A Problem Squared - 092 = Broken Hands and Alexander's Bands

Episode Date: September 2, 2024

🌈 How many colours are really in a rainbow? 🕰 How often is a broken clock is right?  💼 The Any Other Business briefcase is open. Briefly.  🏴‍ And we have a NEW POST CREDITS easter egg.  To hear more about the brilliant and terrible Emily In Paris premier, listen to this episode: https://podcasts.apple.com/gb/podcast/point-break-pt-1-season-04-episode-01/id1720149980?i=1000666164553 If you want to see where the Mattparker asteroid / minor planet is in the universe, you can locate it on this NASA database: https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=314159&view=VOP A full circle rainbow was captured by aerial photographer Colin Leonhardt while flying through a rain shower just above Cottesloe Beach in Western Australia: https://s.abcnews.com/images/International/cn_round_rainbow_australia_jc_141010_16x9_992.jpg?w=1600 And if you want to be even more wowed, you can see a reflective rainbow (or six!) in Norway: https://apod.nasa.gov/apod/image/0709/sixrainbows_nordvik_big.jpg Finally, send your problems and solutions to our website: www.aproblemsquared.com. If you’re on Patreon and have a creative Wizard offer to give Bec and Matt, please comment on the ‘Sup ‘Zards’ pinned post!  And if you like, leave us a review, pass the podcast onto a friend or give us a rating! Every little helps.  If you want even more from A Problem Squared, you can find us onTwitter,Instagram, and on Discord

Hello and welcome to a problem squared, a podcast, which is a bit like chlorophyll. Oh. Yeah. Chlorophyll, the substance in plants. Oh, not the one. Not the chloroform doesn't put you to sleep. Okay. That's the opposite of this podcast.

Oh. Or the people who listen to it to go to sleep, so let's not rule that out. No, I mean like chlorophyll converts carbon dioxide into like energy, carbohydrates using the power of the sun. We convert problems into a quality podcast. Oh, by the power of coffee. Most of the time. I like this. Yeah.

And that, I think that sums up the podcast. New listeners. This is how every start goes, by the way. We workshop the intro as it's happening. So the chemical symbol for the several types of chlorophyll, but chlorophyll A is C55H72O5N4MG. Catchy. And as a mathematician, I'm a lot like that in that I'm also full of numbers.

And letters. And letters, occasionally letters, mainly numbers. There's some algebra in there. Some algebra. And I'm co-host Beck Hill. So chlorophyll provides the wonderful green that plants are renowned for. Yes. And Beck, comedian, performer, writer, also brings a lot of color to the world.

Aww. So there you are. Thank you. Welcome to the podcast. Thanks. Oh wait, you're talking to the listeners. I was like- I did, all of us.

Episode 92, you're finally welcome to the reach. You're finally welcome. You're like, about time. On this episode- How many colors are in a rainbow? Wrong brand. How often is a broken clock correct? And any other business?

So, Beck, how have you been? For you listeners, we now have a conversation. Like we just said what we're going to do in the podcast and then we don't do that for a while because we want to catch up. Yes. It's kind of like hiding the tasty stuff amongst the vegetables. Oh, right. Like, yeah, you got to get through us to get to the solutions.

You're going to consume your chlorophyll. Yeah. That's it. Why does that sound so creepy coming from? It does, doesn't it? Cons consume your chlorophyll. Yeah, that's why does that sound so creepy? It does, doesn't it? Consume your chlorophyll. How's your life going? It's been a while.

It has. We saw each other briefly at big festival. Very briefly. But yeah, that was a very busy time. I was in LA. I've just, we've both come back from the States last week. Eight days ago, we both got back. Is that it? Yeah, I know. It feels like both the time has passed. Eight days ago, we both got back. Was that it?

Yeah, I know. It feels like both of times has passed. I've done two more trips since then. You have? That's ridiculous. I'm so tired. I was in LA, you were in New York. We didn't actually cross over this time.

I attended the Emily in Paris season four premiere. Yes, you said. I've been holding off asking because I knew if I asked you'd say I'm saving it for the podcast. Yes. Now I can ask, how did that happen and how did it go? Okay. So we tried loads of different ways to get, but again, for new listeners, I host another

podcast called Enemy in Paris. It is a hate watch podcast covering that show. And we wanted to go to the premiere. We had some contacts who were within the sort of TV, pop culture industry. And unfortunately they were unsuccessful. And I was like, oh man, because I did that thing where I was like, you know, build it and they will come kind of thing.

I was like, I will go to LA. I'll just get to LA when the premiere is on. It'll happen. Yeah. And it was getting closer and closer. I was like, oh will go to LA. I'll just get to LA when the premiere is on. It'll happen. Yeah. And it was getting closer and closer. I was like, Oh my goodness, we're not going to get there. And I'm so, okay. So I was adding the show notes for an episode that I just uploaded and quoting a bit from the Netflix website. And when I went to double check the link for the website,

they had not an hour or so beforehand added a thing to say that you could apply for tickets to the premiere. But obviously they were very limited. We both applied asking for two tickets. This is my co-host Sam. I told them, I just told, I was just honest. I went, we do a podcast. It's a hate watch podcast, but also weirdly we've now got this strange connection

with the show and it means a lot to us. I came over from the UK specifically wanting to come to this event and it would mean a lot to us and they approved it. That's amazing. Sam did not get approved. He just wrote, we love the show or something like that. Everyone else would have done it. But you hedged your bets. You did both. It was both brilliant and awful. Oh, my gosh. There is nothing quite like sitting in a cinema.

