A Problem Squared - 138 = Crossword Panels and Crossing Channels

Episode Date: July 6, 2026

📝 What is the perfect arrangement of black squares in a crossword?🚣 Does weight on an aqueduct increase as a boat goes over it?🛟 And there’s some Any Other Boardness (not ‘boredness’)Head to our BlueSky/Twitter/Instagram to see some crossword arrangements and some cross-bridge arrangements.Michael Keith’s Crossword research paper: http://www.gathering4gardner.org/g4g16gift/KeithMichael-GiftExchange-CrosswordGrids-G4G16.pdf Michael Keith’s Pilish book, Not A Wake: https://www.cadaeic.net/notawake.htm 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 is bringing An Evening of Unnecessary Detail to New York! The Bell House, Brooklyn on Sunday 12th July. Tickets can be found here:https://www.ticketmaster.com/event/300064B84CCCF567And Matt 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 Squared, the problem-solving podcast, which is a lot like spring boards. And then we provide a solid platform to launch things from, such as creative solutions, clever conclusions, and the occasional joke which we refuse to let die. I'm your host, Beck Hill, a comedian, writer and winner of the 50-meter freestyle event at the 1999 Vale Park Primary School Aquatics Carnival, which I think more than quality. qualifies me to dive headfirst into your problems. And I am joined by your other host, Matt Parker. That's me. A comedian mathematician and a man so committed to the bit that he was recently banned from his local Lido

for flinging pensioners into the pool while yelling, I'm a springboard. It's an analogy. I'd written a non-spring intro for the next episode. Good. This needs to die. It doesn't. And on this episode, I've had a quick look across all crosswords. I've done some research into boat bridges.

Oh, and of course we'll have any other boredness, comma, spring. Hello, Matt. How are you? I'm doing well. I'm back in the UK. That's exciting. Yes. I came back via Singapore, as previously advertised.

Yes, you were doing a talk. That's doing a talk. For my good friends at Jane Street, they run like a... a program for undergrads who want to get into financial coding and trading and such things. Because this was a change to my travel plans, I was in Singapore when Australia's second World Cup game was on. Ah, yes. Have you been watching the World Cup back? Nope.

I only know how certain games have turned out based on sounds of people around me. I'm aware of how that match played out because at American. It wasn't great. ...confront of hours got suddenly interested in football. Oh, that would do it, yeah. I was really looking forward to the game. This is the game where Australia was playing the USA. In the USA.

I was like, that's going to be a fun game. However, because of time zones, it was on at 3 a.m. And I was like, oh, am I going to get up at 3 a.m. to watch the game? And I'm like, I'm pretty sure I am. And then I realized, no. None of the channels in the hotel were showing that game. There was no streaming service that I could get to work that would show me the game. I tried to VPN to watch it on SBS and that didn't work,

which I think basically every Australian was doing everywhere. SBS being an Australian channel. Back when I was living there, it used to be one of six channels and it was the one that you watched if you wanted to see boobs because they would always play foreign films. They play foreign films, yeah, yeah. So I realized if I wanted to watch the game, I would have to find a local sports bar or somewhere. I don't think there's a city on earth that hasn't got at least one Aussie themed pub or bar.

So I was like, right, I'll just do a bit of research, find somewhere that's showing it and just wander over and see it. I found one place. Now, Beck, I need you to help me out here. I'm not going to name and shame here. I'm sending you the image that this bar had on their Instagram about the World Cup games they were showing. Okay. World Cup, 26 fixtures.

And then they got a big list. Now, just from your reading of this list, given they say Saturday, 20th of June, United States versus Australia, 3 a.m. And then they say live and loud, at and then the name of the bar, would you say it would be logical to assume if you walked across the city to get there, they would be showing that game live and loud at 3 a.m. Yes.

Oh, well, I have some terrible news for you. I got there closed as anything. Like 20 minute walk across Singapore. Oh, and for the record, there are not many cities, even as a lot. large Caucasian male where I would just set my alarm to walk out at 2.30 in the morning across the streets of a city, but Singapore is just a weirdly safe in that regard city. So I'd found this place about a 20 minute walk away, wandered across the pretty quiet streets, got there, and it was completely closed. So I'm now standing at 3 a.m.

next to a close bar in a foreign city. I'm assuming that they didn't stay open until 3 a.m. on Saturday night slash Sunday morning when there was no match on. Ah, right. Yeah, you think maybe it was a, the Saturday they've written there. Would have been a Friday night. Quote, unquote, Friday night, but 3 a.m. Saturday morning. So you think maybe they opened up the following night and they're like, whoops.

Yeah, that's what I reckon. I reckon that someone there misread it. There was like an Aussie backpackery kind of place that was open and pretty busy. But the game wasn't on. It was just like loud music and like young people enjoying themselves. But then on the horizon, a Turkish bar was open. And they had two screens outside the bar because a bunch of outside seating basically on like the footpath, the sidewalk. And they had two screens on showing the game.

