A Problem Squared - 139 = Music Down Where It’s Wetter and Throwing Darts Better

Episode Date: July 20, 2026

🎺 What would standard instruments sound like if they were played under water?🎯 What would be the average score of one randomly thrown dart?📂 And there’s some Any Other Business.AOB.zipOn THIS episode of A Problem Squared we’re joined by a special guest! The musical (and regularly quite wet) musical maestro Steve Pretty tells us all about playing music underwater - illustrated with some very bubbly examples. You can hear more from Steve on his podcast Origin Of The Pieces: https://www.originofthepieces.com/ and hear more from Steve playing shells here: https://www.shellectronica.com/ This is the episode of Steve Pretty’s Origin of the Pieces with Richard Sabin, whale expert and lead curator of mammals at the Natural History Museum, who shares sonic insights from the deep. https://www.originofthepieces.com/episodes/episode-31-earwax-echoes-and-entropyTo find out more about Aquasonic, the ethereal underwater quintet, head here: https://www.betweenmusic.dk/aquasonic Here is the full Hermeto Pascoal video in all its glory: https://www.youtube.com/watch?v=lZbfNtDCHdM And here’s the video of a hydraulophone https://www.youtube.com/watch?v=tgU0OZkGhGI World Darts Federation Playing and Tournament Rules for all the specifications used in Matt’s working out: https://dartswdf.com/storage/uploads/fb05b306-c92c-4f08-b512-affb092a1b3d/2018-02-28_WDF_Playing_and_Tournament_Rules_rev20.pdf The fabulously detailed Data Genetics blog post on where to aim on a dart board to maximise point-scoring - A Geek Plays Darts: http://www.datagenetics.com/blog/january12012/index.html Here’s more about the (excellent) book that Bec shouted out: Gemma Correll - Anxietyland https://www.gemmacorrell.com/ If you’d like to see Bec’s Work In Progress shows, her upcoming gigs can be found here: https://www.bechillcomedian.com/tour-gigsMatt has extended his tour - and he’s filming the show in London in October - all the dates, venues and tickets can be found here:http://standupmaths.com/shows Join 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 a problem problem underscore problem underscore squared dash final underscore the 139 dot zip. This is the problem solving podcast which is a lot like a dot zip file in that it's just secretly a bunch of things all bundled together and if you send it to your parents they probably won't be able to open it. I'm Matt Parker, a mathematician, performer, and a bit like a dot-XLS file. Oh. In that, I'm weirdly old school.

Love doing lots of calculations in a spreadsheet, but also stop after 65,536 rows. Yes, famously so. It's famously so. Many people are saying it. I'm joined by Beck Hill, a performer, writer, comedian, and a lot like a dot WMV file in that she combined. It defines visual, audio, and text-based entertainment, but yet, for some reason, doesn't work on a Mac. Oh, God, you've got me. Thank you.

I'm really proud of that joke, by the way. And on this episode... Music, is it better down where it's wetter under the sea? Oh, my goodness. I've worked out how to be average at Darts. And there'll be some any other dot BMPs. Yeah. It's dot Aob.

Is that a file type? Is it? It's an audio object file found on DVD discs. Whoa. Wow, there you go. We're an audio object file and we're outdated like DVDs. Beck underscore hill. WMV.

How are you doing? I'm great. I have recently consumed from some very enjoyable, what I say, media, I suppose, art. And it's made me very happy. And so I just wanted to talk about two things I've enjoyed. We're not sponsored. I've just really enjoyed them. Great.

Yeah. Other art are available. Yes. I saw the Beetlejuice musical, which I have no real, I think I saw a it like once or twice when I was younger. I feel like I never watched it the full way through or I did and I just never really understood it. I don't think I've ever seen the film.

It's one of those things where I remember really enjoying the stop motion moments and sort of the direction. But like a lot of films that people think of fondly, I do remember going, oh, it's a bit creepy. Like, not good creepy. Like, oh, no, this makes me uncomfortable. Yeah, yeah. So I sort of was a bit like, eh, but I went for a friend's birthday because they really wanted to go.

And I'm really glad I did. They fixed the plot. Oh, good. They fixed the plot in the musical and they've made it very, very, very funny. I would not take children. It is not child friendly, but it is a very, very funny show. And I was pleasantly surprised. Oh, there you go.

And as someone with audio processing disorder, and I've talked about this before, I find musicals quite, difficult to enjoy because if most of the artistry is within the way the lyrics tell the story, it's sometimes very hard for me to. So I haven't seen Hamilton because I feel like a lot of that information will be very fast and I won't understand the storyline because I can't process that information when it's in the form of a song. Yeah, it's out of respect for you that I refuse to sing on this podcast. Yeah, thank you. I appreciate that. You're welcome. Whereas with this, I found that I was actually able to follow it along. The dialogue's really, really solid.

