A Problem Squared - 093 = Game Solutions and Neil Distributions

Episode Date: September 16, 2024

⭕ Circle Can Tic-Tac-Toe ever be fun?  🤏What is the smallest Neil called?  🗞There’s some Any Other Business!  ⚔️Battleships continues.   If you want to read the works referenced by Matt in his tic-tac-toe / naughts and crosses problem, take a look below:  Solo Noughts and Crosses, James Grime: https://www.tandfonline.com/doi/full/10.1080/10724117.2023.2168434 Tic-tac-tome, Willy Yoners: https://www.amazon.co.uk/Tic-Tac-Autonomous-Playing-Book/dp/1594746877 A Knowledge-based Approach of Connect-Four The Game is Solved: White Wins, Victor Alis: https://disi.unitn.it/~montreso/asd/docs/connect4.pdf  Is ultimate tic-tac-toe solved? https://www.joachim-breitner.de/blog/604-Ultimate_Tic_Tac_Toe_is_always_won_by_X Please send your problems and solutions to the website: www.aproblemsquared.com. If you’re on Patreon and have a creative Wizard offer to give Bec and Matt, please comment on the ‘Sup ‘Zards’ pinned post!  And if you like, leave us a review, pass the podcast onto a friend or give us a rating! Every little helps.  If you want even more from A Problem Squared, you can find us on Twitter, Instagram, and on Discord

Welcome to a Problem Squared, the problem solving cod, codpast. Codpast? Oh my. You can't get any cod past you. That's the most PG response you could have given me. Thank you. That's a teacher. Yes.

I was going to say a podcast much like a duck. Oh, like a duck. In what way would you say, Beck, a Problem Squared podcast is like a duck? In that it seems very calm. It does. But underneath we're scrambling. Scrambling to write intros. Love it.

Yep. You're Matt Parker. That's correct. A mathematician and comedian. Oh yeah. And a big fan of Drake. Which is a... That's both very funny and horrific.

A male duck. No, I get the... Also, you're not... That's not the half of the joke I was worried about. I realize that Drake might be problematic. So that was a joke. Matt is not a fan of Drake. He hates Drake, in fact.

I did recreate the Drake meme. You did. Yeah, that's what I was a joke. Matt is not a fan of Drake. He hates Drake in fact. I did recreate the Drake meme. You did. Yeah, that's what I was thinking of. So there's a less problematic version of it if anyone wants it. Yes. And I am Beck Hill, a comedian, podcaster. And if I tip my hat, I would say, malarady. What? Malarady. You duck around all the obvious, like anything easy and you're like, well, it's an obscure. Yeah, like I'm Quackers or something.

Yeah, exactly. Yeah. Yeah. You're buoyant. Yeah. Things roll off you. Heck, Bill. Oh, yeah. Yeah.

You're always, I was going to try to do like a comedy Bill joke. Water for ducks, Beck. That's the one. That's the one. See? We got there in the end. Nailed it. Look, this show is more collaborative than it. We assemble the show in front of you.

Yeah. Yeah. We're like the Heston Blumenthal of podcasters. We come out to the table. Yeah. Other podcasts are out the back, planning ahead, rehearsing, writing. Not us. This is a deconstructed podcast. Yes. It's an open podcast. Rehearsing, writing, not us.

This is a deconstructed podcast. Yes, it's an open podcast. And on this episode. I've worked out if tic-tac-toe can be fun. I'll be looking at sizes of Neil. Oh, of Neil. Yeah. And we will conclude with any other beekness. Because ducks have a beak.

Yeah. Well, it's a bill, isn't it? Is it a bill? Would you call it a beak? Is it different to a beak? Any other billness doesn't work quite as well. No. Oops. We'll solve this problem in, in any other burs. Any other birdness. Birdness. There we go. Hey Matt. Beck. How are you? I'm good. And I'm aware of the pressure here because last time we had a catch up, previous episode.

You had a lot of news. I had a lot of news. We were recently in the States and Lucy had work in Boulder, Colorado. Ah. Wonderful place. Better than the lesser known Shire, Colorado. Because if you're Boulder, you're Boulder Shire, Colorado. Because if you're Boulder, you're Boulder. Oh, oh. So Shire, like you're shy.

Yeah. But you said Shire, like a location. That's good work. Yeah, thanks. Yeah, yeah, yeah. So she was in Boulder, which if people haven't been, wonderful place. I had not been before. Lucy's been there a lot because it's kind of the center of massive solar physics to some extent in the US. Is that because there's mountains there? It's because a lot of physicists and particularly space scientists are into outdoor activity.