Great. And I couldn't believe it. And so I got a beer. It felt like a kind of classic Southeast Asian, pretty minimal bar. Like you sit outside of the table in a plastic chair and there's some TVs around. Yep. And so I was able to sit there at three in the morning. It was perfect.

It was like quite a nice temperature to be outside at 3 a.m. and I sat down at a table by myself. And there were a few other tables of people watching the game. But then my table became the table for people who decided to wander the streets of Singapore and three in the morning just in case somewhere was showing the game. So it was me, another Aussie guy and an American all sat on this table. And all of us had gone out and gone where we thought the game was going to be on and it wasn't. It sounds like the setting for a play.

You know what? It did feel a bit like that. And then we all ended up like moths to a flame drinking tiger beer outside at three in the morning watching the game. The game itself was a letdown, but that little adventure was a lot of fun. Well, that was riveting. Thank you, you know, twists, turns. I never thought anyone could make a story about trying to find a bar that showed the Aussie match into a 10-minute story. And you did it, Matt. Unless producer Laura edits it down, in which case, well done for making that story so succinct and enthralling.

What are you talking about? That took me like 40 minutes to explain to you about it. I wanted to recreate the feeling of watching the game in story form. Yeah. Where you finally get to the end and realize there was nothing. That was it. Oh, that was a waste. No, I like that.

I love the camaraderie that comes in those moments where you realize you're in the same boat as a bunch of other people. Especially that one of them was an American because there's this vibe of like, we're in this together and yet against. And yet. Yeah. So, Beck, have you had any exciting 10 minute-esque story situations that you'd like to bring up? I mean, I did spend three days deep cleaning a bathroom, but I don't want to really relive that. Was this a problem of your own making? No, it was deep cleaning someone else's bathroom that I had offered to do because when I'm using someone else's facilities, like semi-regular, I want them to be ones where I feel cleaner coming out of.

Yeah. And I wouldn't say, look, if it was super disgusting, I never would even set foot in the bathroom. But as a suffer of IBS, I spend a lot of time in bathrooms. You do. And I eventually said, hey, if you buy a bunch of cleaning stuff, I will deep clean this bathroom. More for myself, really, than anything. I guess it doubles up. It's productive and helpful and also deeply selfish.

Yeah, exactly. And I've done it for multiple people. I've got friends in Dublin who they used to always. like put me up whenever I was staying in Dublin. And so I deep cleaned their kitchen for them. I think I deep clean their bathroom at one point. This is not a new thing, usually by way of thanking people if I stay somewhere, I give something a deep clean if I'm there long enough. I think there's something really nice in setting out to do something and then afterwards seeing the result. Yeah. Because there's few things in life where you can immediately see the product of

your work. So there was something very enjoyable about that. I didn't need. need to take three days. I definitely prolonged it because I've been putting off other stuff. And at risk of getting into like way too personal territory, there's some stuff I was like probably mentally dealing with as well. But I did in totally different news. Yeah. Elon Musk has become the first trillion. Well, there's a gear change. It's a real, real gear change. But I had a realization the other day, which was what is? What is? if he does for capitalism what he did for Twitter. Oh, that's interesting.

So with Twitter, he was like, I'm going to become the king of Twitter. I'm going to be the most, the biggest, I'm going to change the name, and it's going to be mine, and I'll have all the followers. And then he did that, and everyone went, you've ruined it. And a bunch of people left. We're all leaving. Yeah. Yep.

What if he accidentally starts the socialist revolution? Like, what if just enough people go, oh, well, I guess. money's not really worth anything. If you can have a trillion of it, we'll just stop using it. I would love that so much if he accidentally changed a system that has become so corrupted that I think a lot of people, not all I understand, but a lot of people are in agreeance that there is, that the system is inherently flawed and problematic. But it's been hard to change because we all live within the system.

I love the idea that him doing this has proven how dumb this system is and we can just like stop using. That was his real superpower, the whole. Yeah. I mean, potentially it could expose the ridiculousness of, and I'm going to walk this back in a second, by the way, of the way we do IPOs when companies first go public on a stock exchange and the way the evaluation works for that. and the ramifications of these index funds who have altered rules to add space X in early. So you're right, the whole notion of what it means to be worth a trillion dollars. Could have some knock on effects there. But that said, the Tesla share price has been so detached from any meaningful real world logic for so long.

I've lost any confidence that these things will ever, ever snap back and have any ramifications. But I'd be very happy to be proven wrong. I understand that this is like the most political we've ever got on this podcast. And I'm aware that there are strengths. I just think it'd be really funny. Everyone went, okay, well, how about instead of using money, we just like do skill swaps. Everyone's like, wait, money's made up, and then that's it. Yeah, yeah, exactly.

And on that bombshell, uh, Patreon.com slash a problem squared. Now that money's worthless, why not give it to us? The reason that I wanted to say this at the podcast is because in the past, I would have tweeted about this. But no one will see that tweet on account of him ruining Twitter, which further proves my point. Yeah. I got to go on to the Today program, BBC Radio 4 Today program, to explain how big a trillion is. Oh, yeah. And you just go, it's big.