The musical styles really lent itself to being able to actually hear and understand the lyrics. But the staging, everything was incredible. Everything was incredible. The actors, the stagecraft, there's a lot of magic elements that they've put in for effects and stuff. It's really great. I really, really thoroughly enjoyed it. Oh, and it was written by an Australian, the lyrics and the music by Eddie Perfect, who is, I wouldn't want to be quoted elsewhere saying this,

but I would say if you can't get Tim Minchin, get Eddie Perfect. Actually, I would say I really like Eddie Perfect. And if anything, I probably was much more familiar with his stuff. So that was good. That's one recommendation. And the other thing was Gemma Coral. I followed her on Instagram for a very long time.

She's an incredible artist and illustrator. she's done a graphic novel sort of memoir, which is all about her anxiety and her experience with her anxiety called Anxiety Land. And she sent me a copy. And quite often when I get sent a copy of a book, I find myself being like, now I've got homework.

Yep, yeah. I sort of put off opening the package for ages because I was like, I'm so stressed right now. And then I rip eventually, I was like, you know what's good for de-stressing, sitting down and reading something? Yep. And so I started reading it, and it's really fantastic, and I found it. We've had very, very different experiences in our lives,

but there was a lot in there that I really related to and a lot that also helped me understand stuff about my friends who have anxiety. I just really wanted to recommend it, Anxiety Land. It was a very quick thing to get through, and at the same time, incredibly insightful. and weirdly made me feel very, very hopeful. So, yeah, I wanted to give that a shout out. Oh, good. I mean, it's just amazing that you almost didn't open the package

because you were too anxious about it. I know. And actually, I think I'm going to have to message her to tell her that because at least she'll understand why there's been such a big gap between her sending me the book and me saying anything about it. I mean, you'll struggle to find someone more sympathetic to that problem. Yeah, yeah, exactly. How about you, Matt, have you been?

Well, you know what? But in the same theme, I also went to a new form of entertainment. Go on. Because I enjoy watching sport. I'll watch all sorts of sports. I had never been to an Australian football game. Really? I'm reacting like that.

I don't think I've ever been to an Aussie rules football game. I've now seen one more than you. A friend of mine was going to a Docker's game. So in WA there are kind of two teams. there's the Eagles and the Dockers. And I don't really follow or support footy, as in Ozzy Rules, but I will on paper be a Dockers supporter. When they made the grand final back in 2013,

I did go into London to watch the game at like 5 or 6 a.m. or whatever it was, something ridiculous. I went in, watched the grand final, and we lost horribly. But I'd never seen a live game. And so my mate was going for work. He had to like take some people for work. And he nominally follows the Eagles. But he was like, well, we're going to take them overseas visitors.

We're taking them to an AFL game. Do you want to come along? And so I was just kind of there to keep him company. And I was like, yeah, I've not never been to an AFL game. And I will explain Aussie rules football for any of listeners who aren't familiar with it. Oh, yeah. It's more foot-based than American football. but more hand-based than British football.

It's like we took an American football and decided to play basketball with it and occasionally you can kick it. You're not far off. You're not allowed to throw the ball, but you are allowed to punch it. You can dribble it. You can dribble it.

That's true. You can bounce it. But you can't throw it to someone. You can catch it. Catching it is a whole thing. People love catching it. it. Yeah, you're right. You do. You punt it with your fist. I forgot about that. That's how long it's been since I've watched Ozzie rules. Yeah, because you're not allowed to throw it to someone.

You can't hold it in one hand and then you punch it out of your hand with your other hand. You use your other hand like a polo mallet. It honestly sounds like I'm making up an Aussie game. But no, you just, you punch the ball at people. It's like now play croquet with your fist. Yeah. And then you get to jump on a guy and that's a game. There you are. When you explain it like that, I'm surprised I don't watch more of it. I'm not to say.

So anyway, I went along to see a game and the dockers are just having such a great season. They've won like their 12 consecutive games or something ridiculous coming into this game. Their top of the ladder. Like real dream season. and I was like, oh great, perfect time for me to get back on the bandwagon, go see the docker's play. Incredible. I go along and they just start losing. Just awful start to the game.

At one point, coming in pretty much at the half time, they had 28 points to Geelong, the other team's, 56 points. They literally had half as many points as the other team. And I was like, oh no. and I sent a selfie of me at the stadium to my mate Trent and Trent was like I didn't tell him I was going but he knew immediately what was going I think he was listening to the game at home in the UK and he was like they're doing so well and the one time you're there they're losing and he was like you have to go and like buy a hat or something and so I ran down at half time to the merch place and I

bought a hat here it is here's my docker's hat I ran down and got it's the cheapest hat they sell. I wasn't, you know, not made of money. Very stylish. It's just, it's a standard cap with a docker's logo on it. Okay. And once I bought the hat, they started their comeback. They scored like 10 goals in a row without missing and then they won the game. So the hat, you know, I know correlation isn't causality, but you need correlation to have causality. And the hat, I think the results speak for themselves. So I've now got my lucky hat, which is pretty exciting. Do you think, does it work for other sports? I will find out. I, I didn't wear the hat when I went out in Singapore at three

in the morning to watch the game where we did not do well. So it's your file. I'm hoping if we can make the grand final again, I'll wear this hat. I'll find somewhere to watch the game and I'll put the hat to its test. So there you are currently one for one on the hat. So they are. So they That's my, it's my hat update. A lot of sport updates of late. I think there should be more hat updates. Well, we'll see how this hat goes. So far as like it.