Yeah. Full of people who love a show that you hate. And every time they laugh at a joke that you think is dumb. You're like, no. You're like, who are these people? So we got to meet some of the cast members. That's ridiculous. And I have a gold dress. Yep. The gold dress I got from a charity shop in Godeming. It's this beautiful formal dress.

The Godeming gold gown, they call it. They do call it that, don't they? Yeah, the Godeming gold gown. And that's my go-to formal attire. I wore it to your book launch. I left it at home. Ah. And I realised I don't have a formal and they did say it. It's like chic formal attire. And Sam was like, look, well, there's some good thrift shops in LA.

We'll go there. And the first one we went to, it's an LGBTQI plus supporting charity flip shop and it's called Out of the Closet. It's very funny. Great, great stuff. And I walked in there and pretty much the first thing I saw was a floor length sequin gown. Not gold, black. Black this time. Black sequins?

Yeah. For evening wear. For evening wear. That was $35. Not quite eight pounds, but you know. 35 bucks, that's normal. Still. It's like probably 28 quid or something. Closer to zero than the real price.

Yes, exactly. So I got some truly horrific hot pink metallic heels, which I immediately complained about upon wearing. I got the biggest blisters on my toes, but they looked both terrible and fantastic. And I imagine you've discussed this at length on Enemy in Paris. Yes. Yeah, if people want to hear us talk more about the premiere and what went down, and if anyone has been watching the show, then yes, check out Enemy in Paris.

Amazing. I don't know how I can possibly compete. How can you compete? I can't. You had a book launch. I did have a book launch. My book's now out in the US. So actually I've got two tiny bits of news.

I am now a New York Times bestselling author. Oh my God! We made the New York Times bestseller list. Yeah, yeah. Paul D. Parker! Isn't that phenomenal? Because you told me when you found out you were the Sunday Times bestseller.

And you couldn't, you were, you were like. Well, as we record it's still embargoed but when this episode comes out that would be public not oh wait on this Monday yes it'll be this weekend well yeah yeah like yesterday day before that two days ago yeah yeah on Wednesday and then curl that's amazing yeah so congratulations thank you I'm so excited. Like I've always sold, like obviously I'm super proud of something well in the UK and it was a huge achievement. It's great, but I could never match it in the US. And this time people bought the book. So yeah, it's not even the funny one. I know.

It didn't even have the, um. Well, getting tricky with it. Yeah, that was it. Yeah. Although if you go into, I think it's Waterstones Piccadilly, they use that as their table label for my book display. And it's still there. They're still getting tricky with it table with all my books on it. Great.

So get down there quick before they put that away. Yeah. And yeah, phenomenal. And now I'm very grateful the New York Times has a top 15 because Love Triangle was the 14th best selling book in the US. Wow. So it doesn't matter best selling, I'm on the bestseller list. That's so many books.

So many books. Is that like of a nonfiction list or just of all? Nonfiction, yeah. That's so cool. Yeah. I'm very excited. Yeah. Amazing. So thank you so much. Everyone who bought the book. I hugely appreciate it. What made it possible. I was like, Hey, if we all buy it, we might get the New York Times bestseller list. And I was like, I don't know if it's possible or not. And then we made it. So, uh, so I think that's my only significant

news other than the, um, internationalical Union named an asteroid after me. Okay, what? So I feel like that's the whole list. What? Of things that have happened. Hang on. How, why, you've got a square and now you have an asteroid. Now we've got an asteroid.

When's this going to end? So, well, I think the asteroid will outlast us all. Yeah, can you imagine if the Earth would kill? This is it. When it happened, I was like, that's great! And my brain went, the sentence, all life on Earth was destroyed by Matt Parker, has just become slightly more likely. Like before it was pretty much zero,

and now it's reasonably non-zero. Okay, so how? Yeah, yeah, yeah. Now, this is not like the Namer star or anything like that. You know, someone didn't buy you a tin from W. H. Smith. No, no, no. So the International Astronomical Union, which ironically, my wife is a member of, is in charge of naming everything. Is that ironic or is it nepotism? It is a good... She... So the result came out because they had their massive meeting, which they do a meeting every three years where they do like big important stuff.

And obviously the Matt Parker minor planet was, no, it would have happened anyway, but we didn't go because we were in the U S so it was in South Africa this time and Lucy was invited as an invited speaker to go down, but we couldn't make it this time because we were both doing stuff in the States. And she's refused to say if she would have recused herself from the vote. Right. She'd been in attendance. Yeah. I think this this was done by a subcommittee but anyway so yeah occasionally they will name unnamed asteroids and they named I think like 40 or time for this round and but how is your name to be nominated some people nominated me who have asked to remain anonymous but they are undergraduate

students at a university and then one of them realized you can look up as words have been discovered in different places haven't got names they all they're all given an index so this is the and key hold your jokes the small body database lookup now the index is 3 1 4159, which people at home may recognise as the first six digits of pi. So you got to choose the number. No, they realised no one had named the pi asteroid. So they were like, yeah, they're like, no one's named pi. It needs to be some of maths related.