It's big. Yeah. And as you know from my previous sample pie show, I did the same thing on BBC News like 14 years prior when the UK national debt first hit a trillion pounds. Right. I know they're different units, US dollars and pounds. But I think it's very interesting that over the span of, you know, a decade and a half, a trillion pounds has gone from something that, you know, a nation state can't pay off to a single human has. I think that's an interesting little journey. How much of that is to do with inflation?

Oh, like a billion percent. Well, shall we solve some problems that we feel a bit more within our reach? Let's solve one trillion problems. Yeah. Our first problem comes from Adam on the problem posing page, which is a problem squared.com. And Adam says, I have a crossword conundrum. I mean, that's not the sort of problem we're supposed to, you know, solve. Yeah, what's seven down?

Yeah, yeah. I'm obsessed with puzzles. Yeah, actually, I have that problem too. Okay, yeah. I recently had to block Pusmo. Anyway, that's another story. and recently gone into the idea of making crosswords myself. The main issue I found when building my first puzzle

was that I kept wanting to add more and more blacked out squares to just reduce how many long words I needed as I thought it would be easier to connect a lot of smaller words than fewer longer ones until I suddenly had two little squares to even make a puzzle. Okay, so hang on. Just so that I make sure I understand. Adam reckons that a crossword would be easier

to make with lots of small words. Yes. I mean, whether or not that's a true assumption, we'll have a chat about, but that's what Adam is attempting to do. I've never made a crossword, but I, I am, the way I think I would do it is I would just come up with all the words

that I want to have as the answers. Good first step. And then figure out how I can make them all fit together and then put in all the black squares. Yes. I think that is what Adam is trying to do. But Adam is putting in the black squares first. Yeah. Which to me feels like a harder task. It is a harder task, but it's not strictly wrong in terms of how crossword puzzles are made. So I think they've got the right idea, but hopefully we can, we can get them a nudge in the right direction.

So Adam goes on to say, my problem is, if I had a grid of 10 by 10 and wanted to add a perfect amount of black squares to reduce the amount of long words I needed to think of without limiting the space, I have to work to the point of it being impossible again, or even just blacking out the whole grid, is there a definitive number to use? Love the show and hope you can help. And they've put a little smiley face. I've never really, I do a lot of cryptic crosswords, but I don't think to count the quick. cryptic in the Guardian is smaller than, you know, the normal one. Yes. I'm guessing they're the same number of squares each time? They should be the same number of squares each time.

Obviously, that can vary. Yeah. I think standard ones are around 15 by 15 boxes. Okay. But obviously, you can make a mini size you want. I don't have to be square, but that's convention. Yes, yeah. Now, do you want to hear the two things I didn't do trying to solve this?

problem for Adam. Sure. Why not? Number one, I didn't talk to my friends who actually make crossword puzzles. Yes, because I think between us, we probably know quite a few. Correct. Right. And that would have been one way to go about this. But that's not quite what Adam's asking. And secondly, I didn't get my big list of all the words and analyze how long they are, how they overlap, all that kind of thing. I didn't get involved in, you know, the English language in any way trying to solve this problem. Because honestly, as much as I love those ones, I was like, this is becoming your teeth. A hundred percent. And I'm like, I've got a, every now and then as a treat, I'll do in all the words. But I'm trying to, try to spread them out as much as possible.

What I did do, however, was have a look into just the geometry. the combinations of where and how you can put black squares in a crossword grid such that you end up with a valid arrangement that could then have the words filled in to become a puzzle. So just a straight black and white, where can you put the squares such that you will end up with a crossword puzzle? And it turns out different countries have different conventions for what you have to do to make something a valid crossword puzzle. Oh. Particularly American crosswords have a very different layout to British crosswords,

and it's because of the rules and requirements for how the words overlap. Okay. I'm so invested in this. I'm strapping in. I've sent you an image back with no context and we'll share this on social media people want to check it out. It's got a side by side of an American crossword and a British crossword. Oh, okay.

Have a look, yep. Yeah. This makes so much more sense. Tell us about the differences. Okay, so the British one is correct. Okay, this is fascinating. The British one typically will sort of more closely resemble a bad chessboard. It's got a feeling of like a lattice with glitches in it.

Yeah, if you think about a finished Scrabble game and then you just filled in all the grid that doesn't have letters on it, that would probably more closely look like a British crossword. Yeah. The American crossword, it's so American because what's, they've done is they've tried to cram it in with as much, like, we're going to put in all the word. Like, there are so few black squares in the American crossword. There's only 32. I just counted them real quick. You're saying this is a format that American crosswords do. So it's basically where almost every letter of every word intersect. Yes. In fact, you've stumbled exactly across

what is the major difference between a British crossword and American crossword? So there are some common rules across all of them. So first of all, and again, you can break any of these rules. If you want to make your own crossword, go nuts. I'm just saying these are the kind of accepted conventions across the two countries. In both cases, the grids are square with odd sides. So it's 15 by 15 or 13 by 13, etc. It tends to be in newspapers.