Well, I think you've put a pretty good cap on our intro. Shall we jump into the episode? Let's do it. First problem today has been sent in by someone named Calc Dave. I assume that's a Dave who loves Calc. Don't tell me if that's not true, Dave. I'm really hoping it's calcium, Dave. Cal, it could be calcium Dave.

Yeah. Bloody loves some milk. I think it's a big fan of calendars. CalC Dave. Calendar collector Dave. That's who we got. I prefer Milky Dave. Milky Dave.

I'm not saying that again. Calc Dave. And Calc Dave went to the problem person page at a problem squared.com and said, with summer here and pool parties happening. Calc Dave was thinking about underwater music. That's a heck of a pool party you went to, Calc Dave. And they have a few related questions, which they have enumerated with letters.

Question A, are there any instruments that are designed to be played underwater? Question B, how would standard instruments sound underwater? When they elaborate that wind instruments would have a hard time. Yeah, good point, Dave. But what if you stuck the end of a trumpet? in and kept the mouthpiece above the water. Would a guitar, drum, piano or string instrument actually sound okay or would they need modifications?

And finally, question C, would you consider designing a quartet of instruments that would sound really nice being played and listened to underwater? Beck, you've taken Calcdaves A, B and C, and you've done some research. Well, by research, I mean... Yeah. The cornerstone of this podcast. Yeah.

You know how I accidentally knew an expert in misophonia? Yeah. When we had a question about misophonia. And let's not forget the original dentist Sophie. Precisely. So my friend, Steve, is not only a musician, but I have on many occasions seen Steve play a conch. not just play a contra, but like turn it into a playable instrument.

And I thought, oh, I bet Steve will know something about this. Because he's also got an incredible podcast called The Origin of Pieces. Yeah. We'll put links and stuff in the show notes. I was like, I bet he could answer this question. And so I sent him a voice note with these questions said, hey, do you know anything about this? Not only did Steve send me a voice note back saying that he was actually a great person for me to ask this question.

to. But also, he was on holiday in Italy and he spent some of his holiday time listening to a voice note from me about underwater instruments and taking the time to respond to it. And I really appreciated that. And I thought, do you know what? Let's just get him on the show. And he's here right now. Hello. Funnily enough, I was actually kind of at a pool party. Yeah, I was on holiday with my family and my kids basically spent the entire time just in the pool. So I was literally basically at a pool party when you called. And you didn't like immediately grab a trumpet and put this to the test? I didn't immediately grab and trumpet and put it to the test.

But part of the reason for that is that I have already put this to the test on several occasions, in fact. Amazing. Yeah, Steve, actually, before we go any further, will you introduce yourself to our listeners? Yeah, so my name is Steve Pretty, and I'm a musician. I'm a trumpet player and a composer, and I have a podcast, as Beck said called Steve Pretty on the origin of the pieces. about kind of where music comes from and the people who make it and the reason it exists and how it evolved and things.

But I also, I should say, I'm a very keen scuba diver. So I've done a lot of scuba diving over the years and I have, I supplement my extremely lucrative income as a jazz musician and composer with an even more lucrative income as an underwater photographer,

which as you can imagine is such a lucrative business. This guy hates money. I love spending money on making... I really love spending money. Yeah, exactly. I'm fascinated by water and sound. And as you mentioned, Beck, I also, increasingly these days, seem to spend a lot of my life playing shells, again, because I hate money.

Kind of conch shells and similar shells like that, I played like a brass instrument. So I do that a lot. I've just come back from a festival where I actually hardly played trumpet at all, mainly play shells. But also, because of this interest in the underwater world, A few years ago, I was supposed to be musical director for a show at the Royal Albert Hall called Sea Shambles. Now, unfortunately, the show had to be cancelled because of the pandemic, so it didn't actually go ahead in the end.

But one of the things I wanted to do in that show, and genuinely, I was researching how we might do this, was to play and conduct the music of the show from inside a tank with a scuba gear run. So we were genuinely, myself and the producer were researching how this would happen if it could happen. As part of that, I went to the London School of Diving and I convinced them to let me use their training pool to film and record a load of underwater sound stuff with my friend Helen Chersky, a scientist. And so she and I strapped a load of scuba gear

and I took a load of instruments down and played them underwater. So I have actually done some of this myself. So I didn't need to know what a trumpet sounded like underwater because I have played one five metres down which is quite fun. All right. Well, I guess this might help answer question B that Milky Dave sent us.

Stop calling them Milky Day. They wanted to know how would standard instruments sound underwater. So what instruments did you play underwater at the diving centre and what did they sound like? Being a trumpet player, I played mainly brass instruments and win instruments, but also some drums.

