that are all getting very excited. So you're pretty safe there. Even if you've moved it to the side,

chances are, depending on the size of your microwave, you're probably going to have the edge of it near the center of the microwave anyway. The edge of your other one, if you're using my mom's microwave, then means that you might get some of the sides that are a little bit, the side might get cooler. But because it's, it's all turning on the one axis, you're not like leaving those space. You're still in that point. You're still in the like the bit that's not getting cooked as much, regardless of, it's not like a spirograph, like you've not got a different anchor that's moving in it. You're not like teacups.

Yeah. You should have a teacup ride microwave spinning plates. Yeah. Put a lazy Susan on the side. Yeah. I mean, you'd be able to fit smaller things in the microwave, but it would be way more fun. It would be. But also, I mean, as as folks will see if they see that as well, the popodons all get

pretty much uniformly cooked. And I will for argument's sake also show the one of the smaller plate. that's in the center and when it's on the side, it's a very, very similar amount of cooking slash non-cooking. So it's much of a muchness. But I am aware that not all microwaves are created equal. You touched on this very early on. Microwaves, you talked about in the context of defrosting,

but in general, microwave is a case of blasting energy in a few spots and then giving it time to diffuse out. And moving it around spreads the spots out a bit, but never completely. And so microwaves always involve heating and waiting, heating and waiting. I believe, like a lot of microwaves don't even bother altering how much power is going into the magnetron. They just vary how long it's on or off. So the power settings are power settings. They're actually time settings for how long it's on.

Which is easier than changing the energy, I believe. So I don't know the numbers, but let's say you have, you put it on full power. It's on for 30 of the 60 seconds. You put it on half power. It's still on full power, but just for 15 of the 60 seconds. Is that kind of thing? Yep, yep, yep. So all microwaves are turning power on and off and giving it time to spread out in the meantime.

The other way of doing it, commercial microwaves don't have a spinning plate. You shove the food in, everything stays still, and they still get evenly cooked because when I was looking at buying a microwave, I felt on the rabbit hole of well, what are the options? And the commercial ones cost more money. What if I want a microwave, an emuute? Exactly. They're not necessarily bigger. They're more robust and they're less likely to break because they're designed for being run like a normal household microwave. You run it for a couple minutes a day, maybe an hour if you're in the 80s and want to do an entire cake. Have you ever used a microwave for an hour?

Yeah. You can do stuff where you bake a cake in the microwave. It's dumb, but it exists. Okay, if you're listening to this and you have used your microwaves like for an hour, like food. I mean, I'm interested if it's not food, but can you write in anything? A problemsquare.com, click solution and tell me what you were cooking for an hour in a microwave. We want to hear about it. But the point is you're shocked at the notion of a normal microwave for an hour.

If you're in a restaurant, your microwave could be running nonstop if you're constantly putting different food items through it in a commercial kitchen. So it needs to be out of run like nonstop way more. And anything that moves wears out. And so commercial microwaves do not have any moving parts. But they still have the same problem. So instead of keeping the waves still, like the standing waves and rotating the food, the whole microwave moves. Still.

Yeah, the microwave. So they use the magnetron. I can't know if they use multiple magnetrons or they vary one. And by messing with the phase of the waves, they're able to move the standing wave points around in the microwave by altering, I believe it's the phase of the waves coming out of the magnetron. And so it's a very clever way of not needing to move the food because you're altering the waves so the standing points move around.

through the food. But it's obviously just more expensive to make a microwave to do that. It's cheaper to just put a motor and spin the food. But if you care more about long-term durability, you go with the moving waves. So what you're telling me is you, you realized in the process of this that you actually don't microwave that much at all and then just didn't bother. Basically, yes, yeah. I fell down a rabbit warren of learning, well, it was just like a whole bit bigger. All the ridiculous crazy physics in commercial microwaves, but then I realized, which is at the moment zero, and I'm fine. It's only mildly inconvenient every now and then. But if people write in, you know, I write off microwave cooking because I saw all the terrible

microwave cookbooks of the 80s, maybe it's come a long way now. Maybe there's all sorts of fabulous things you can do in a microwave that I'm just missing out on. So please, write in, Tell me. Can I put another challenge out there for the listeners? Yes, please. I saw online someone claim that you can't microwave snow. So the idea was if you put very cold snow in the microwave, it won't melt because it's ice. It's frozen, right? And I wasn't convinced, I'm like, surely enough of it will be liquid.