Secondly, the arrangements of black squares is always symmetric if you do a 180 degree rotation. On both types of crossword. On both types of crossword. Really? So if you look at the two I sent you, if you were to turn them around, like spin them 180, they would look the same. I've never noticed that. Yeah, isn't that interesting? I know there's a lot of people listening going, how did you not know that?

But I'm sorry, I'm too busy looking at the clues. Yeah, you just you solve it too fast to even stop and process. I really don't. Every word has to have at least three letters. So you can't have anything shorter than three. Some crosswords will have a house rule that four letters is the shortest. But that's an exception. They never have one smaller than three.

So three letters is the minimum length you'll find in any major crossword in either country. Yep. The surrounding edge cannot be completely black. No one outside edge is totally black because that would mean it's not a square or it's got a border semantics. And this is an interesting one that never crossed my mind, but now I know it. I'm like, yeah, the region of the white squares where you're going to fill the words in is one big contiguous region. You don't have two disconnected blocks of words that have no overlap between them, no path between them. All the solutions are all one big tangled mess.

They're not in two separate chunks as such. When I've done American crosswords, I've found them weirdly, while they look much harder to put together, because you've got to find words that completely intersect. and spell out other words, if you were to line them all up next to each other, it would be hard to set, I imagine. But in doing it, I find them so much easier because if I get all of the down answers, but don't know any of the cross answers, I get them anyway. You get the answers for free.

Yep. Yeah. And I don't like that. I like the feeling of when you're doing a British crossword, and you still have no idea what an answer might be, even though you have all the other answers. So the big difference, which you've just basically described, comes down to which letters are what's called checked.

And by checked, it means they appear in more than one word so you can check it's correct because it's got to match two words. And so it kind of rules out guessing. And in American crosswords, every letter in every word must be checked. So it means every single letter in an American crossword is in two different words, which is why it's basically almost all white squares to be filled in. And it's just a couple black ones to give a bit of structure. So words begin and end.

But every single letter is used in two different words. And so the whole thing is just a block of on top of each other overlapping back to back words, basically filling the whole grid. I'd be interested to know, because, what's quite fun with, at least with, I don't have enough experience with normal crosswords, but with cryptic crosswords, the British ones, usually quite often you'll find that there was some theme. The setter will use a theme that's sort of like a little surprise. Like, oh, look at that. All of the answers have a number in them. You know, so it could be, there could be one for,

Like one of the answers could be wait and you're like, oh, the number eight's in there. You know, so you can count all the way up to 10 or something, which is very enjoyable. But I don't know if that would be as easy to do with an American one. Oh, you just got so many words and they're quite constrained because they have to overlap so much. you've losing a lot of freedom for what words you can pick. And I know friends of mine do the listener crossword. I don't know that one. And have set listener crosswords.

Oh, it's not a cryptic crossword in the traditional kind of sense, but it's like a lot of meta puzzles hidden in the crossword puzzle style puzzle. Right. Your friend of mine, Katie Stakell's and Paul Taylor, a big listener crossword fans. And they do so well. at it, they get invited to the dinner for people who are great at crosswords. Amazing. And again, could have asked them about making crosswords because they do, and I didn't.

Now, speaking of British crosswords. Yes. Different rules on how many letters have to be checked, as in there have to be in more than one word. So the rules are, for every word, half the letters need to be checked. So in America, it's all of them in the UK. It's half of them. And traditionally, if it's an odd number of letters in your word, you round up for the number that are checked. But some crossword setters will allow rounding up or rounding down.

But the dominant convention is you round up. So a five-letter word needs three checked letters. On top of that, you cannot have three or more unchecked letters in a row. so the longest span in a word that's not checked is two. However, that's not allowed to be at the start or the end of the word. So the first two can't both be unchecked and the last two can't both be unchecked, but two in the middle can be unchecked as long as overall half the letters in the word are checked, which I know is a confusing sentence with me saying the word checked a lot.

But the point is, this is why they look very different. because in the American one they all have to overlap everywhere. And so you just get whole chunks of white where you've got to fill in words. And in the British one, because half of them are checked, you get what you were referring to before looking a bit like a chessboard because you kind of get these spaced out grids of blacks so that the words are forming a lattice structure around them. It's not every second letter, but for a lot of it it is.

Or on average it's every second letter. it's every second letter. So that's why you get that kind of vibe. Do we know if Adam is British or American? Actually, that's a good question. Did Adam spell anything that would give that away? Let's have a look. I can't see any words that Adam's written, which would definitively put them in one country. Because Adam says that they have, they want to add more and more blacked out squares. That suggests to me that it's not an American style crossword. Yeah, I think you're completely correct, Beck, because Adam wants to add more and more black squares, they're definitely thinking of a British style crossword puzzle where you have lots of blacked out squares and fewer words overall.