Yeah, well, they had the idea first. So they were like, let's get an asteroid for math. And then it has to go to a meeting of the IAU who then have a big vote on it. They had a big old vote and then they approved it. Yeah, there it is. Look at that. Asteroid 314159, Matt Parker. I love that. It was discovered in 2005. It was discovered by the University of Arizona as part of their sky survey. It's interesting that they don't get to name it.

I feel like if I discovered something, I should be able to name it. But then again, that is very colonial. I saw it first. Yeah, that's very white of me. No, so there are very strict rules for naming things in space and different categories of things. Has to be named after a nerd. Has to be a nerd. Is there one called Butts?

I don't know if I can search by name, but what I can show you is you read through what some of the other ones are So we got someone who was a geographer. We've got an Italian astronomer Was a so this is someone who lived from 1858 to 1917 and they were a master fisherman He was the founder of modern Hungarian fish farming. Oh vales. Yeah. Hmm. So I've joined the ranks of a Hungarian revolutionary fish farmer. Yeah. Yeah. I can show you a picture of... Oh, it looks just like you.

It's got a good rhythm balance. Matt is showing me an image of some dots. Yep. So there are only four images of it. And in each one, it just moved slightly compared to the background stars. So the stars don't move, but this is an asteroid, so it's orbiting the sun. And so these are shots taken a reasonable amount of time apart to show it actually moving. If you put a cap on it, would anyone recognise it?

That is a reference to our Patreon bonus show, which if you would like to listen to, you can sign up as a Patreon supporter today. As far as we're aware, we can't like it's a dot. We don't know what it looks like. If people want to do an artist's impression, I would love to get some artistic impressions of Matt Parker. So I have something to show. Of what your eyes are going to look like.

Other than a moving dot. We haven't seen enough of it to work out its shape based on how its brightness changes, but based on the brightness that has been observed so far, it's nominally, they're pretty sure it's probably between 700 meters and one and a half kilometers in diameter. That's a fairly healthy range. Yeah, it's big enough. I've flipped between asteroid and minor planet because they describe the same category of

object. I think it sounds a little bit grander to have a minor planet. Oh, absolutely. Yeah. So that's my minor planet. Those are two very exciting bits of information. Well done for not divulging them to me any earlier. It wasn't easy. Exciting stuff. Well, I only found out about the New York Times a couple of days ago.

That was easy. But the asteroid, I knew it was coming a little way off. So yeah, like a lot of asteroids You see it coming. Thank you everyone Now let's solve some problems. All right First problem was sent in by someone named Leandra They went to the problem posing because the name is Leandra. Yeah, yeah, nice. It's very funny. It's funny if you see it written down.

I appreciate this is an audio medium. They went to the problem posing page. It's Leondra. Leondra, sorry. Yeah, as in Deondra. And some number of people who go by that pseudonym said, how many colours are there in a rainbow? And they've added one qualifying fact. They say that we can see, I guess.

Still quite a concise problem, I have to say. And, Beck, you've looked into this. Yes, I have looked into it and there- How many colours did you see? There are several answers that I can see, I guess. Uh, so in order to answer this, I wanted to do a little bit of research about rainbows before. We have, we talk rainbows.

What? Yeah. Why did we talk about that? I got really into rainbows when I was writing love triangle. And I can't remember if we were just talking about that or it was a problem that came up on the podcast. You know, I think I'd probably show, I'd probably rather ducking you showing you the stuff I was putting in my book. Cause I did a whole thing about the geometry of rainbows. This is good though. This is good because you can

fact check me. I will. So to the extent of my ability. Okay. So the way rainbows work is that if you imagine a cone where you are the pointy bit. Yep. And the base of the cone is the opposite side to the sun. Yep. Then basically the way that the sun is reflecting off of the shape of the raindrops means that you're getting that refraction. If you picture like the way that prisms work a bit like that, but because the raindrops are round or teardropped, then it means that the wavelengths of light are reflecting within it at different angles and then dispersing.

So that's why you get this rainbow fuzzy effect as opposed to when you're looking at a spectrum from a prism. Yeah, I think it's a very similar mechanism in both. Yes, except the other one will clearly divide up the colours as like the wavelengths are much more clearer, the differences between them. Yes, I suspect the difference, I think individually it'll be the same, but the rainbow will be the combined light from lots of raindrops. So that's probably why it's a bit less clear as opposed to a prism, which is giving you a single prism splitting the light. Yeah. Yeah.

And that you get different rainbows depending on the size of the raindrops and what sort of water it is. So sea water, you get a brighter rainbow, I believe. Yeah, it depends because the reason we get the rainbow is when lights going from one medium to another, it changes speed. Yes. Does light go faster or slower in saltwater? I'm going to say slower, but that's a total guess.

So, yeah. Seawater has a higher refractive index in rain water, so the radius of a rainbow in sea spray is smaller than that of a true rainbow. Got it. So the light's changing less. Oh, maybe it is faster in there. There you go. Yeah.