They love skiing, they love mountain biking, they love all this stuff. So their conferences are like in Hawaii and all this kind of stuff. I'm in the wrong job. Wrong industry. Yeah. So a lot of them live and work in Boulder. And so there's like, obviously, there's a university there, University of Colorado is in Boulder.

You've also then got the Southwest Research Institute and the Northwest Research Institute, which are completely unrelated institutes. Oh, wow. So Lizzy was out there working with her colleagues and I thought I would hire a bike. We're going to be there for a week in this phenomenal cycling in the Rockies. Like you're right in the Rocky Mountains. Yeah, nice.

Cause we landed and went straight to one place where we were staying over the weekend and I went from sea level. I'd been New York and San Francisco all on the coast to a place in the Rockies, which is eight and a half thousand feet altitude. Yes. And we got there and we're like, why have we got headaches? It turns out we had mild altitude sickness. I never even considered that.

Just like, I mean, Boulder itself, like Denver, the plane is already 5000 feet altitude. Wow. And then you got the Rockies. But the cycling, once you got over feeling ill, was absolutely incredible. Except when we picked up my bike, because we flew into Denver and then I got the higher bike from Boulder, then we drove into the Rockies. I forgot to get the helmet.

We had driven a reasonable distance and Lucy went, oh, have you got a helmet? And I was like, oh, I forgot to, because it comes with a helmet. And at that point it was right on the cusp of, can I be bothered going all the way back to get a higher helmet or not? And Lucy very correctly said, we should go back and get the helmet. Turned around, we drove all the way back, went in and they're like, oh, what size are

you? What helmet do you want? They got like a rack of them and they happen to have the exact helmet that I have at home that I'd recently purchased. It's like one of the safest rated helmets. I like her brain. My brain's. I like the career critical. You'll be, you'll either. Can I be stuffed to wear a helmet? Yeah.

And if I am going to wear a helmet, it's the best. Yeah, yeah, yeah, yeah. It's all or nothing. I don't want the illusion of safety. I either want to know I'm not being safe, but just wearing a hat or I'm as safe as possible because I go all out. So you might think, was that the coincidence? No, it wasn't. I'm just wearing a hat or I'm as safe as possible. Yep. Cause I go all out. So you might think, was that the coincidence?

No, it wasn't. I was initially startled that they have the exactly the normal helmet I have at home, different color, but otherwise identical. And I was like, that's amazing. Got it. Go back in the car. We had to retrace our steps driving out. Got all the way to the place where we were staying in the Rockies several hours drive later pull up roll the bike into the garage hanging on the

wall in the garage of the place we had rented was the identical helmet what yeah wow right there and so I took a photo of both helmets we'll put that on the socials and that's my story oh. Yeah, I feel like we're maintaining a caliber of story People have come to expect what we've learned from this is that you like a very popular helmet Yeah, yeah chances are that any place that you go anywhere you go will have this helmet. It's the gyro, gyro helmet. Yeah. Well, we'll put a link in the show notes. Good helmet.

That's how I've been. Lovely. Thank you. How have you been? I've been good. I went to an antique toy fair. And oh yes, you sent me various photos from an antique toy fair. Yes.

There were a few things I thought might tickle your fancy. Oh yeah. There was a display case that looked like it was for one of the original slinkies. Yes, it was a slinky display steps. Yes. There was a calculator of Lego that was still working. Many cursed objects. Oh, I can imagine.

But I picked you up something because they didn't just have toys there. They had other sort of collectibles. Because you sent me a bunch of photos and I was like, where is this toy fair? And you like named the suburb it was in. Glendale. There you go. You're like Glendale. I'm like, ah. I was there for I think about four hours.

Wow. I just zigzagged. Amongst all the tables. I was in my element. I was very happy. I didn't really buy anything because I packed very light. Did you buy anything for you? I bought a troll. A troll, of course. Yeah, I bought a troll though. Great. Yeah. buy anything because I packed very light. Did you buy anything for you? I bought a troll.

A troll, of course. Yeah, I bought a troll doll. Great. Yeah. But I picked up something for you. As the world expert on trolls. Yes, exactly. And you got me something.

I got you something. Oh my goodness. I bought this within seconds of walking in. Oh really? And this is not something you sent me a photo of? No. I walked in, I went, I wonder if, ah, they do have that. All right, I'll get that.

You've handed me a small bit of card that seems to have a coin in the middle. A coin that says one cent. And on the back is, is it the Indian cent? It is. It is. It's the Native American, if you remember on that episode that we talked about. When pennies will be worth the most. Yes. Wondering what was on it before Lincoln? Yeah, the Lincoln sent. Washington, whatever it was.