So I'm going to assume we're dealing with British conventions. And while we're admitting shortcuts we're taking, Adam did say a grid of 10 by 10. I mean, the fact that crosswords are odd seems weirdly universal. And all the research I've come across only looks at odd-sized grids. So I'm very sorry, Adam, I'm going to have to look at nine by nines and 11 by 11s. And you can decide if you'd rather go up or down. Given Adam's seeming obsession with simplifying and having fewer words, I suspect nine by nine might be the solution we're after here.

What's the reason for having this many rules? Like in my brain I thought I would just, you know, lay it out like a scrabble thing and fill in the blank spots. But obviously that wouldn't fit the conventions. Why are the conventions there? Is it just because people who like crosswords really like rules? I think that's it. I don't know why. Because you're right.

It could just look like a game of Scrabble without needing to be symmetric or any of these other things. I mean, I can see why the overlaps. you can have different opinions on how much checking should there be and how many clues should you get from other words as you go along. And America, it's like as many clues as you can, loads of overlaps, you'll get whole answers for free. Whereas the British one of always getting exactly half of it checked means you never get a word for free.

You still have to work it out and fill in the other half of the letters. best case scenario. And I think people just like conventions and rules and challenges. So, you know, constraints are fun. That's true. But if people know better, let me know. So the question now becomes, if we're just focusing in on British ones, how many possible arrangements are there that Adam could choose from?

And so I looked it up. And if you start real small, a five by five grid, there are only seven possible crosswords. And these are, like you could then flip them, mirror image, etc. These are like seven, you know, unique possible arrangements on a five-by-five grid. There's only seven different arrangements of white and black squares that fulfill all the rules. And so they're the seven at the top. And they've got different colors depending on what symmetries they've got.

and then underneath you've got the other versions if you rotate or flip them. That were also equally valid. But that seven at the top, as you can see, they're the seven different unique possible arrangements of white and black squares that make for a valid five-by-five crossword puzzle. My favorite is the second from the last that looks like a dollar sign. I think that's fun. It does.

It does look a bit like a dollar sign. It looks like something you would draw in the back of class on your desk. It does. It does. It does. if you were colouring in squares to make something. That's what you're doing your maths book. Yeah, yeah, yeah.

But Adam wanted bigger than that. If you go up to a 7 by 7 grid, 7 by 7, there's 182 possibilities. 9 by 9, there's just over 17.5,000. Oosh. 11 by 11, just over 15 million possibilities. Oh, my goodness. 13 by 13. There are 14.6 billion possible ways to fill in.

to fill in a crossword that are all distinct and are 15 by 15. The actual size that is used in most newspapers, there are. We don't know how many possible crosswords there are. Wait, what do you mean? We've only got as far as the one before. It's too hard to work out and no human. has done it yet. What?

Yeah. Ain't that something. Adam, if you love puzzles, I've got a puzzle for you. Well, exactly. So I'm taking all this off research by someone named Mike Keith, a fantastic mathematician. Some people might know them as the person who writes long-form text in pylish, which is where each word has the number of letters as the next digit. in pie.

It's pretty special. I both love that and hate that. It's so good. In fact, somewhere I have a copy of Mike's book, Not a Wake, which is not three letters, a one letter, wake, four letters. It's a whole book. I think it's the first 10,000 digits of pie where every word is the length of the next, anyway.

So I saw Mike speak about crosswords at the Gathering for Gardner conference. earlier this year. And when I saw this problem come in from Adam, I'm like, I know exactly who to talk to. So I emailed Mike. Mike sent me through all this stuff. He sent me the paper of the research, which I will share. We'll link to it below if you want to go into all the great details. And it's not super dissimilar to when I was trying to find five words, each with five letters, to have 25 distinct letters, which was a whole thing a while back, where you're looking for combinations of white and black

that have certain constraints between them when you put them together to make a whole board. And what Mike did was write some code where every row was represented as a binary number for zeros and ones where there are blacks and whites, and then was able to search through all of those and check which ones match all the rules. And when they sent that going on a regular computer,

solving the 7 by 7 grid took 1.4 milliseconds. The 9 by 9 grid. 0.79 seconds. Okay, a little longer. The 11 by 11, 36 minutes. And the 13 by 13, 84 days. Have I suggested that he put that on a YouTube video. and see

Let everyone optimize the bit out of it Yeah, yeah, yeah, yeah Because as soon as you say something With some level of conviction Because you've done the work Oh boy, are you opening up the floodgates Now, Mike has altered the code

So it can be run in parallel On multiple processes at once And you got the 13 by 13 down to four and a bit days but the 15 by 15 solution will take multiple years. There's possible other ways to maybe constrict it down to make it a faster search, but at the moment there's no way to do it. There's not going to be a multi-year computation to get a number that no one really needs. Oh, Matt, who really needs any numbers?