I took some drums underwater. The first thing to think about with this question is that wind instruments, like the way that we are used to hearing sound is obviously through the air, right? And so what we're doing when we're listening to sound is listening for vibrations in the air. Now, of course, underwater, it's a very different medium. It's much denser medium. So water transmits things in very different ways, which I'll come to in a minute. But the first thing is that wind instruments obviously use air to create the sound. So you use buzzing lips like that.

and then that vibration through the air, starting those lips vibrating, is what creates the sound, right? And then a trumpet or whatever amplifies that. So the problem is, because they expel air, what you're getting is just a very bubbly sound, like, because you're having to push air through the instrument and then against the water. So it's quite hard to play physically, because you have to blow quite a lot harder in order to get the air out through the instrument to start with, and then it creates essentially a bubble. So you get like a bubble of sound that goes, blubbblub, you know, so it does sound a bit like a trumpet, but a kind of bubbly trumpet,

if you like, like blowing bubbles in, you know, putting your lips in water and going it sounds a bit like that. Yeah. So the sound of the air in the water is louder than the sound that the instrument makes, so to speak. Yes, because what you're hearing is you're hearing the because I'm breathing in air right from either from above the water if I just put the end of the bell of the trumpet in the water or when with scuba gear on I'm breathing in air from the tank and then blowing it out right you're hearing the air kind of bubbling out into the water so and then that I suppose creates its own pressure ways within the water so you do you can still hear it underwater really I think in order to get a proper representation of how sound works underwater you need to not use air as a media

if you know what I mean. You need to use water as the medium to start with. And so the drums work a bit better. Drums work a bit better because that is a vibrating skin that then physically moves the water, right? That does work. But what you find with water is just a very, very, very different experience of sound as anyone who's put their head under water, you know, will know in the bath or a swimming pool or whatever. Everything sounds very, very, very different. And there's a couple of reasons for that. It's much, much denser. But another reason, is that sound actually moves much, much faster through water than it does through air. It's much more efficient transmission medium. But you get this what's called a kind of filtering effect, right? So the high frequencies get filtered out much more quickly because the energy gets absorbed much more quickly with the shorter wavelengths. And so high frequencies get lost very quickly in water. But low frequencies, conversely, can cover enormous distances.

On my podcast, I did a whole episode interviewing Richard Sabin from the Natural History Museum, who's the guy who put the hoped the whale skeleton in the Hinty Hall. I talked to him about whale song and underwater sound. And one of the things that I found really interesting with that is that whales use sound in a couple of different ways and different species of whales use them in different ways. But if you think of like dolphins and other sort of small dissertations who are communicating with echo location, that's all very, very high, right? the kind of clicks and pops and stuff.

And that's because you can get a very, very accurate picture because water is such a good transmission mechanism. You get a very accurate picture by sending out like bursts of very high frequency sound. And so because it's so dark at depth, they're getting a 3D picture back into their ears and then their brains of what the bottom of the ocean looks like or schools of fish or whatever.

And the amazing thing about that just briefly is that they can send out a burst, but then everyone else receives it. So it's this sort of amazing way of sharing a sound, but which is really sharing a vision. So they can kind of send out a burst of sound. And then the whole pod get back that vision of what, does that make sense? Yeah, it's like taking a photo and putting it on Instagram.

So all your friends see. Yeah, exactly. Exactly. Just like this podcast. We make a sound. And everyone gets to receive. That's why it's called a podcast. It's, I see.

Very nice. Very nice. Thank you. Thank you. Just stop listening now. everyone, it's not going to get any better. But the other way, of course, is by like, if you think of Whale Song, and that is mainly very low, it's more like, you know, they're sending out bursts of sound through the water, if you like,

as the transmission mechanism rather than air as the transmission mechanism, if that makes sense. But it, but it's significantly faster, right, so air is about 340 metres a second speed of sound in air, and the speed of sound in water is 1,500 metres a second, so, you know, it's a lot faster. Firstly, for things like echolocation, it's much more efficient. But also, once you get to the low frequency sounds, they can transmit over enormous distances. There's evidence that whales are communicating over hundreds or even thousands of miles because they're using extremely low frequency sound,

lower than the limit of human hearing, which is 20 hertz, in other words, 20 cycles per second. Wales are going below that and are able to kind of transmit messages or whatever to each other over really long distances using that kind of subsonic frequency. So we know they're speaking English, but it's so low, we can't hear it. Just very low, just like, yeah, like an operatic base, but more. Yeah. Whale opera.

Now, whales don't speak English. They speak whalick. Well, not Welsh. Come on, Matt. I thought about Welsh, but Whalick made me giggle more internally. So I wondered that. Part of the reason to talk about dolphins and whales is that water, interesting things to sound in that compared to what it does in air so it filters out

the higher sounds and almost magnifies the lower sounds and so what that means is it's for one reason and another it's much harder to locate particularly lower sounds where when you're in water especially as a human because of the denser transmission substance you're in in other words hear water it sort of surrounds you a bit more than it it does in air, particularly in those lower frequencies. So it all feels a bit strange, listening to music underwater, listening to sound underwater. This might be a physics way to think about it, but are you saying because the high frequencies fade out faster, water is kind of a low pass filter where we're just kind of losing the top end, if you will? A hundred percent. So yeah,

it's exactly that. And that's what we describe in music production and mixing and things as well, a low-pass filter, or the way it can sometimes act, particularly in the sea, where you've got lots of different topography of the sounds bouncing off different rocks and different things in the sea and different densities of water and salinity and all sorts of things.