Lincoln sent, yeah. Yeah, we're trying to work out what it was and looked it up and went, oh, it's a Native American. That's what it looks like. And there I went oh I'll pick that up. It's a 1902 that's quite late in the game. This is definitely a bit more worn than some of the more collectible ones. Okay I've got an adequate photo we'll put that out on the socials if people want to check out the gift Beck bought me that's worth one cent. At least. At least. Face value. Thanks, Bec. You're welcome.

While I was there though, I bumped into Grant, better known as The Toiley. Ah! Who did our- The Toiley paper Grant. Yeah, did an evening of unnecessary detail at Flappers in January. Yeah. I bumped into him and his partner, Susanna. Yep. She was there because she's working or doing a favor for a woman who bought

the house that was being used for the external shots for the Brady Bunch. She bought that house and then as part of some TV show beforehand they'd done a thing where they made the inside look like. Oh, it actually looked like the inside. The inside of the Brady Bunch house. made the inside look like. Obviously, it actually looked like the inside. The inside of the Brady Bunch house. What the sets looked like. Yeah. And the owner of the house, basically, it's their life-size dollhouse. That's how much money they have.

So they're like- They just bought a house. They were like, oh, let's preserve it as if it were the Brady Bunch house. So let's fill it up with- Contemporary toys and bits and things and- Yeah. So Susanna was there picking up. Her job was to buy old stuff to put in a life-size model of the Brady Bunch house. Yeah.

Ah, love California. I know, right. Anyway, meanwhile, I'd like a mortgage so I can get a studio flat so if anyone wants to lend me some money. If you could be sufficiently. Can lend me a penny. Period accurate. You could live in the house. I mean, yeah, if I can...

I would. I would happily. You have to research like lingo to make sure you don't use any anacharistic words. Yeah, daddy-o. See? You're in. Our first problem comes from Alec who says, It is well understood that Tic-Tac-Toe is a solved game. That is, all of the possible game states can be brute-forced crunched through to determine

how to win all the time. This is because the total number of game states is a small enough volume that is trivial for modern computers. My question is, what variations or combinations of variations of tic-tac-toe are necessary to design a game that isn't solvable by brute force? Examples, playing the game in three dimensions, which would be three by three by three instead of just three by three, or super tic-tac-toe where each cell contains a smaller game and you have to play in the super cell corresponding to the sub cell that was just played in. Alec, I don't know what that last sentence was. I'm sure Matt will explain it to me.

Well, get ready. Yeah, here we go. So first of all, Alec is talking about the notion of solving a game. Yes. And that means that it's no longer a game. So for example, Knots and Crosses, you can play Knots and Crosses as a game, but if both players know how to play, it's no longer a game because it's guaranteed to end in a draw. Yep.

And optimal play in Noughts and Crosses isn't that difficult. So we say a game is solved if we can prove that the person who goes first, Yes. or sometimes second, is guaranteed a strategy that means no matter what the other player does they win at that point no longer a game we're not playing it we've solved it yeah if you go first you're most likely either going to win or draw well if you assume both players play perfectly yeah the options are either for any game like this and these this excludes games where there's like random chance because obviously that changes things and it's separate to saying there are good strategies.

It's saying there's an ultimate way to do it. You follow the algorithm. You'll definitely win. Or if everyone follows the optimal play algorithm, it's definitely a draw. So those are generally the options in games like this. Either it's definitely a draw or it's definitely a win. Interestingly. I would argue that it's a psychological game in that you keep playing until one of the people playing either gets bored enough that they just try somewhere else to see whether you're going to go somewhere else differently or they just lose concentration.

Well, that's what happened two nights ago after a talk, I was signing books. Someone said, Oh, could we play a game of tic tac toe? Or knots and crosses, I should say. Or knots and crosses. Yeah. I'm flipping between the two actually in the UK, it's knots and crosses. In America, it's tic tac toe. I forget Australia, which one's that? Dingleberry.

Dingleberrys. Good game. It involves sheep. We said knots and crosses growing up. Okay, right, good, good. So- Hugs and kisses I think I've heard people call it. Really? I guess circles and crosses, yeah, fair enough.

So, I did a weird open, I opened in a corner and then they didn't take the center, they did a weird move but we were deliberately licking around because it's not fun if you play it properly. Yeah. It walking around because it's not funny if you play it properly. Yeah, it's boring. Cause it's soft. Is there a way to play where both parties deliberately try to avoid winning? There's there there is there's anti-Norton crosses where you have to play such that whoever makes three in a row loses. Yeah, that's fun.