Hey, let's not pull on that thread. Look, as I say, every number counts. Hey. Thank you. There's an open challenge here. If people, like I said, I'll link to the paper. If people want to give it a go and see if they can improve on the approach that Mike has put together. Because it is both hilarious and frustrating that we have the answer all the way up to the one before the one we actually use.

So it would be nice to have the actual. definitive answer for the 15 by 15 grid that's actually used. But because Mike has found every possible arrangement of black squares for all the sizes, all the odd sizes up to 13 by 13, we can at least give Adam a good range to pick from depending on what Adam wants to optimize for. They want to add the perfect amount of black squares to reduce the amount of long words they need to think of without limiting the space they have to work with. Well, one thing that Mike did do was analyze

every single solution for the ones that had the best distribution of different word lengths. And I think that might be what Adam is after. So actually, I'll share with you, Mike's finding. Yeah. Here is for every size from 5 by 5 up to 13 by 13, the most even distributions possible. Yeah, because it's not like what puzzle could Adam make that has the fewest words.

No. I mean, Mike found those as well. We've got the fewest words, but that doesn't feel like the challenge Adam's after. Oh, this is coloured very pleasingly. So these have extra colours in them because Mike was labelling how the, arrangements are symmetric and Mike stopped putting all the outer black squares on. So you can imagine filling in the rest to make it the full square. These are just all the internal bits we could actually fill in the letters. If they're going to go nine by nine, there's two pleasing options. There's one

arrangement where there are two words of every length of three letters, four letters, five letters, six letters, seven letters, eight letters. Now that three is shorter than is sometimes allowed, but could work there. Or there's another nine by nine arrangement where there are two of everything from five letters, six letters, seven letters, eight, and nine. So they're really nice even distributions, not too many big ones, not too many small ones, a good number of words. So I would recommend either of those two nines is what.

what might be my suggested solution to Adam's problem. And of the two, I prefer the one on the left that's got the smaller words, which also seems to match what Adam wants. If I got the full data set off Mike, we could search for the ones that have the minimum possible maximum word length. That would be a whole bunch of small ones. I thought that Adam was still like some longer ones, but not too many, which is why I've gone for the even distribution ones.

But I'm very happy for Adam to request otherwise. Hmm. I never would have thought that this would have interested me as much as it did. Oh, good. But I feel like I know a lot more about crosswords now. Not since my story about watching football and Singapore. Oh, boy. It's been a real roller coaster.

Well, thank goodness for people like Michael. Exactly. So, Adam, I'm going to send you my suggested 9x9 one, and you can report back on if you'd like me to optimize differently. And I'll ask Mike for the whole data set. So I could search on it. But I'm going to send you my current candidate for your arrangement. I don't feel like I have the power to ding this right now. No, this is still a work in progress.

Yeah. But well done. I'm impressed. No further questions, Your Honor. Excellent. Our next problem was sent in by someone named Harry. Can I call you Hazer? I'm going to call you Hazer here.

Hazzar here went to the problem posing page at a problemsquare.com and said my colleagues and I were taking a lunchtime walk along the canal and came to an aqueduct where the canal becomes a bridge going over a railway. Okay, just so we're all picturing that correctly. You're walking next to a canal, which is a, you know, boxy river. And then it's got to go over something. So there's a bridge that is the river. Okay, got it.

Hazard and Pals wondered, does the weight on the bridge increase as a boat goes over it? So just to paraphrase that, if you're in a boat going down the river, when you go over the aqueduct, bit of the canal, are you increasing the force that's going through the bridge? Wow. There's a problem. Beck has decided to dive in. Yes.

I loved this problem. I love the problems where there are things where I would genuinely pose that question myself and then probably forget about it later. Great. Yep. And it's always really nice when you're like, oh, an opportunity to look into something that I would also ask. And I had a suspicion that I knew the answer, and it's always very pleasing

when you think you know the answer and then you research it. And you're like, ah, it seems to support my theory. And I understand that that would also bring some bias. But I did look for any arguments against it. and was yet to find any's. Because I saw this problem and I had a thought about what I think the answer is. And then I was like, I'm not sure if I'm right and it will take a bunch of research. So I then moved on with my life.

But I'm glad I'm glad you were more dedicated than I was to actually do the research required to solve this for us. Well, here's the thing, Matt. Would you be surprised to find out that the sort of, of question that I am drawn to is the sort of question that many people have already asked, and so therefore it's quite easy to find quite a lot of information about it. I struggle to believe that. So this was one of those ones where originally you read it and you're like, of course it would add weight. No, wait, no.

Hang on. No, it wouldn't. Ah, ha. Yep. And I did do it wrong in my head to start with because I was thinking about like, okay, well, when I get in the bath, the bath is heavier. Yes, that's true. And I was thinking of it in those terms, like, okay, well, there's a boat.