Yeah, snow is pretty watery compared to, like, above ice. But then I was like, you know, technically, I guess, if it's all perfectly frozen. So I went as far as messaging it over to Steve Mould to say, can you check this for me? And I haven't done any work of my own. Look, I got enough physicists in my life without having to work it out myself. I just hand these things out to them. Yeah, that's what you are to me, but maths. Yeah, I know.

Yeah. I accept that role. And I then outsource my physics. But if anyone's got any thoughts on that, let me know. And eventually something gets back to me. I'm not sure how, but. It's a box of bees. It arrives in the mail.

I'll have your box of bees and your leftover food. There you go. So my answer to Jethro is if you're using my mom's microwave, it doesn't really matter. And chances are that the size of your micro, versus the size of your lasagna means that there's going to be spots regardless. You're better off just shugling the food around itself, like giving it a little mix or something or stopping and then turning it around a little bit and then putting it back in.

That would be my answer to do that. But I would also say, well, why not run an experiment? Why not cover a plate in popatoms or marshmallows and take a look at where your microwave's hotspots are. Or a thin layer of cheese. Yeah, you should get to know your own microwave. Find out where the hotspots are. You can do it with the sort of paper they used to use in like old receipts and maybe faxes like the thermal paper where it changes color with heat.

If you want to do it real boring, you can do it that way. I mean, all microwaves have hotspots and cold spots, but it gets moved around enough. It's fine. Just be patient if you need to let it sit for a bit. Yeah, that's my answer. Well, Jethro, let us know if that answers your question. Let us know how you're going with your microwave investigations. And normally I would say, let us know if this solution gets a ding.

But in this case, we'd want one of these. Our next problem comes from Stu and Angus, who say, hi, Beck and Matt. Greetings from Geelong and Blah to the power of three. We started around 30, I think. Then as soon, I'm guessing that's episode 30. And as soon as we realized the callbacks, we went back to number one and followed ever since. Ah, lovely. Stu says, here's a question that delves into the fourth dimension.

Oh, no, I hate this already. Son, Angus, 12. Ah, Angus, what are you doing? You're killing me. Came to me asking about the distance from opposite corners of a cube and whether if the cube was, okay, so we've done distance to the opposite corners of a cube before, but Angus goes on to ask whether if the cube was one meter, to each side, would the diagonal equal root squared?

So we got some paper and started working it out. We quickly realized he meant a square. That's the sort of thing I would do. We quickly realized he meant a square, but we carried on and found that the cube diagonal was root three. Makes sense. Three dimensions. Love it.

That got us thinking. 2D cube. Diagonal is root two. Three do cube. It's root three. Matt, I've tried understanding fourth dimensions from many of your videos. The understanding just doesn't stay in my brain, but does this pattern continue?

Ah, does the longest diagonal in a one-meter-sided four-dimensional square equal root for five dimensions, which I think is, five dimensions is when they squirt water at your face. Yes, yes. And the seat moves around while you're trying to work it out. Yeah, yeah, yeah, yeah. And so on. I don't have the understanding of math to work it out, but if there's anyone up for it, I'm sure it's back. Good work, Steele-Hungus.

I mean, Matt, I mean, okay, fight for it if you must. Stu, I love you. Thank you for making me feel included. Stu also loved the Christmas card. We sent out the Christmas card to our Patreon supporters, and it was a battleship game. Stu says they made a photocopy so they could actually play it with someone else.

And as a side note for Matt, Australia's first large-scale saltwork. is here in Geelong. Field trip anyone, the end? The Coldwell family, they've signed off officially. That's very sweet. I would go to a saltworks. If you want your problem solved, don't distract me with a saltworks.