But again, that's also there's so few states and actually Alex right here that these sorts of things are solvable. Yeah. Chess. So chess is not solved. Oh, okay. Because partly there are just so many possible states the game can be in. Yeah. Partly it is a very complex game. So you don't have to have analyzed every single state for a game to be solved, because you can just come up with a like a strategy, which guarantees a certain outcome. Let's say there's a game where you play by putting a mark on some kind of massive grid. And as player two, you can just always do the exact symmetric move or whatever player

one did. Yeah. and guarantee a draw by the end of the game. Because you stay in lockstep with them for whatever the rules may be. That's now a game where we haven't analyzed every possible state, but we can say it's solved because we know there's optimal strategy for player two such that they will always draw. Yes.

However, one way to do it is just to analyse every single state. And in anti chess, so that's chess where you win by the other person taking your king. Oh, wow. Now anti chess is interesting because normal rules, except if you can capture a piece, you have to. Okay. And that means you can force your opponent to do a certain move. That's much more doable. Yeah. Yeah. And that's why anti-chess has been solved to an extent. There's a winning move. I think the first player to play moves the fifth pawn from the

left in. And with that opening move, it's like, it's quite complicated, but there's a series of other moves. No matter what the second player does, the first player will win anti-chess. And the reason we can analyze anti-chess, but not regular chess is that forcing element. Cause you can control several moves of your opponent in a row by just constantly feeding them pieces. And that makes it possible to solve. Whereas regular chess, forget it. Now, Noughts and Crosses, solved. You can actually buy, you can play Noughts and Crosses against a book. It was a book written by Dennis Gurrier called Solo Noughts and Crosses. Okay. Where you just look up the current game state, You look up the board in the book. Yeah. And it will tell you where it's move was.

What move the book wants to play. Okay. And then you do your move and look up the next one. Oh, I like that. There was a more modern version called Tic Tac Tome. Nice. It's like an idea. It's like a very boring choose your own adventure book.

Yeah. Well, that's what I was going to say. Definitely better name. When you were describing a solvable game, that's where my brain went. I went, oh, it's a choose your own adventure book. Yeah well that's what I was going to say. When you were describing a solvable game that's where my brain went. I went oh it's a choose your own adventure. Yeah 100% yeah. Yeah. And you'll never beat the book. I can't remember off the top of my head if either of those have deliberate mistakes where you can win. My friend James Grime owns a copy of both and he brought solo knots and crosses along to a show one time to show

it. We played it with the audience. It was that evening of a necessary detail. Of course it was. Huge amount of fun. And my friend Matthew Scroggs built a computer out of matchboxes and beads that learns to play Noughts and Crosses. I've got a video on my channel about this because it's basically machine learning. But again, there's one matchbox per possible state the game can be in. And it's such a small number, you can just have a box for each. And then there's this complicated system where you train the match boxes to play by adding and removing beads if it wins and loses to reinforce. Bees? Bees, yeah. So regular knots and crosses, very simple,

very boring. People have come up with a bunch of other variations on a theme. You can do knots and crosses where instead of having just the three by three grid and absolute edges, you have it where it wraps around. So you can use what are called the super diagonals. Yeah. Big fan of a super diagonal. You talk about it in your double talk. I do talk about it in my double talk. Yeah. I love the super diagonal.

You used a super diagonal in your graph for time in the last episode. We did. There you are. Yep. Good old super diagonals. Where the line went from 0 to 12 and then started with 0 again. Perfectly good diagonal. One of my favorites.

They're super. So super diagonal knots and crosses or you're playing it on a cylinder You can't draw anymore. You gotta win. So take out the draws, which is nice fun because there's no way to arrange all the Hugs and kisses such that there are not three in a row before we go any further. Yeah Connect for is like a bigger version of tic-tac-toe or not. Yes Yeah. Connect 4 is like a bigger version of Tic Tac Toe or Knots and Crosses. What is that a solved game? Connect 4 is solved.

First player wins. The first move is they have to play in the center. And then no matter what the other player does, it's a guaranteed victory for the first player. But it's such a complicated strategy. I was going to say, because I've played against you. No, I don't know. It's not. I don't think it's, don't think it's human doable.