You actually can get more of what's called a band-pass filter, which is where you get different bands being amplified and other ones being squashed. Like a sort of DJ filter, right? When you think a DJ will go, like, and the underwater sound, yeah. Exactly, yeah, yeah. You know, whenever you're listening to sound in the air,

you're hearing the original source sound, but you're also hearing whatever it's reverberating off. And actually, I think the best way of experiencing this and the way that I've experimented with a bit as well is by getting an underwater speaker and playing sound that you've made in the air with whatever conventional instruments you like. played underwater if that makes sense because that way you get a sense of what

music that you're used to hearing sounds like underwater. So here's the real question. Do you have any examples of these sounds being played underwater that we can listen to? I certainly do because I've got some hydrophones which are underwater microphones which I record stuff with. Because of that pool session was for the Seashamble show, we did record it all and I took all my underwater filming gear and all my hydrophones and everything down. So do you want to, do you want to check it out how that sounds? Yeah. Yeah. Okay. Here we go. I mean, yeah,

you weren't kidding. That's what. That is some, some musical bubbles. Yeah, it's basically musical bubbles. But again, it's because the air is, you know, I'm blowing air through. And so you're hearing the air bubbling out of the trumpet. Right. Normally in when you're listening to a trumpet in air, you're hearing a stream of air coming to you, right? But here, of course, the way that the physics of the air and in the water works is that it bubbles out. It sounds how I imagine boiling a trumpeter would sound. I think like that, but a lot more angry. I'd be quite furious at them and started boiling me.

So, yeah, and you can hear the drum. It sounds a bit more like a drum, but what you find is you don't have the sustain. In other words, you lose the sound more quickly than you would in the air. So I took a big bass drum down. If you hit it with a big beta, you normally get boo, whereas in water it just kind of goes doop, doop, because the water sort of muffles the vibration much more quickly. But yeah, so A, are there in instruments designed to be played underwater?

Not very many, partly because we don't listen to music underwater, so there's not that much of a market for it. Strings do vibrate underwater. They dampen much more quickly because it's much denser. So you could conceivably get a string resonating underwater. There is a project from Denmark called Aquasonic, who did a kind of quartet of instruments underwater that they'd made especially. It's kind of interesting. I don't know anything about it.

I don't have any connection with them other than just when I've been interested in this stuff in the past. I've sort of looked them up. Do you want to hear a bit of them? Yeah, yeah. Does sound very otherworldly. If you're going to write underwater-style music, because there's something about the underwater world that is very, very otherworldly. I think you probably, as a musician, you're inclined to write, you know, write in an otherworldly sense in some ways as well. It's nailed the vibe of being underwater.

Yes. Yeah, yeah, yeah. I should say we're watching a clip. All the performers are underwater. They're not using breathing apparatus as they're in tanks. I'm guessing they sort of lift their heads up occasionally and they turn off the microphones in each tank so they can take a breath. There's some instruments that I've definitely never seen before, which appear to have been made for this. And then others, which sort of looks similar. There appears to be a violin that have been maybe modified to be played underwater.

But also we do have an answer to question C here as well. It's not quite a quartet, but it does look like there is a quintet. It does. I mean, I should say, if anyone wants to fund me making an underwater quartet, I am all in. It may not surprise you to hear.

There's a couple of other underwater instruments or water-related instruments that are quite interesting, one of which is called the hydrolyphone, which is where you have essentially a water pump driving water through some sort of tube or similar with some holes in it. So it just looks like a kind of slightly pathetic fountain. So, okay, so the actual tube is not underwater,

but water's being forced through it. Water's being forced through it, so it's not an underwater instrument per se, but it does use water directly. Yeah, it pushes water through, and then you cover you the holes, with your fingers. So in other words, you block the water from spurting out. It's like a messy flute. Yes, exactly. And that pushes water and I guess air through a kind of little resonating chamber, I think. So that's quite fun. I think there's a couple in sort of public parks and stuff in the US,

which are quite fun. Oh. And we're going to have to play a clip of one now. Which sounds, it sounds very sort of pure and flutie. It's quite nice. I like to think that someone got a leak in there, like a plumbing leak or something. and they were like, ah, and all these holes kept appearing. They were trying to cover them up with all their fingers. And then someone was walking past. They're like, that sounds amazing. It does sound quite sort of almost the opposite of the underwater ones.

It's like if you play underwater, you get creepy. If you play water above water, then you get sort of fantasy, mystical, lovely. Yeah, I think it's just the way that it's pushing air and water through the particular resonating chambers. So it's really just about how you design those. I think the water is sort of the thing that's driving the pressure. Oh, so it's basically just a wind instrument, but you don't have to blow it with your mouth. I think so. Quite fun. But probably my favourite sort of water musician, if you like,

is a Brazilian musician called Hermeto Pasquo, who sadly died last year. And he is a very quirky musician who plays all sorts of stuff. including he played a piglet on one of his albums. He played a live piglet. He's very, very respected in kind of jazz and world music type circles because he's an extraordinary and very, very virtuosic, amazing composer and musician, but also not afraid to muck about with water and piglets and like. But there's the video that I saw many years ago,

and in this video he's playing a flute, and he ducks under the water. It's in a beautiful kind of waterfall in Brazil, I think. in Brazil, I think, and he ducks under the water and fills the body of the flute with water, which because of it shortens the length of the tube with water, but then, of course, the water flows out. It kind of has this effect where, of course, a shorter tube means a higher note, so it goes, woo-hoo, whee-floo! Like a slide whistle. It's exactly that, but because he's such a great musician, he does some really interesting stuff with it.