That's what I'm saying. Now, that's interesting. I mean, I feel a certain affinity for that because they were Australia's first large-scale saltworks, whereas I am Australia's most recent. Small-scale. Small-scale. A scale.

A scale. Oh, as we speak, it's drying out as we speak. Oh, can't wait. Wait till I get to Australia, then we'll check in on the salt. Right. So here's the problem, Beck. I discussed the diagonal in a cube back in episode 1-2-1. And we are currently in episode 131.

So it's a palindromic number to the previous palindromic numbered episode. It's a face off and an edge off. It's an edge off. And a vertex off. Vertex off. Do you remember the name? Popquares? Yeah, I do.

Space diagonal. That's the one. We did a whole thing about the space diagonal. I used it as an excuse to talk about no-put shapes and ru-pit shapes. Yeah. It was a lot of fun. We discussed oil abricks with face diagonals as well as space diagonals. Face Diagnos.

That's the one. And I thought I did a pretty good, comprehensive job. But then Stuart Angus here, another parent-offspring combo, have come rolling in, asking about how long the space diagonal is. And I realized I didn't actually talk about Pythagoras or anything to do with that. Last time, in 1-21. So now I've got a second chance in 13-1. I'm going to set the diagonal straight. Oh, he's going to make it about triangles, isn't he?

Always with the triangles. What makes you think I'm going to make it about triangles? Now, first of all, to recap Pythagoras, which is why I brought with me this triangle. So, shut up. Matt is holding up a triangle. It is a right angle triangle. Thank you.

Well, well, well, well, named. A, B, and C to help work out the hypotenuse. And I've labeled A, B, and C is the hypotenuse. A and B are the shorter sides. A is longer than B for people imagining this at home. And I've also got next to me here, my whiteboard. I think I feel like I've mastered the podcast audio format here. I'm going to stick my triangle on the board.

There we are. And I'll take a photo of this, everyone, so you can check it over. want or you can just listen along. I'm now going to draw in A squared, B squared and C squared. Because when we say it, A square, B, C squared, we mean it as a number. You square whatever the length is A, is in by multiply it by itself. But geometrically, and this is how this was originally proved way back in the day, you would take a square that's A by A. So that area there, that's the area of A squared. I'm going to label that, A squared. Here's a square that's the size of B. So that's B squared.

And then you've got a massive square here because C's the biggest side on the triangle. Now, none of this is to scale. Currently, there's C squared. The Pythagoras theorem has been around for a long time. It was discovered multiple times by other people, pretty Pythagoras. People knew it worked for a very long time.

There's evidence of it from, you know, China, way back in the day, Egypt, long before Pythagoras. When it kind of became a real mass thing was proving it always works. People knew it worked. because it always did, but no one was able to prove there wasn't some mysterious right angle triangle for which it wouldn't work. And so I thought I'll just start with a, this was my favorite ever proof, because what I've

shown you here, you've just got to believe me that the area of A and B squared, if you were to combine those two areas, you get the area of C squared. And you'll sometimes see demos in science museums, like hands-on museums, where these are like empty acrylic containers full of water and you can rotate it up the Y way and all the water from A and B perfectly fills C. I said that's clever. Yeah, it's very smart. But it's just showing you in that specific triangle it works, not it works in general.

So what I thought I'd do, this is what I used to do when I was a teacher, which is probably why you're getting classroom flashbacks. And you're a teacher who has a whiteboard. Yeah, exactly. Yeah. This is a very teacher coded session, if I'm being 100% honest. What I can do now is instead of just one of these, I've actually quickly knocked together four identical right angle triangles. So here's the second one.

Now I've stuck magnets on the back of them. The magnets are dangerously overpowered for the task I've given them of sticking to a whiteboard. Because I only had like high powered neodymium, whatever they are, magnets available in my house. So that's what we're working with. So those triangles are there forever. Yeah. Okay. So I've arranged them into two rectangles. Ah. And this bottom here, this is a B by B square, right?