If you end up with a mix of large and small raindrops, you can sometimes get the, it's really rare, but you can sometimes get a twinned rainbow. A twinned rainbow? Oh, one from from big drops one from little drops yeah it's not to be confused with a double rainbow not a double rainbow so most double rainbows that people see the fact that all rainbows have a secondary rainbow many times we can't see them because the light is too faint and the way that a secondary rainbow is formed this is now okay, so this is where you get to correct me if I'm wrong.

So if I'm at the point of the cone and the cone is facing out for me like a big megaphone, and the circle base of the cone that's opposite to the sun, that diameter is gonna be the rainbow. Due to reasons, the cone is at an angle. So if you're looking, it's not straight in front of you. No, it's an angle based on where the sun is behind you. So it's, well, it's always at the same angle.

I think it's 42 degrees. That's how wide the cone is. So that's how pointy the bit near your face is at that angle. Oh, okay. But yeah, because the sun is above the horizon, that cone is always going to be pointing downwards. Yes. Which is why when you see a rainbow, it looks like a bow, but in reality, if the earth weren't in the way, it'd be a massive circle, which is why you were able to see circles from, you saw a circle rainbow or at least you saw some good photos of circles. There's an excellent photo taken by a photographer in Western Australia that I will put on the social media and you can see, only can you see the full circle because it's taken from

an aircraft looking down. So, yes, not in the way. I've just realised why it was that we were talking about them. It was because you had a couple of photos to choose from, but you knew it was going to be black and white. And you were asking which one I thought looked the most impressive. Oh, great. I can't remember. We might have put it on a Patreon chat or something. Maybe it was just us.

Might have just been us. Sat around my dining room table. Look, a little look behind the scenes for all the listeners. Yeah. So we can share, I'm pretty sure we can share that image because it's from an aircraft looking down you can see the full circle of the rainbow because the cone, the earth's not getting in the way of the cone. And you can actually see the double rainbow, you can see the second one out.

Yeah, so the double rainbow, which I got very excited when I think I managed to wrap my head around it. Yep. So the double rainbow, because it refracts within the raindrops, you've got the really bright one where the light is exiting them. Most of the light is exiting from one side and that's the rainbow we see. But there is also light that escapes in reverse, like it bounces backwards in the opposite direction. And that technically means that that one isn't centered around the center of the cone that you're looking at. It's actually centered around the sun, but it's so big that it goes beyond 90 degrees. It comes back around the other side.

So it is like an inside out rainbow, essentially. It is an inside out rainbow because the colours are reversed. Yes, I'm going to say yes. It's easier to say, it's a mirror image. It's easier to say that. It's reflected back the other way, so we get a lot of sort of smaller mirror image. I'm going to give you a slightly caveated yes. What's your caveat that you're just not sure of?

Yeah, well, it's not complicated, but your standard rainbow, the edge is coloured in gradients, and then the inside is actually brighter than the rest of the sky around it. Whereas the second rainbow is the other way around, the colours go out, and then the sky on the sky around it. Whereas the second rainbow is the other way around, the colors go out and then the sky on the outside's brighter. So if you ever see a double rainbow, the gap between the rainbows is darker, yes.

Yeah, which has actually got a name. Oh, I didn't know that. The dark space between two rainbows is known as Alexander's Band, named after the ancient philosopher Alexander of Aphrodisias. Funnily enough, while I was in LA, I saw a free concert in the city.

By Alexander's band? Well, not far off. It was being held by the Parks and Rec department. It was being hosted by Chris Kirkpatrick, whose name I only know from Eminem singing Chris Kirkpatrick, you can get your ass kicked. Right. name I only know from Eminem singing Chris Capatric, You Can Get Your Ass Kicked. But he was one of the lesser known members of Nsync, a band The Calling were on first. Basically, it was all these 90s boy bands that you probably don't remember the names of, but might remember they had one song. But I looked them up, because I was like, because the lead singer, I couldn't work out

if he was an old looking young guy or a young looking old guy. And I was like, how old is this guy? And then when I looked them up, the calling, the main singer, his name is Alex Band. His name is Alex Band, but he's called his band The Calling. Why don't you call it the Alex Band? It's right there.

It's right there. Absolutely furious. So now I'm going to call them the band between the band. The band between the two rainbows is now called The Calling. The Calling. Love it. If someone wants to put that in the Wikipedia, please don't. OK, so that's Alexander's band. Oh yeah, we're solving a problem.

Yeah. Yeah, yeah. The secondary rainbow is technically centered on the sun. It appears on the same side of the sky as the primary rainbow because it has a larger angular size of over 90 degrees. So around 127 degrees for violet and 130 for red. I think that's over complicating. The reason I had caveats is I think that's over complicating the way it's explained.

But I don't think it's invalid. Yes. It's not necessary to know all of that. You don't have to imagine it as a giant inverted cone centered on the Sun. You can still picture it as a second cone centred on the viewer. Yeah. However, I found it more fun because of the same reason that when you went to see the eclipse in Antarctica, but you guys were on the other side of the South Pole. Yes. So the sun looked like it was moving in the opposite direction. Well, yeah, the shadow went the wrong way. Yeah.