It's really complicated. I've actually, you've seen me learn stupid things for more ridiculous reasons. There was a paper called a knowledge based approach of connect four, which was written by someone called Victor Ellis, I think in the eighties from memory. Yes, 1988, they were the first person to crack it and solved first person will win with optimal strategy. Wow. And you know what? Interestingly, you've now pointed out because Connect Four, if you ignore the gravity aspect, if you imagine you can just put the pieces wherever you want. Oh, yeah, I forgot about gravity. Exactly that makes a mild difference. I forget the exact

ratio of a connect 4. But surely the gravity element then actually makes it more solvable because you're being forced. It changes it. So connect 4 is played on a 6 by 7 grid and you have to get 4 in a row. Whereas Tic Tac Toe is on a 3 by 3 grid, you have to get 3 in a row. And all 3 of those numbers, you've got a something by something grid, an M by N for 2 numbers, M and N grid. And then you have to get some number K in a row. That's actually, that is the generalized version of Tic Tac Toe. It's an M by N grid with K in a row to win.

And there's a whole family of these. They're called MNK. Is it MNK? Yeah. Is there a reason for those letters being used? No, there's a bunch of conventions and maths for different letters you use to mean different things. So X and Y are often used as real values that continue between numbers, whereas M and N are often used as whole numbers, as is K.

Oh yeah, I know about N. Yes, we often use N for number. Once you've used N, people would also use M. For Bumba. For Mumba. I forget why, K? But K, i, j and k are often used as whole numbers. L is not often used because it looks like a one. So M and N is a common way to do a grid and K is a constant that you might include. So, but there's no hard and fast rules. You can do whatever you want, but if you follow the vague conventions, it makes it easier for other people to understand. Yeah.

So it's called an MNK game. Why does that? It's making me think of a multi-level marketing MLM. MLM. Yes. Like an MLM game. Whoever starts playing first, is most likely to win. likely to win. So different combinations of M and N and K either give you solved or unsolved versions of the game. Yes. So 3 3 3 as we know is soft. There's another version that's

called I'm going to mispronounce this, Gomoku. That's on... That sounds Japanese. It probably is Japanese. Let me have a look. Almost certainly Japanese. That's five in a row on a 15 by 15 grid. Sometimes bigger grids. But that's a 15, 15, 5 game.

Oh, it looks a bit like Go? It's played on a Go board, yeah. But different rules. It's a... Well, there are different versions of how you start the game. But then it's basically Connect 5 on a go board. So no gravity, which distinguishes Connect 4. So Connect 4, which is our 6-7-4 game, but with gravity.

And different games can have different additional rules, like connect for has gravity. Gomoku has sometimes you start by placing like two whites and a black at the beginning because the first player had such an advantage. They changed it. So you've got to put two whites and a black and then you're the second player. You can now decide if you're white or black. Ah, and there's other rules like you can either add an extra

black or you can, you can add one of each and send it back to me to decide what I want to be. Okay. So you can add all these extra twists. 15, 15, 5 is solved. First player will win. Actually, it's impossible for these sorts of games to if you're playing second, your best case scenario is a draw, assuming everyone is playing with optimal strategy. Yeah.

And the reason for that is these are a category of games where there's no downside, there's no penalty for making a move. So it's not like you can put an X somewhere if you're X and later on it gets in the way. Yeah. Or causes is your problem. It's like in chess you can move a piece forward but that piece can be taken out of the game or you've no longer got it available to go somewhere else instead and all that whereas in knots and crosses. They're all the symbols are equivalent for each player you've got an infinite supply them, and they're indistinguishable. So for these games, if you imagine a massive grid, an N by M grid, we're playing some crazy version of it, and I get to go first, but you're like,

ah, you're player two, and you know there's a winning move for player two, so you're like, got him. I could get my first one and decide, you know what? I wanna be player two, and I just put my first move in some arbitrary far flung corner of the board. And I'm like, surprise, jerk, now you're player one. Because I've thrown away my first go. And hypothetically, even if I did later have to use that spot, it's great. It's already there.

And then instead of doing that move, I can put that one in some arbitrary other space. So because player one could opt to become player two, and which makes the other person player one, you could then throw away your spot and back and forth. So it's impossible to have an optimal strategy for player two because player one could co-opt that strategy instead. Got it. So we know for all these games, either it's guaranteed draw or it's guaranteed player one win. So to answer the problem as stated for completeness,

they wanted to know at which point is it not solved? Is it fun again? And I can confirm that if you're playing on a board, which is something by four Of any size by four And that something is between nine and twenty-nine. Yeah. Okay, so if you're between nine by four and twenty nine by four and you're playing Connect four first to four but without gravity so tic-tac-toe, but four in a row wins All of those boards are currently unsolved. So there you

are. So if you want to play a version of the game where you're guaranteed nobody's going to have the perfect strategy, a four by nine board of knots and crosses, four in a row wins, have at it. There are other versions where you keep playing and you just get a point for each time you get four in a row, whatever it may be. So then you have a score by the end. And you have to agree in advance, if you're playing this where it's four in a row to win, and you put in a symbol and you complete a row that's like five or six at once, does

that count? And some versions of the games, it's four or more, and others are exactly four or whatever the number may be. OK, so for instance, if you had three going up. Yep. And then space. And then a diagonal. What I mean is if you've got like one, two, three in a row. Oh, but hang on, it's not gravity.