So you can see this square here, that little square at the bottom there is B squared. Yeah. And this one over here, this is A by A. So this whole corner up here, that whole section there is A squared. This total area here, the square A squared, the square B squared, and four triangles. Remember, A squared and B squared and four triangles. I'm going to take the triangles out.

Put them up here. So now you can still see the kind of space where they were, and I'm going to remove A squared and B squared, but I'm going to leave the outer perimeter there. So that's the entire box that was A squared, B squared, and four triangles. I'm now going to put the four triangles back in a different way. I'm going to put one in each corner. One's going to go there.

One's going to go over here like this. One's going to come up there like that. And if I put one in each corner, I've now got a single square in the middle, but that's side C. So that whole middle square now is C squared. And it fits in the same perimeter box as before. So the same area can either be four triangles, A squared and B squared, or it can be four triangles and C squared.

and C squared, which means that the C squared must be exactly the same area as A squared and B squared because it's the same amount left over from the four triangles in both situations. It doesn't matter what right angle triangle you pick is in terms of how long A and B are compared to each other. You can always do those two arrangements, which means for any right angle triangle, the two smaller squares, A squared and B squared, must be the same areas as C squared. And so that's a geometric visual proof for Pythagoras's theorem. Now, that was a proof of Pythagoras' theorem. And the whole point of why Pythagoras is wonderful,

apart from the practical applications, blah, blah, blah, is the fact that it was one of the first things that was like a really nice proof for the sake of proving it's always true. And Pythagoras's theorem has now been proven, oh goodness, well over 300 times. I think we're coming up on 400 different ways you can prove. that Pythagoras's theorem is true. It's a bit like, you know the meme about running doom

on any kind of ridiculous hardware? People try to get doom running on like vacuum cleaners and calculators and all this ridiculous stuff. You know, cannot run doom is, it's nice thing. There's a rough equivalent in math, which is, yeah, but can it prove Pythagoras? And so every proof, it's very funny if you inadvertently prove Pythagoras again.

And there's a whole book. Great. Actually, I got a copy somewhere. There's a book of like, there's just different proofs of Pythagoras, which I think is amazing. And we know it's true, but we keep proving it over and over again. However, we only proved it there for a square. And what Stuart Angus are now talking about is a cube.

So our right angle, triangle, if A and B were both the same and we say they're equal to one, then the hypotenuse is now the diagonal of a square and its length is the square root of two because hypotenuse squared must be one squared plus one squared which is two. So the hypotenuse squared must be two so its length must be the square root of two which is by definition the number if you square it you get two.

They were then looking at a cube and they worked out that the space diagonal is the square root of 3. And that's because Pythagoras actually works in any number of dimensions. So for a square or any kind of rectangle, A squared plus B squared equals C squared. But for a cube or any kind of cuboid,

A squared plus B squared plus, I don't know what we're going to call the next one. We've already used. Let's rename the diagonal D and have the three sides of the cube, A, B, and C, A square plus B, plus C squared, still equals D squared. It works exactly the same in three dimensions. In fact, it works in four dimensions.

And I know we got distracted with hypercubes last time. But if you did have the space diagonal across a four-dimensional cube, A-square plus B-square plus C-square-plus D-D-squared, would still equal E-squared. So it doesn't matter how many dimensions you're in, Pythagoras always works, and it doesn't get any more complicated. You just square or your orthogonal directions, and if you add them all together, they equal the square of the space diagonal going from the beginning to the end. And that's it. You can do that in any number of dimensions you want to do. And it's still a super straightforward computation. Never gets more complicated than square roots, which is interesting.

you'd think in higher dimensions, you'd end up with like higher powered roots, but no, square root every single time. And so they're absolutely correct. In 4D, it's just the square root of 4, which happens to be 2. In 5D, it's the square root of 5. Because if all your directions are 1, you're just adding up 1 squared plus 1 square plus 1, 1 plus 1, 1 for each dimension. it's always just the square root of the number of the dimension you're in will give you the space diagonal in the cube and for more complicated cuboids

you're still not doing anything more complex than squaring and adding the numbers together than taking a single square root so in conclusion stew and Angus you're 100% correct good job spotting the pattern you are bang on

it's always the square root of the number of dimensions if you get bored you can try and show algebraically why it still works in higher dimensions. It's not super complex to show, but I'll leave that as a fun activity. Yes, hand up, young Ms. Hill. Yes, I have my hand raised. Thank you. Present.