So obviously it'll be heavier, like, case closed. Yeah, job done. Thanks for going. But then was like, well, hang on, if I got in a river, does a river become heavier? Yeah, yep. Technically it does, but like by such a small amount, it's sort of imperceptible. And the reason for that, if I was to get into an amount of water, because it's, displacement. You know the Archimedes principle. He was in the bath. He noticed that the water level

went up. Yep. You displace the amount of water equal to the mass of the body that's floating. I think that's it. That feels right. Can I give you one of my favorite puzzles that covers the same concept, which I often go to and I want to think through these things. So imagine you're in a boat in a lake, closed lake, closed system. And the water is perfectly smooth. It's. It's just you sitting in your boat with a giant pet rock that you've brought with you. And after a while you think, you know what, I'm sick of this rock. So you take the rock out of the boat, you put it in the water, you let go, and you watch it sink to the bottom of the lake. Once that's finished and the water's calm again, is the overall level of the lake higher, lower, or the same as when the rock was in the boat?

Right, yeah. And I'm pretty sure the answer is the same. It will be lower. What? Because when the rock is in the boat to be floating, it's got to displace an amount of water with the same mass as the rock to be buoyant. And the water is less dense than the rock. We know that because it sinks. So the rock in the boat is pushing the boat down to displace a volume of water larger than the volume of the rock, but equal to the rock in weight. Right. And then when you put the rock in the water, it's now only displacing the amount of water equal to its volume, which is smaller than before.

Got it. And so the water level goes down a tad. I always have to think it through again, because I couldn't tell you the answer straight away. I've just got to go through that reasoning. But that reminds me of the difference between to be buoyant, you're displacing the mass of water equivalent to the object versus if you're in the water or at the bottom of the pool, you're only displacing the volume equivalence. Right. So I like it.

It's a fun puzzle. There you go. That is very clever. And luckily, not quite what we're talking about here. It's close. It's close, but also far more complex than what we're talking about. because we are just talking about buoyant things. And so if this bridge was closed,

then adding a boat to it would add to the weight in the same way that if I get in a bath, I'm adding weight to it. But because it's an open system, the boat is already in the water. Yep. So that has already been taken into account with the weight that the bridge has to carry.

Yep. I'm sure there's a world where adding multiple new boats to the system is problematic, but that's not the problem I'm solving. Plenty of space. I think it's interesting that by this logic, if you've got a long canal and you put a boat in at one end, when you put the boat in at the beginning of the canal, it increases the weight on all the bridges all the way down the canal at effectively the same time just by getting in at one end. Yeah, precisely. And I imagine that's why, if it's a big enough system, it's sort of being built to take that into account, right? Like if new boats are being added.

Yeah. I mean, I think we're arguing the exact details of it in a technical sense. I think in reality, and I could be wrong, I suspect the mass of the boats or the buoyancy, leather boats being in the river, the mass of them, is so small compared to the mass of the water, it doesn't matter. Everything's built, just whole. the water and then the boats are a trivial increase on that. Yeah, that's probably also true. And again, it makes more of a difference because it's an open system. It's got to carry that much water that is like connected to all the other water as opposed to just the amount of water that is over the top of the, what was over the top of? Railway line. Essentially, when the boat passes over the top, the amount of water being displaced as to where it is, is being pushed away from the bridge. So the weight isn't changing as the boat passes over. I did see some stuff about, there's an interesting photo which I'm

always hesitant to share images that look quite interesting these days because I'm like, okay, but is this a real photo? Is it real? But it was one of like a super yacht going over a very, very, very large aqueduct going over a motorway. And a lot of the discussion was about, I mean, the super yacht as it enters, there could be like a small wave and that would mean that you would get like a little bit of a change, not a huge amount, but you get a little bit of a volume change as the wave enters the bridge. And then it would get lighter as the wave moved away from the bridge. because you're sort of moving

sweeping the water off basically. If you're moving at speed, agree. But then a lot of people did also point out that I'm pretty sure if you were trying to drive, if you're trying to what do you do with a boat? Captain? If you were trying to captain sail. Yeah, if you're trying to get a boat

a super yacht that's worth millions and millions over an aqueduct that is going over a motorway I don't think you're going to be going pretty fast. I think you're going to be going pretty slow. I was thrilled to see that once I realized, oh, hang on, this is different if it's not a closed system, that my argument seems to be supported.

Well, Beck, I think you're right because you agree with what I thought. So I'm going to say great argument, well supported, evenly distributed load. We will see if. anyone writes into complain, I guess. I think that's the only remaining test. Yeah, it wouldn't be, there have been moments where I've backed up a claim that I had. It all made sense and then had several people of authority explain why I was wrong. And actually, I quite enjoy that.

So. If not so much we got to wait. I mean, I guess Harry can ding to say if they're happy with that explanation. So Harry, has a. Let us know if you're happy with that. But I think more to the point, we need to wait for the absolute. absence of anti-dings before we're confident we can go ahead with this. That's fair.

So if you're happy with that, don't send us an anti-ding. There we are. That works. Yeah, great. It's a real cliffhanger of an episode. I'll tell you that. And now we come to any other boredness, as in spring boring? Oh, that's better. This old episode's any other.