Yes. Yeah, sorry. Yeah, the shadow went in the opposite direction because you're looking past the pole of the side that you're on. And for the same reason, I love that. Like, yeah, you could say, yeah, it's kind of a mirror image of the other one. And it's a bit weaker because of that. This.

You like the giant inverted cone. I love the inverted cone. I love the idea that it's not actually a second rainbow. That's just a bit above the first one. It's actually a flipping massive one that's like inside out the next time my ice cream falls off my current. No no it's still fine it's just a giant inverted kind yeah I scream is still inside. giant inverted cone. Yeah. And the ice cream is still inside. Exactly. Yeah, there's someone having an ice cream and you slap it onto the ground.

This is the new, the glass is half full. Yeah, yeah, yeah, yeah. No, the cone is just inverted. Yeah. The cone is just inverted. Yeah. So let's get back to the colours. Without thinking too much, if someone said, what are the colours of the rainbow?

What would the QI answer be? Oh, the klexinable one. Yeah. Well, I know, I think this might be a UK thing. The mnemonic Richard of York gave battle in vain. Yes. Is like your way of remembering it. So, Richard of York, so red, orange, yellow, gave battle, green, blue, in vain battle. Green, blue in vain.

Indigo, violet. Yeah. So, those would be your classic ways of dividing the rainbow up. I learned from Roy G. Biv. Roy G. Biv. The idea that there's someone called Roy G. Biv. Catch ya. G, middle name G.

Biv. I was always annoyed by this because for me, indigo and violet are like pretty much the same. Agreed. to say purple so is Isaac Newton who coined it as seven originally. He said it was five colours in the rainbow which were noticeable they were red yellow green blue and violet and that was kind of what. and violet and that was kind of what the public consensus was. Although some scholars have noted that when Newton was saying blue, they would have meant closer to cyan. Oh, okay. Closer to this actual sky blue.

You know, when you think about a bright, bright sky blue, not like the bluey bluey. The dark bluey. But that sort of bright one. That is technically more cyan. So I believe, and this is me talking off the top of my head, from things I've half remembered, orange wasn't its own separate colour in the English language until

Shakespearean times. And it was named after the fruit as in like, hey, it's that red that looks a bit like that fruit we all know. Orange has been around longer than you're giving credit for. Oh. Yeah. So the word orange entered Middle English from Old French and Anglo-Norman orange. The earliest recorded use of the word in English is from the 13th century. Oh, wow. And referred to the fruit.

Oh, I take it back. The first recorded use of orange as a color name in English was in a description of clothing purchased for Margaret Tudor. Okay, I wasn't far off. The 1500s Shakespearean. Yeah. It was still before Newton. Now the reason that we have orange and

violet yellow red why we have rogbiv is Not because you can see two other colors, but because he wanted to Match the number of notes in a scale. Oh, did you know this? No, so ancient Greek sophists thought there was a connection between colours, musical notes, the known objects in the solar system and the days of the week. Oh, seven planets, seven days. Yeah, good job. And they're not counting, obviously, small body planets.

They should include the minor planets. Minor planets. And now my day of the week, please. Yes. When is that day? So to truthfully answer the question, it's however many colours you can see. Most people will say that there's seven colours. And this is where we get to how colour perception works.

If you want to put a limit on there, how many of the colours can you name? Yes. Let's say we're like, right, Roy G Biv, lock it in, that's the winner. You're like, yeah, it goes red, then it goes orange orangey red and then orange and then yellowy orange, then yellow. Yeah. And so on. And then greeny yellow and then all the way down. I think if I was to describe it, looking at a picture of a rainbow, and we've got to also bear in mind that I'm looking at on a computer screen.

So I'm getting a different colour from if I saw it. Yeah. Exactly. Yeah. So I'm getting a different colour from if I saw it. Exactly, yeah. So I'm going to look at the darkest part where the rainbow is meeting the edge here. Oh yeah, yeah. Yeah, I could go six. And this was also taken by a camera, I assume on some kind of CCD.

And the CCD might be sensitive to more colours than the human eye. So potentially, it's picking up ultraviolet and showing it. It should have filters to stop infrared, but technically might be showing as infrared. It's very obvious it's brighter inside the rainbow and very dark in the calling here. Yes, that's right. In Alexander's band. Right. I think I can name, I can see eight. Eight? Yeah. If I had, I would know what ones to go for.

So, I've got purple, blue, teal, green, yellow. You're packing them in the middle there. Orange, pink and then like a rouge. Yeah. I would say the colour I'm least confident of is red. I feel like it's kind of orange most of the way out and I'm convincing myself it ends in red. I will say the one thing we're kind of skipping around is it is a spectrum, ha ha, as in the frequency of the light, which we perceive as color, changes.