Can you demonstrate the four-dimension version on the whiteboard with magnets and triangles? I do have an excellent proof, but I don't have enough magnets. Actually, what you would end up doing is if you do the proof of 4D, it's the same 2D proof. Just you chain a couple of them in a row. And so visually, it's not nearly as nice. That's disappointing. It's the same thing over and over. But you take the output of the previous one into the next one.

I've got one more question. My hand is raised. Oh, yes. Hand is up again, yes? Yes. Can you demonstrate it using bees? Yes. Maybe you didn't hear me when I said A squared plus B squared equals C squared. It's actually B's squared.

Yes. Didn't I articulate B's quared? Sorry if I wasn't saying that clearly. No further questions, Your Honor. Excellent. I rest my case. Well, I give that a ding squared. Yay. Well, we did have two problem poses for that one. So between Stu and Angus, we might get a square ding back. Let's see what happens.

And now it's time for bonus other business, the old bob, we call it. Peter posed us the problem for other uses for fancy glasses for episode one, two, six. They said, I'm writing to let you know, you may consider the special vessels problem solved. on several levels in fact. They've listed the levels for us starting at zero. I love that. Excellent work. So as a little recap, we were asked for advice on other uses for fancy glasses

if you're no longer putting alcohol into them. So level zero, cold brew coffees and teas are a brilliant idea. The Ford planning required and the following anticipation are enough to make any mundane situation sufficiently special. I agree. And I actually did this the other day. I made the cold brew coffee with mint that I had suggested on that answer, mixed in with some Mugave syrup and served it in a, I didn't have any martini glasses, but I have a champagne flute here.

So we had some, it was very, very, very fancy. It felt very nice. Then level number one, all of our guests from now on will be served in the fanciest glass where practicable. I like that. Anytime you're a guest, you get given a very fancy glass. That's a really nice. That's just a nice little tradition. over my house you get a fancy glass.

Don't be limited by practicality. Yeah. Level number two, snacks will also be served from glasses. It's actually easier to pass them around this way. Agreed. Little handles. Level number three, and most importantly, you inspired us to do our own experimentation.

Can't be bothered making a whole Sunday? Just plop a scoop of ice cream into a wine glass. Pretending that black currant juice is fine red wine is just as much fun now as it was when I was eight. I love this so much. Although we're still working through the complete list, as suggested, I feel I can present you with your ding. P.S., we are regular visitors to the largest annual pottery market here in Chechia,

a contributing factor to the original problem. It's held in September, and if I spot ceramic champagne flutes for your morning coffee, I will find a way to ship them to you as a thank you. Thank you. I would also be worried about getting to us safely by doing so, You'd be more than welcome to pass on the artist's details if they have an online store. Adrian's got back to us who posed the pantry item labelling problem back in episode 127. They say thank you, Beck, for your pantry item names.

And thanks Matt and Laura for playing along. I mean, I don't think we get thanked enough for our service of playing along with whatever Beck is doing. Yeah, it's fair. Adrian here has followed Beck's suggestions. I'm sorry to hear that. and they've gone for the more descriptive versions. They've used the following. Spaghetti about it.

Flower power. Mr. Bean, Queen of Spades, and Colonel Sanders. They also like some of the other ones. They may use them in the future, but those are the ones that have gone into service so far. They say thanks again and all the best.

Thanks, Adrian. Yay. We also heard from OSA, who suggested, who suggested doing the Ask Me Anything episode. Great suggestion. So they've come back to say a ding for a wonderful AMA. I feel so much more enriched after hearing your stories.