This whole show is any other boredness. So, in any other boredness, two people, Christian and DePeterson, both on Instagram, started chatting about ways you can remember the digits of the number E, which we discussed in a previous episode. We talked about the number E, not specifically remembering the digits, but they both have a mnemonic you can use. As long as you start by remembering that E begins 2.7, so Christian says, you remember it as 2.7 Ibsen-Ibsen, as the Norwegian playwright Henrik Ibsen was born in 1828. And indeed, E goes 2.7-1828-1828. So there you are.

Given you've already previously memorized the birth year of Henrik Ibsen, you're in business. Whereas DePeterson says that the seventh president of the US, Andrew Jackson, and EDA start 2.7. So I'm going to just to build on DePeterson here, US has two letters. So that's two. The seventh president, so that's seven, Andrew Jackson, was elected for their first term in 1828 and he served two terms so that's 2.7 1828-1828-1828. I'm just going to quickly look up the Wikipedia page for the year 1828. While you do that I'm going to say I find an easy way to remember it is 2.7 and then remember 1828 and then remember 1828 and then just

do it again. Oh, interesting. Whereas I remember it as the Uruguay number, because that's when they gained independence from Brazil. You're always saying that. You know, I always say that. We also heard from David. David wasn't writing in with sort of any other business. This was technically posed as a problem, but it gave me something that I wanted to ask the listeners. Right. Okay. That works. So David said, I like this podcast. but just can't stand or sit in one place while listening. I need something physical to be doing whenever I listen to a podcast or audiobook. Not sure if this is a common problem,

but do you know what physical activities I can do when I can't think of any chores I need to do? Oh. So I was drawn to this because I also can't sit still and listen to something. I get very ants in my pants. And so I have to go for a walk. And that's sort of what I tend to most listen to podcasts. is usually when I'm walking somewhere. So instantly I was like, yeah, go for a walk.

That felt like a very obvious solution to me. But I understand that I did listen to a lot. Actually, I listened to a lot of YouTube videos while I was cleaning the bathroom. Oh, there you go. So I do understand the needing to be doing something at the same time. And I also find that they're sort of one or the other. Like I can't do a chore without listening to something and I can't listen to something without doing something. Right.

So I prefer to listen while I'm going on a walk or something like that. But I'm intrigued. It made me wonder, are there any physical activities or any interesting places where people listen to this? Like, are you doing something right now where in your head you might be like, yeah, I just listen to it while I work? But is your job, like, honestly, if you were to explain it to someone, it's quite weird and it's a physical activity? If you're listening to this while you like quality check, I don't know, dishwashing bottles to see if the lids are squirty. Or you're like operating a crane. Yeah.

Yeah, yeah, yeah. I want to hear from listeners who listen to this while doing, do you listen while doing surgery? I mean, I wouldn't recommend it. Yeah, you. What are you doing right now? Yeah. What are you doing right now and how are we distracting you? We know that some of you listen to this to fall asleep.

So we're not looking for people who are in stationary positions. We're looking for activities. And then we'll let you know, David. Great. Let us know by going to the problem posing page at a problemsquare.com select the drop down for solution. Or if you want, start a discussion on our Discord and our Reddit. Or feel free to discuss it in the comments of our Instagram posts.

Why not? You could also discuss it on our Patreon page. Oh, wow. Good segue. Which brings me to the final section of the show where we will choose three of our Patreon supporters at random and thank them by kindly mispronouncing every one of their names as best we can. And on this episode, those Patreon supporters are... C. Hrist. I. Annenkel. Anto.

New York City. Arta. No, I've got a conundrum here, Beck. Mm. Because the next name is just open square brackets, insert clever name here, closed square brackets. You can't say it. You can't describe it.

I know, bar. Here's the thing. Because I get all the Patreon names and I pick three at random each time using a spreadsheet, alphabetically open square bracket this is the first name that appears every single time and whoever this is

has been supporting us on Patreon for a long time so every time I see this name and I always think when will this name get picked and it finally was and I was so excited so I just want to break with tradition

and insert clever name here well done your time has come. Congratulations. Or as I like to say. All right. Brighetins. Ert Clay.

Vernum. A heret bracket. Well done. All right. All that's left to do is thank you, the lovely listeners. Please tell everyone who you. think would enjoy this show. I'd like to also thank my co-host Matt Parker. And if this podcast is a springboard by a pool, then that makes our producer, Laura Grimshaw,

our lifeguard. Oh, that works. Yeah. She's got a whistle. She tells us off whenever we're bombing. That's so true. Yeah. No running by the podcast. No eating in the podcast. That's my impression of Laura. And our listeners, really. That's the end of, yeah, they're all in agreement on that one. Bye. Now, where did you go last, Beck? You're over there.

Ah, what are you trying to do? The greatest trick I ever pulled was convincing people that I had a plan. I'm going to stick with my plan, which is just to mirror image whatever Beck does. So, can I go into thank you? Yep, the mirror image spot for Beck. move. Brilliant.