If you ignore quantum effects, it changes continuously. Yes. Changes continuously yes so it's just one continuously changing variable and our eyes just happen to be sensitive different amounts of different bits of light and. What we've kind of learned as colors we then perceive them as different colors but in reality is one continuously changing spectrum of colors and we're kind of semi arbitrarily drawing lines. And I will give it semi arbitrarily as opposed to completely arbitrary because of just how sensitive the rods and cones. I guess this is color. So it's cones detect things and the way our brain perceives them. I think we have got some biases. Maybe that's why we see more colors packed in the middle because our eyes are more sensitive to that middle greenish part of the spectrum. But that's me, 100% speculation. Or as they say in the business, spectrum-lation. Nice. So yeah, to get to the delicious stuff hidden in the vegetables.

In conclusion. There's just as many colors as you want. I know. Colors are in the eye of the beholder. I did toy with the idea of us renaming the colors and coming up with a better mnemonic. If any listeners want to do that, you're welcome to send them in. Based on what colors you view in a rainbow, the best thing to do is once you're outside and you see a rainbow see whether you can count different amounts than you could on a computer screen. Well, Beck.

I don't want to speak on behalf of Leandra or Lee or Ra or all of them. Or Leandra. Or Leandra. Or all of them. Or La Andra. Or The Andra. But I feel like even though your answer was as many as you want, you've answered the question, you've solved the problem, and it is down to what they said with that we can see,

because it's whatever you want to choose to perceive. So I'm going to give you a... Or it's five. Give you a ding... Oh, it's five! Yeah. It turns out the answer was always five. Could be two if you had, you know... Well, I'm going to give you a ding. Even if just for, it was an exhaustive answer.

Could you ding a rainbow? Oh my goodness. That's very funny. Our next problem, our dinglet, our wing ding. Yep. Tiny problem, tiny answer. Yep. Is from Charlie. God, Chaz. Or Charlie, tiny answer. Yep. Is from Charlie. Got Chaz. Or Charlie, if we're... Oh.

They say a broken clock is right twice a day. However, I realised that this is based on the assumption, which is capitalised. Assumption. That the broken, they've used double quotations there. Yeah. On both sides. They've gone all in. Broken nature of the clock means it has stopped.

But what about other breakages? What about a clock running fast or running slow or a clock with a broken cog for the hour hand that counts minutes all day but only advances the hour hand for half the day? How many times a day can a clock with various breakages be right twice a day? My semi-mathematical mind is picturing some very interesting functions being plotted, but I still can't work out if there is an answer other than TWICE. I love your pronunciation of double quote marks. Yeah.

Yeah. Interesting question. I mean, is the phrase a broken clock is right twice a day or is it a stopped clock is right twice a day? Like, I don't know if I didn't look up if there's a dominant version of that expression. I feel like a stopped clock is the less exciting version of the expression. And in that case, yeah, twice a day. It's been attributed to both Lewis Carroll. Yep. A.K.A. Charles L. Dodgson, who might be the person.

Yeah, might be Charlie, yeah. And writer Marie von Ebner EsEschonbach, an Austrian writer. And what form does it appear in their writings? So the earliest match was a quote in The Spectator magazine in 1711. Oh, wow. Even in the 1700s, dress fashions were ever changing. If one maintained a single clothing style, it would become passé, but eventually it would return to the mode so cyclical fashion idea. It's the same sort of thing if you hold on to your clothes eventually they'll be fashionable again.

So in this sentence they say if instead of running after the mode they would continue fixed in one certain habit the mode would sometime or other overtake them as a clock that stands still is sure to point right once in 12 hours. And then it goes on. It's a real, real snappy version of that phrase. Mm. So I was then thinking, well, how would we explore all the other ways in which a clock could be broken? If we're going to generalise this, as Charlie says, to not just not moving, but moving in all sorts of ways. Mm-hmm. to not just not moving, but moving in all sorts of ways. Now I immediately think about it as a graph. And Charlie didn't help because they said that they're picturing some mathematical functions being plotted.

And I was like, yes, that's exactly how I imagine it. So I'm going to draw it now. Because what we can do. Because when Charlie said that, I was like, I don't understand. Right. So here's on my horizontal, let's say X axis, that's time. And that goes from the beginning of the day, which we're just going to call zero, all the way through to midnight over here, which is 24 hours later. Okay. But then on our vertical axis, we have the time showing on the clock, which goes from zero again, same zero, which I'll label

twice, all the way up to this time only 12 hours Okay, and in theory we know that 12 would be like halfway through the day Yeah So if I was to plot a real clock, it just always shows the time it is so it starts showing zero I mean zero is synonymous with 12 So it starts there and then it goes up in a linear fashion on exactly a 45 degree angle until it shows 12 at midday. And then it goes all the way up to the very end of the day at midnight and then reappears over there. And that just repeats over and over.

Yeah. A stopped clock. So let's say our clock is broken and just shows six and then never moves. So it's a flat line, always showing six, doesn't matter what the time is. Oh, it's correct. There it is. Oh, it's correct. There it goes. Oh, it's correct. And then obviously at the end, it loops back to the beginning again. But the reason this is helpful, I thought, is now you can imagine other red lines representing other ways a clock can be broken. And importantly, every single possible broken clock you can imagine will be represented

by some kind of line on this space. So we don't have to think about, oh, is it because a cog is broken? Is it because the hour hand is too heavy? Oh, the minute hand goes backwards. Right? Anything you care to imagine is represented by a line moving around on this. And if the line goes down, that's the clock running backwards. And if the line goes down, that's the clock running backwards. Okay. And if the line goes up, that's the clock going forwards. And for anyone trying to sort of picture this in their heads, there will be photos and video on socials, but if you're unable to look those up right now.