I keep forgetting there's no gravity. That's what it is. One, two, three in a row, then space, then another one, two, three, and you put one right in the middle, you've just made a row of seven. Does that count? If you're playing connect five, yeah. Yeah. Agree in advance changes the rules.

Yeah. But if it's exactly four, anything between a four by nine to a four by 29 grid, connect four, no gravity. 4x9 to a 4x29 grid connect for no gravity. And your opponent cannot have a built in strategy because it's as we are aware unsolved by humans. So this might also be a fun way for people to display their collectible coins. 100%. Or pennies. If your friend has bought you several pennies you can do that.

You can build a little a little rack. You would have to or like a little game board. Little game board. Yeah. Game board. Finally, ultimate tic-tac-toe. Our ultimate knots and crosses, which Alec referred to in the original problem. We're going to have a quick game. Okay. So do you want to describe to the listener what I'm setting up here?

Matt seems to be drawing a traditional Tic Tac Toe three by three grid. Yep, like a hash. He's now just got a different colored, he's just got a different colored pen and now he is putting the same thing into each of those squares. So here's the way the game works.

There's a pen for you. All right. So first of all, you've got to pick a cell So here's the way the game works. There's a pen for you. All right. So first of all, you've got to pick a cell and then you make a move on the game within that cell. I'm going to do a circle. Yeah. I like circles and I'm putting it in the very center.

Middle middle. Okay. Of the one in the middle. Now the bonus rule that makes this interesting. So to win this, you've got to win three games in a row. Okay. So by forming a line is winning all three of those games. Got it. Yeah.

Except you have to go in the equivalent game that matches the smaller spot the previous person just went in. So on this sub game, you went in the middle. Yeah. That means I now have to play the middle game. Okay. But if I go top right, you now have to play top right. Oh, got it.

Okay. So you can, not only are you putting your move in to win each sub game, there's the meta strategy of you're directing where I have to play next. Now what happens if I go, so if I go middle again. Then I got to play back over here. Yep. So I've gone back. I've gone in the corner, put a zero in the middle.

Matt has put an X on the other side for the sake of symmetry. I've gone in the middle on the other diagonal. Ah, let me see what has happened here. Yes, but you got to give me an idea. But you might. So on the center one now, I've set it up. So there are two places that would win for me in a normal game of knots and crosses game over because you can't Block both. Yes, so I wouldn't send you do not want to send me back to the middle because I'm gonna win if I go back

There I see so now you can't take any more center spots All right, I'm enjoying this we will Down Now best I'm aware this version we're playing has been solved if once someone wins a game, you still have to keep playing in that now pointless game if you get directed there. But there's another version where if you direct someone to play on one that's already finished, it's a free move they can go anywhere. I see.

And that's not solved. Okay. To the best of my understanding. And that was a lot more fun than regular Nolson crosses. But even if it, like, even if this was solved, it's like Connect 4, the optimal strategy is not human doable. So we know it is not. Or if it is, you've got no friends to play with. No, if someone's got like a massive encyclopedia of instructions they're referring to.

Yeah. Yeah. So even if, you know, things like Connect Four, it's still fun to play, even though it's solved and this falls into that same category. So I highly recommend if you do need to play Knots and Crosses in a pinch, go for Ultimate Knots and Crosses. Alec, I hope that solves your problem. I'm going to give that a ding from Dingleberry. From Dingleberry?

Yeah. Oh, the game of dings and dongs. Yeah, there you go. You don't want to see what the symbols are. They take longer to draw. Our next problem comes in and they've started Hi Beck, Hi Matt, really covering their bases there. So Matthew Smart says they're a big fan of the podcast and of you guys in general, they're

still hedging their bets. So Matthew's problem, they have two work colleagues, both named Neil. They are father and son, but they're American. Americans love reusing the same name for different generations. To differentiate them, we call them Big Neil and Little Neil. That could be workplace bullying. Matthew complains that's not particularly original and they were wondering how many Neils could you have with ever decreasing sizes and what would the smallest of all Neils could you have with ever decreasing sizes? And what would the smallest of all Neils be called?