Appreciate the camaraderie you all have with each other. Here's to another two to the power of seven episodes. We also heard from a few other folks with Rock Day suggestions because we had so many suggestions for Tooth Day. I knew there was a built-up excitement. about Rock Day that we just hadn't tapped into yet. And now, now there's been a fissure. It's breaking forth.

Yes. It's been chiseled into their minds. Exactly. We're chipping away at it. We're fracking their brains. So Leah wrote in and said, Dear Beck and Matt, as a geologist, I love, I love some credentials. Love me some credentials.

As a geologist, I'm ashamed that there were not more Rock Day suggestions in the most recent episode. Well, first, Alia, that's not your fault. You shouldn't be ashamed. Leah said, here are a few alternate ideas for your consideration. April 5th or May 4th, depending on which date notation you prefer. 4.5. To celebrate Earth's age of 4.5 billion years. Oh, that's nice.

January 3rd or March the 1st, either one. To represent the three main classes of rocks, igneous, sedimentary and mediation. metamorphic, which are all really one as they can transform into each other through the rock cycle, or August the 14th, obviously, or like 814 or 14 or 148, to represent the atomic numbers of silicon and oxygen. These are the two most abundant elements in Earth's crust and are found in almost every rock. Leah then signs off rock on. We also heard from Griff, who he said blah, blah, rock day should be second of May since

that is Dwayne Johnson's birthday. No deal. Oh, I think that's my favorite. Also, because, oh my gosh, Matt, Matt, Matt, Matt. Because we were thinking of combining Tooth Day and Rock Day. And Dwayne was the Tooth Fairy. He played the Tooth Fairy. Tuesday and Rock Day needs to be the 2nd of May.

Oh, my goodness. That's a very good argument. I was pretty excited about the 4th of May for 4.5 billion years. I acknowledge we'd have to change the day once every 100 million years, but that's a small price to pay. But you've got a pretty good Dwayne the Rock Johnson argument there. I would want to revert this back to Leah as the official geologist. Yeah. And say, Leah, how do you feel about it being Duy Johnson's personal?

instead. Now, I would like to cast my vote for 4th of May, but I'm open to being outvoted here. I will also run this past dentist Sophie to see if she approves of Dwayne Johnson's birthday being Tuesday. It gives us enough time to bring together all of our questions and answers for our special rock tooth episode. Leah, let us. know how you feel about Dwayne the Rock Johnson's birthday. I'm going to ask Dr. Sophie if she's okay with sharing that as a tooth day.

Do you know what? Because it wouldn't be Rock Day. It would be Rock and Tooth Day. Like it's combined. So we're, you know, you could still have an official rock day on May 4th. But rock and tooth day. Yeah. The tooth of May.

The tooth of May. And looking at the release dates of the podcast, the closest episode would be May the 11th. 11th, the Monday of release. Ah. So that does take in the fourth and the second. Always produced. So it's a week, it's a whole week. There you go.

Tooth and Rock Day goes from the second of May to the fourth. It's a season. It's a festival. Well, thank you so much, everyone for listening to Some Problems Squared. The Pelondromic. podcast. We also like to thank specifically our Patreon supporters who pay the bills round here. And to do that, every episode we pick three of their names to be gloriously hand-mispronounced by us, which this episode shall include.

Marriage and Vigel. Tej Kedon. Exunder Scorlapper 7 She pronounced the underscore as underscore good work

great great All right You've been listening to me Matt Parker and Beck Hill And finally we thank Producer Laura Grimshaw who as a palindrome

keeps us on the level The palindromic word there was level. Toot, toot. You boob. Okay, Beck, we are not in the same room as each other or indeed our game of Connect 4. Laura's holding up the thing for us. Laura's literally holding it up.

This is Backside that I'm showing me. Oh, Beck's looking at it. And what color was Beck? Back is green. Okay. Okay. So, because I can see your face through it, Laura. You just said.

I'm going to put my eyes there. There they go. Screen grabbed that. I screen grabbed it. So I would like to put it in column. Yeah, column, column right eye, please. Right eye. Yeah, that's the one.

Yep. Oh.