So next is my axis graph, and then the lines are sort of going up in a diagonal direction as they go towards the 12 on the left-hand side, and then it stops because it's reached the top and then starts at zero again halfway through and it goes up so you've got two lines on this graph and then the other one is going from the y-axis that line all the way across all the way across yeah so if we want to have a broken clock that's never right we have to get from this side of the graph. From the beginning of the... The beginning of the plot, to this side of the plot.

That is pointing to the... To the end, without crossing the green lines that represent the correct time. Okay. So all we have to do is dodge those lines and the clock will never be correct. Right. So what we could do, I'll do this in a dotted line, is set the clock running, but actually gradually increasing in time. And then I'll be like, Oh no.

And then we got to the top and because it's a clock, we sneak on the bottom. Okay. No worries. So Matt has plotted a line that goes from six, roughly going, Oh, it's at now seven, eight, thirty, nine. Yeah. It's all over the place. Then it gets to 12.

It's running faster than the real time, slightly, some of it. Yeah. But actually, what I've kind of done is if you just had a clock running at the correct rate, but it's a couple hours fast, it will never be the correct time. Yeah, but would you count that as broken? That's where the question comes in. Yeah, what counts as broken? That's that's where the question comes in

Yeah, what counts as broken because if it's running a couple minutes fast or slow, it's never correct Can I make a suggestion? Yeah, I don't think this works immediately as soon as I start to actually think about it I suggest it if the clocks almost reached the time of day, but then started moving backwards Yeah, you'd still end up hitting it because what you'll do is you'd be flat, flat, flat, flat, flat, but then you'll start to go down. You're still going to hit. So again, just trying to describe what Matt is pointing out. So it'd be going from, uh, it's six o'clock all the time.

Yep. We start to get to actual six o'clock in reality. Oh no, it's about to be six o'clock soon. What are we going to do? And even if you start trying to decrease the time on the clock, it's just going to hit the correct time sooner. Yes. Yeah.

There's no way to dodge. Because then that's not flat and like, so that's not only flat anymore if it starts to move. Yeah, it'll be sloping down. So it stays six o'clock, but then it decides it's going to go backwards. So the clocks can go backwards in time, but they can't go back in time. Yeah. So if they went back in time, this line't go back in time. Yeah, yeah. So if it was...

If they went back in time, this line would go back in the other direction. Yeah. I think that's cheating. So right, because the X axis is actual time. I mean, it could. This is where my brain is broken. For completeness, if the clock managed to sneak around, what would happen is as you go forward in time, and a clock clock anti-clock pair would spontaneously peer,

then you'd have three clocks for a while and then the anti-clock would collide with the original clock and annihilate and you'd be left with just the new clock. I think that checks out from a particle physics point of view. Yeah. What if it was stopped? Yep. And we're, we're getting towards? The correct time. Correct time. Yep.

It's almost six o'clock. Yep. And then suddenly it starts to work again. Yep. But it's like an hour behind. Yeah. Maybe it might wibble wobble. It's an hour.

It's 58. What I would love is it gets almost the right time and then rapidly accelerates all the way up to 12 and around the other side and comes back in here. It sprints the long way around the time to avoid hitting the real time. Yeah, but it would still cross over at some point. No, it would be like, so it's, your clock's broken at six. I think I was starting to get there with my, with, cause it's the same sort of thing, isn't it? The idea that it's behind the actual time.

Yes. So that means we would hit 12 o'clock in reality while it's still saying like 11. Yeah. So you can be running slow, but also inconsistently, which thus makes it broken. Broken, not just the wrong time. Yeah. And therefore it wouldn't ever be right. Yeah.

You can definitely have a clock that's broken and never right. But just got to dodge that time without falling afoul of our categorization of broken versus just showing the wrong time. Yeah. You can also have fun making it right more than twice in the day. So if you ran it slow, you could arrange the speed it's running at so that it's only right once in the day. If you run it fast, you can be correct any multiple number of times in the day you want. My favorite of which would be to have it running 61 times too fast. Because what that means is in normal time, every time regular time goes forward by a minute, the minute hand would

go forward by 61 minutes and be back where it should be at the right point. Yep. Now the hour hand, we're ignoring that for now. Oh, you could run it. So every minute it does 12 hours in a minute because then it would be just a blur of movement. Yeah. But technically it would come back into alignment once a minute. So it would be right every minute, but never it would come back into alignment once a minute.

So it would be right every minute, but never in between. Yes. And very difficult to read. Yes. But very funny. So here's a fun broken clock. The hour hand goes forward at the correct rate. That cog's fine.

That's what I was going to ask about because Charlie sort of made a similar suggestion. Yes. Someone's reversed the cog and made it too big for the minute hand, so it goes backwards 61 times too fast. And that would still tell the correct time every minute that you care to look at it. Ooh, nice. But you just, but you wouldn't know when that moment was to look at it. And then she'll got a reference clock next to it.