They pointed out they hurt the parameters, will stop the problem from taking infinitely long to solve. It's not a concern most problem posers have to be honest. They want to say, here are the rules. You can only put one word in front of Neal. So you can't have smaller than the previous Neil. That's good. I guess it rules out kind of, you know, recursive Neils.

Yep. You can only use words in the English language. You can't generate gradually bigger and bigger nonsensical words to describe the Neils. Kind regards, Matthew Smart. Beck. By the way, first of all, Matthew said hi. Hi, Matthew. Secondly, have you found a way to work out how many meals there are and what would the smallest one be called?

Yes. Good. Maybe. Maybe. Well, no, I'm back down real fast. Maybe. Maybe. Well, no.

I'm back down real fast. Um, I, I liked this one because it reminded me of something that I've mentioned on the show before my grandparents on my dad's side, we called them Nana and Dad Dad. That was actually because that's what my dad used to call them when he was a baby. Oh wow. And that stuck around. And I think they were like, oh, we'll call them. Generational nicknames.

Yeah, to differentiate from grandma and grandpa on the other side. So like, oh, nanana, dad, dad. And that meant that my great grandparents, the two who are still alive when I when I was born, were big nanana and big dad. Got it. I realized there's something missing from this, which is they've got a big nail and a little nail. Yep.

They don't have a zeroth nail. No, it reminded me of in Bob's Burgers, they got a character who's called regular sized Rudy. Yes, yeah. I really like that. Out of context, that's just what they call that kid in the show. Yeah. Because there's a big or a small one out there, right? Well, yeah, because I don't think they ever introduce the other side. Yeah that I'm aware of yeah which is even funnier.

Yeah well first you need a default Neil yeah in order to what is your standard issue metric Neil yeah and then you can have a bigger Neil or like a reference Neil yeah. I realize that you I feel like this is cheating. So I'm going to say it, but then immediately discount it. You could have, obviously it's words in the English language. You can't have numbers. So you wouldn't have numbers have words, one, Neil, two, Neil, three, Neil, as in it starts to sound like you're doing very poorly in a game. I like to add that numbers are also apparently valid words.

Yes. And I'm the first person to solve most problems by using numbers. Sure. But I do agree. We should discount just numbering the Neils. Because then even if you are spelling it with letters, when you get to 21, if you were to say you're not going to allow dashes, you can say it's two words and then you're limited. Five, five, five. So I think that's potentially out.

You could do divisions. So you could do, so rather than little you do half nil, quarter nil, eight nil. Okay. Yeah. I think we're also going to discount that for the same reasons as the numbers. Okay. I think it's all it is on. It's slightly in this area. I think it skirts into being over the line of being allowed.

Right. I think we should go into metric kneels. Like a mega kneel. Yeah. A kilo kneel. Exactly. Exactly. So I think it should, you start at just kneel.

Regular kneel. Yeah. And then above kneel, desi kneel. So you could do kilo kneel. Kilo kneel. You could buy a thousand. If you're just doing thousands, yeah. Kilo kneel, mega kneel, giga kneel, doing thousands, yeah. Kilo Neil, Mega Neil, Giga Neil, Tetra Neil.

Yeah. Yeah, it's very funny. Going down. Pentanil. Yeah, Millinil, Microneel, Nanoneel. Nanoneel is my favorite. Nanoneel, that is lovely. Yeah?

Yeah, yeah, yeah. Who doesn't want to meet Nanoneel? I'm coming on board, Neil. Nanoneel sounds like it should be a nana. It sounds like a nana, yeah. Yeah, so. Hey, check out who's here. It's Nano Neil. Why you just missed giga Neil? Liquid measurements. Right. Yeah. You could have like a millineal, which is almost a millennial. Millineal. A millennial. That's nice. Yeah.

So I need to ask how many words do we have for, let's say, length or- Or size or something. Yeah, in metric. Oh, in metric. Well, we can just keep generating them. You just take the Greek prefix, check it on. But I guess because they've got to be in the English language. We can't start inventing.

No, but you're getting dangerously close. You basically go back to what we've just discounted, which is numbers, because after a while you're just going tetranil, pentranil, sextanil, septanil. Yeah. Octanil. We get a little bit lazy after a while, don't we? Oh, yeah. Like a lot of things in counting, we start creative and then really phone it in. It's like once you go past 10, 11, whole new number, great name, love it. 12. Okay, pretty creative.