He also does water percussion with his hands, I've seen him do, and you do sometimes see this. I've seen this, even in like classical music sometimes, in contemporary classical music, bowls of water with close mics on, where you can use water droplets and you can slap the surface of water and get some quite interesting. By cupping your hand in particular way,

you can get some different textures and stuff. Well, we'll play a little bit of Hermeto Pascal. I love that. I think you answered those three questions quite solidly. But if anyone wants to find out any more about underwater instruments, underwater sounds, or indeed any sounds or instruments, where can they find you? So probably my podcast is the best place to start, and that is Origin of the Pieces.com. And episode 31, I think it is, is with Richard Sabin from the Natural History Museum,

which is an episode all about underwater sound and underwater music. Wales and all that stuff. But just another quick shout, although it's not technically music that's played underwater, the stuff I do with Shells, my project with that is called Shell Electronica.com. And the process of making an album of Shell-based music, Shell-based music. Yeah, we'll put those links in the show notes so everyone can go ahead and listen to that episode and many more. And also check out Shellotronica. Steve, that's very comprehensive. You've hit A, B, and C. Back by extension, gets all the credit for fighting you as an expert.

I'm sorry, these are the rules. I was about saying I don't make them up, but we literally made them up. And I think, I mean, obviously we'll go back to Calculator Dave to get the official word and if this solves their problem. But I'm prepared to give you a ding or more accurately a... That was me saying ding into a glass of water. You've been waiting to do that for so long. I've stopped drinking water so I would have enough left.

I sacrificed my own hydration. Today's a hot day. I don't think it was worth it in hindsight, but here we are. Well, thank you so much, Steve. Thank you, Milky Dave. Calcium Dave for sending in your problem. And thank you everyone for letting me take the credit for Steve's knowledge and hard work. Our next problem comes from the front.

frog. Ah, the frog. And the frog says, on a standard dartboard, okay, multi-talented frog then, on a standard dartboard, what would be the average score with one randomly thrown dart given the relative areas of all the different scoring segments? Ooh, that's a good question, the frog. Good problem, yeah, look at that. So I was like, okay, well, this is fun. I like darts.

You can get a dart frog, can't you? That's a type of frog. A dart frog? A dart frog? Isn't that a poisonous... Hang on. Oh! It's really colourful.

I see what you're doing. A poison dart frog. Yeah, poison dart frog. Yeah, you're right. Good link. And I was like, you know what? I just got to look up, this would be a very straightforward problem. I'll look up the dimensions of a dartboard.

Because for those of you who aren't familiar with this, or you've only got a vague memory, your standard dartboard, you got the bull's eye right in the middle, very small circle. That's worth 50 points. You then got a slightly bigger ring around that. that's worth 25 points then you've got 20 like slices of pizza like sections coming out of the middle

and they're worth between 1 and 20 there's one of each there's one with 1 2 3 all the around and each of them contains then one section where you get double whatever the number is and a different section where you get triple

whatever the number is and so I thought I was got to look up the dimensions of how big how big is the bullseye, how big is the ring around it? How big is the triple section of each wedge? I'll just calculate the area of every single region.

Use that to weight the points you get and then calculate the average score you're going to get. So I looked up the World Darts Federation playing and tournament rules 20th revised edition. From 2018, I think is the most up-to-date version. of the rules. And I scrolled down through that. There's a bunch of things about where you stand, how you throw, what the scoring is, blah, blah, blah. And here we go. Section 4.01, dartboards,

the WDF approved specification. And they got a bunch of things. Interestingly, they do specify the type of board. There's a bull in the middle. There's a ring around it. There's the treble and double rings, all these things. The numbers 1 to 20 in a clock pattern. I can't see anywhere here where it specifies the order those numbers have to be in, which when I was discussing this previously is looking at possible other arrangements, which I think is fascinating. But that's a whole separate side quest I'm not going to get into now.

And they specify how thick the wire can be between the sections and all of this. And then right down the bottom here, they've got all the measurements. The inside diameter of the bull has to be 12.7 millimeters plus or minus 0.2 millimeters. There's the tolerance on it. Then the diameter of the ring around that, 31.8 millimeters plus or minus 0.3 millimeters. And so they've got all the dimensions, accurate to 0.1 millimeter, which is pretty nuts. And then they've got all the tolerances of how fine. away from that official distance you can be.