Yeah, yeah, yeah. Which mildly defeats the purpose. So you know it's right, you just don't know when. I mean, that's what's missing from the saying, a broken clock is right twice a day, but you don't know when. Like it's right, but only if you independently know the time. Matt just used one set of quotation marks. I'm trying to bring the average back down to the correct level.

Okay, yeah, yeah. Yeah, no, but it's true, isn't it? Because people will say it is in like, you know, well, even if it's not working now, we'll work at some point. Yeah, it'll work in the future. Yeah. But you won't know when until it's too late. That's not useful. No.

So in conclusion. What if the clock is half melted like someone's garlic style? Like a deli clock. Yeah, that's a good point. What if it does operate in a different dimension? What if this graph was a cube? I like the one, the clock that can go backwards and forwards in time, because then you'll have different numbers of clocks appearing and disappearing on the wall.

Yeah, exactly. And the number of clocks at any point is what hour it is. So you count up the clocks. You have to count them up. Ignore what's on them. Man. The number of clocks going forward in time at any point. So the answer is, yeah, yeah, you can contrive a ridiculously broken clock if your definition of broken is generous enough.

That's right for any integer number of times a day, or continuously right for a while and then not, et cetera. But I question the premise, is a broken clock right twice a day. If it's only right in retrospect, given you have to know the time to start with. I think the answer is no. The correct answer is to get a binary watch. I'm gonna give this a ding dong ding dong. I'm going to give this a ding dong ding dong. Oh, how are you? Ding dong ding dong.

And we'll put my very confusing plot. I think we should- I'll label it. Stain this with tea and burn the edges because it looks a bit like a terrible pirate map. It does. Well tonight we could do a fire pit tonight. I think that's this year's Christmas card. Christmas comes twice a year.

If Christmas never changes. Christmas card. Christmas comes twice a year. If Christmas never changes. Welcome now to any other business. Yes. Which is a bit like Chlorophyll. Oh dear. In that there's lots of versions of it. So Beck, someone wrote in with a comment on our zeroth floor chat. Yeah, we were talking about in France, my friends live in a building that has a zero level and a zero plus.

Positive zero. Positive zero level. So, Ken wrote in to say they were in Paris recently and on the first night their five year old broke her arm. They said she's recovering nicely now, but they had real trouble having to leave the hospital at one point. And then the next day when they came back in, still couldn't find the ward, really struggled. And it turned out-

If it's one place you want to be obvious it's a hospital to get places quick. They turned out, so Ken knew that their daughter's room was on the ground floor and they knew the department in the room number, but couldn't find it. Eventually, I gave up and asked. Turns out the hospital had two ground floors. What? In this case, they weren't labeled zero and zero positive, but something like RB and RM. What's going on there?

Who knows? I guess two ground floors are a common thing in France. Do not approve. Ken also adds, as a general PSA to any EU citizens out there, you're not covered for health care in other EU countries, unless you have the special blue European health insurance card. What? I had to pay 2200 euros out of pocket for that hospital. Luckily, we had travel insurance, so we'll be able to claim it back.

But honestly, I thought I was covered for that simply by being an EU citizen. I mean, certainly by not being an EU citizen and being a UK citizen. That's changed for us, hasn't it? So yeah, thanks for that Ken. Good heads up. A lot of Americans hearing 2000 euros and thinking how quaint. I know, right? Yeah.

Oh, that seems pretty reasonable. Did they give you an aspirin? That's it for the episode. Thank you so much to everyone for listening. A super thank you to our Patreon supporters for listening. We appreciate everyone pretty much equally, but we really like the fact that Patreon enables us to do this podcast. And for the fine people who voluntarily pay money to make this podcast possible,

we pick three of their names at random to thank them every episode. And they have the bonus reward of us mispronouncing their names, which this episode includes... Ahem. Christ Ian? Yep. Hell-ings? Nailed it.

D. Ang. Oh. Robin. Turt. Lesson. That's a turtle, son. Well, thank you so much to everyone to listening to episode 092 of a Problem Squared. I have been Matt Parker. You're also listening to Beck Hill and our producer, much like chlorophyll, that makes

all life on earth possible until Matt Parker arrives. Oh no! Is our producer, Lauren Armstrong Carter. And that's it for the episode. Bye. Don't want to close my eyes. Don't want to close my eyes. I don't want to fall asleep. That's because that's, um, that's from Armageddon.

Not Deep Impact, the other one. Yeah. Deep Impact by Matt Parker. That's a different film. Matt Parker Gerdin. Oh, you should have called it a Masteroid. Oh, I've got another Masteroid. You have to get another one.

So this was a suggestion from the real Jim Brady. Right. Who wrote into us. Just setting up my ships. Why don't you explain that? Yes. They suggested that as a post-credit thing, we start a game of Battleship, which I like very much. Which we're doing right now. Yep.

So you can play along at home if you like. Yeah. You can play for both of us because you won't know where anything is. We won't listen to your suggestions, but... Oh, who goes first? I don't know. I'm going to say E6. E6.

Miss. All right. A1. A1. Miss. You may have noticed1. A1. Miss. You may have noticed I've got a systematic approach. Until next time. Oh, it just fell out.

No, no, no.