But then I realized, well, hang on, here's a problem, because depending on the tolerance, that will subtly alter how big each region is. And there's two extremes here. There's the one extreme where the single scoring areas are as big as possible, the bull's eyes as small as possible, triples are smallest possible. So everything is, there's like the extreme version where

the overall average is bumped to be as low as possible. Right. And then there's the other version where the bull's eye is as big as possible and the single value areas are smaller and the triples are bigger. So there's also the other extreme where everything is at whatever the max limit of the tolerance is to get the biggest possible average. And so I had to do it twice. So I went through and calculated the minimum extreme version.

and for the maximum extreme version, did all the weightings, ran it through. Does this include how much space the wire between the segments takes up? Now, it does specify the type and shape of wire. There's different ones allowed, how thick it can be. However, my assumption was you never really get the dart hitting the wire and bouncing out. If you hit the wire... You're not trying out enough. it goes one way or the other.

So I just took the wire basically is effectively not there. Because if you hit slightly either side of it, it just tongs into that bit of the board. Yeah. Unless you're me. In which case, it's definitely flinging off. It flies dangerously across the room comically into someone's foot. Yeah. Yeah.

Even though it would be harder to do that than anything else. You bet your sweet butt that I've done it. Yeah. So I ignored the wire. So if people have an issue with that, let me know. But I figured it doesn't happen often enough to worry. I wouldn't know how to discount a one in some crazy number chance of it bouncing out. Anyway, the point is, ran the numbers.

And I can now confirm that the minimum possible weighted average of all regions on a dartboard is 11.43 points. And the maximum possible value is 11.47 points. So it's not a big difference. So there's a tolerance of 0.04 points. So right in the middle, 11.45 points. Does this mean that hypothetically, you're more likely to hit the 11 than in the other number? No, it does not. No.

This is assuming you are, and this is a little bit artificial, but it's assuming you are equally likely to hit any point on the board, but you're not going to miss the board. This is only assuming you hit the board and that every point on the board was equally likely that you would hit it in a circular pattern. I don't think there is a very valid assumption. It answers the academic question of if you weight every region of the dartboard by area what's the average value. And so I can say the average value on a dart board is 11.45.

What it doesn't answer is if you're throwing a dart randomly because I don't think when you throw a dart randomly, it's a perfectly even distribution. That seems very unlikely. Yeah. What you're probably more likely to have is like a normal distribution where if you're aiming for a point,

you're more likely to be closer to the point than further away from the point. But there is still some randomness. Unless you're me. Unless you. Well, that's just me. How big is your distribution, right? So if you're a pro player,

real tight distribution where you're way more likely to hit what you're aiming for, but there's a small chance you'll be a little bit in a different direction. Because even the best players don't hit triple 20 every time. Like they've still got a distribution in their shots. They're not perfectly accurate. And then if you're wildly inaccurate, you've just got a much fatter bell curve. See, you say that, but that's assuming that my hand. and body will be aiming where my brain and eyes are looking.

And so the aim of my brain is quite often very different to the aim of my hand. So the distribution could be wildly in a different direction. Oh, right, right. Yes. All of this assumes a perfectly even circular distribution. No, not even, is in like, it's more likely in the middle, but it's not saying that we're less accurate side to side than up and down. And I suspect that's not true. Would you find that they just, this is a guess, like this isn't a math thing, but would you

find that they're more likely to end in the lower part of the board because of the arc that they, of when you throw them? I know it would change with speed. I suspect you're right. I think we're probably more accurate side to side than up and down. I think up and down, and probably not even up and down equally. Maybe down is more favored. I don't know how.

I couldn't come across any research into this. So if anyone knows, let me know. I don't know how we would work that out. You could maybe get someone to throw darts over and over and over again and track where they land and maybe work out what kind of distribution shape you get at different skill levels. You know who would be really good for that data? Who would be good for that data? Is it you?

No, any of those... You know how there's been a huge surge in, like, fun darts places where they project different games over the top of the dartboard? And you can play different games with your friends and stuff. And they auto score, I guess, right? So they're detecting where they land. Yeah, surely they're collecting that data. Why wouldn't you?

I worry they're only collecting what region. They may not be doing sub-reasons. They may not be doing sub-region precision, but maybe they are. Maybe I'm wrong. If anyone knows, I'd love to hear about that. But when I looked into, if anyone had run the numbers on Dart accuracy, I came across the data genetics, a blog post from many years ago. They ran the numbers on this.

And I've seen a few people who have come up with very similar results, but they've got the best graphics. We'll link to this in the show notes. But they basically worked out for every level. of accuracy for different skill level for different players where you should aim to get the highest average score possible. And so even though that's not what the frog asked, it's 11.45. I thought I would just quickly go through this. So depending on where you think your skill level is, where you should aim. Okay. For the most part, if you think that you get 95% of your shot,

within 10 millimeters of where you were aiming, 95%. That's like the vast majority of the time you can look at a spot and hit within 10 millimeters are there. So what I'm saying is a standard deviation of 5 millimeters, for those of you who are already fluent in such things, then you should aim for the triple 20. That's triple 20 all the way. If you're super tight, like incredibly accurate, obviously, you're. you're going to get an average of 60 points every time.

If you're not so great, if you're closer to 10, you're going to average about 43 points on average per throw, if you're that level of accuracy. And actually, that holds pretty true all the way out to about, if you're getting 95% of your shots within 30 mil, you should still be aiming for triple 20. You'll be averaging a lot less, averaging 22 points, but still, that's not bad.

once you're out to 95% of shots within 40mm, it flips. Suddenly you should be aiming for the triple 19. And the reason is 20 is between 1 and 5. So if you start hitting them too often, they're going to bring your average down. Whereas 19 is between 3 and 7, which are both higher. And actually, further up from 19, you've got 16 and 17 and 8. There's a bunch of other high ones over there. So once your precision gets bad enough, you should flip and start aiming for triple 19.

And then as your accuracy gets worse and worse and worse, the point you should aim for basically moves up and kind of clockwise and spirals in on the bull's eye until once you're terrible, absolutely awful, you should be aiming for just to. the left of the bull's eye. Here's the whole spectrum. If you're very good at darts, aim for triple 20 every time, apart from the end game. If you're pretty good but not incredible at darts, aim for the triple 19. As you get worse and worse, you should be aiming a little bit further up towards like the 16. And then if you're bad, aim for the edge of the outer ring around the bull's eye directly to the left of the bullseye because the left side of the board is actually slightly higher values than the right side of the board but you want to be aiming as central as possible to avoid missing the board so the left side of the outer ring around the bull's eye that's your best spot and if you aim there your average

value is actually worse than the average value of the entire board you would be averaging about 10.9 points per throw. But that's because you can't throw evenly distributed. And so actually, you're pretty unlikely to be hitting, because if you're aiming for the middle, you're going to be hitting the double scores very rarely because they're right on the edge. But in terms of realistic throwing, the best model I can come across is aim there. Well, I'm going to give it a ding because you've answered it. But as always, we will ask the frog. Well, we don't always ask the frog, but We will refer to our listener. Yeah, we will ask the poison dart frog.

See if that's a ding. See if anyone wants to add it to the spreadsheet. And now it's time for any other dot aob, where people send in corrections, follow-on, additions and dings from previous stuff we've done on the podcast. And Beck, what have we got this time? We heard from Julia, the Julia, who contacted us after we did our ASMR episode and begged us to never do it again.

Yep, yep, valid. So Julia wrote back and said, Hello and thank you. I feel very heard and happy that you more than solve my problem with whispering. Hello to my fellow misophonists. It seems like there are many of us. I learned for me it is absolutely about context. Whispering to not wake a baby or in general,

someone's sleeping is no deal. But in other contexts, It just really angers me. So an even bigger thank you for the no whispering guarantee. And here comes a ding, neither whispered nor yelled. I will also check out the recommended book because I love the interview with Dr. Jane Gregory. And if you're interested in listening to that episode, it was episode 137. We also heard from Daniel, who gave us a ding, ding, ding, also on the Misophonia episode.

They've gone and got the audiobook, sounds like misophonia. They say that it despise the sound of people and animals eating so much it makes them physically ill and sometimes have to fight a rising anger. Wow. They try to hide the sounds by wearing earphones or their chainsaw rated earmuffs. Wow. But that amplifies their own internal sound of eating, which is much worse. I can imagine.

Yeah. They basically just chalked it all up to their autism, but they now say that maybe the book sounds like, help. They love the podcast, our various YouTube channels, which are both equally updated frequently, and they have a bunch of my books. So that's great. There you are. Daniel, glad we could help. Yeah, thanks, Daniel. We also heard from Connor, who wasn't writing it about this, but I thought it's very sweet. Just to say, not a problem, just wanted to submit a comment. This morning, I drove by a Beck Hill Road in Montana, and then they've sent a Google Map link. It is Beck with a K,

but I'll accept it. Hey, hey, take it. Thank you. I appreciate that, Connor. I love that. It's nice to know that occasionally someone might drive past a specific sign and think of you. Yeah. Now you know, there's a hotspot in Montana of people thinking about you.

That's hilarious. Yay. Well, that is it for the episode. Thank you so much for listening. Everyone, we appreciate your ongoing support, listening to the podcast, sending in problems, sending us things, etc. Whatever your level of interaction, thank you very much. However, we do like to bonus thank the people who make this possible our fantastic Patreon supporters who go to patreon.com slash a problem squared

and voluntarily give us money. And that holds the whole thing together. And to thank our Patreon supporters every episode, we pick three of their names at random. And so, wow, okay, I'm about to say to mispronounce, and I've just noticed the first name has a character in it. I genuinely have no idea how to pronounce. So well done, whoever this is. So we're going to mispronounce three names that this episode shall include...

Paul Elf Ronzac. Duh, nice, tough. Ander, ew, brown. So thank you so much to them and all our other Patreon supporters. And that's it. Thank you for listening. We'll see you next time. The final thanks goes out to our producer, Laura underscore,

Grimshaw, forward slash. No extension, just a folder that holds this entire podcast together. Aw. Right, Beck, connect. Wait, which color are you again? Are you red? I'm green. You're green.

Oh. I would like to go to the right of the middle. My right. That one. Up there. Bold. Okay, cool.