13. It's basically just three with ten. And then 14, 15, 16. Come on. But then we go the other, like, I've been learning French and the numbering is interesting because in French when you start to get to like 20s and 30s, it starts to be like maths. Now, I've got some bad news. Everything is maths. But yes, the French numbering requires more arithmetic than you would expect if you're not already fluent in it.

Yeah. In French, up to 69, it uses a base 10 system. But when you reach 70, it switches to a vagesimal system, which uses base 20. Yeah. Which is annoying. Super annoying. In Japanese, I like Japanese. Japanese is sort of similar to English in that when you're counting, you get up to like juu for ten and then it's juu ichi, juu nii, juu san, juu yon.

So that's like ten one, ten two, ten three. And then you get up to nijuu, which is twenty. That's two ten. So then you go nijuu ichi, nijuu san, n need you, which is 20. That's two 10. So then you go need you, which need you sound. Did you need you? So yeah, it's just, you're saying it depends if the Neils are French or Japanese. Yeah. Well, in English, I'm just saying the teens are dumb.

Oh yeah. Because we, we go up to 10. Fine. 11. What? We never use 11 ever again. Hey, I love it. 12.

One off name. 13, then- We should have more of them. All of the numbers- There should be another number in the low 3000s that just has a name. I'm so glad I don't have to learn English. Because the more I learn other languages, I'm like, wow. You know, right?

So, I did play around with whether you could have little Neil. Yeah. Smaller Neil. Tiny Neil. Tiny Neil. Micro Neil. Teeny Neil. Yeah. The problem with that is that then it really comes down to the perception of the other person as to whether teeny is more than micro or vice versa.

Yeah. My next question was, because what you could do is just get a set of all the words in English language that mean size. So you got gargantuan Neil, you got big Neil, you got all that jazz, right? Do they need a ranking? Does that matter or they just need to be different? Well, I think it needs to be ranked because if it's different, then you might as well just assign a different word.

Right. So you might as well just assign a different word. Right. You might as well just start at the dictionary and do aadvarknil. And then whatever the next one is. Whatever the next one is, you know, other words. And so on and so forth. So what you'd need to do is get a set of all the words that mean a size. Then you have to get a lot of volunteers and you'd get them all pairs of words and say which one's bigger. And then you do enough of that.

I sense that you're writing some terrible Python code. No, no, no. This would be like a big online survey. We're not going to do this. It'd be a lot of effort for no discernible reward at the end. But if anyone out there listens to this board enough. We will promote it. So.

How many people would we need to fill out the survey to work out? What is depends how? Much everyone's in agreement Would you do it that you give everyone a pair of words and they decide what's bigger and do it that way or? Would you just give them all the words that mean big large? Yeah You give them all and then each person ranks them and then you work out how many are similar I think it'd be easier to do a pairwise comparison. And there's already a bunch of maths done for how you then combine all those results into.

Honestly, I don't have an answer. I was kind of hoping we'd come up with it in this conversation. Oh really? I love your approach. I feel like we've broken the. My suggestion was to go metric. That's great. I feel like it does revert to being just numbers. Yeah. It is very similar to that.

Yeah. It's just a different version of that. Um, yeah, I think this is going to be not yet to be dinged, but potentially if anyone is looking for a project to do in their downtime. I mean, how many? You know what? I'm not saying you haven't done this. English words which mean a size. I'm just getting a lot of words that are synonyms for the word size.

I want words that represent a volume. Yeah. So I'm going to look up large. Because I'm assuming we can go up in Neils as well. Yes. And this is the thing is I'm working on a show at the moment, a kids show at the moment where I don't want to say too much because I don't know what I'm allowed to say. There's always an NDA for a new show, but essentially the main characters parent. So let's say the main character is called Matt. Their parents are called like Mum Matt, Dad Matt.

And then going up the mother's side, it's then Grand Mum Matt. Great Grandma Matt. And then it's very similar to what we have now. However, we were talking about this in the writers room and we think, because we haven't decided, but this might end up being sort of show Bible stuff. But when a new generation is born, they all switch names. Everyone shovels up. Everyone shovels up.

Yeah. So you, you would then become Dad Matt. And then you'd become Granddad Matt, great Granddad Matt. So yeah. And that would just be your name. That's how everyone refers to you. Exactly. Yep. Yeah. So I did just find on a website a worksheet for kids with size words.

And I can confirm if size words is to be trusted, there are 27 kneels. That's so specific. Some of them feel at least 27, let's say. Some of them are pretty straightforward. Little kneel, itty bitty kne, gigantic Neil, enormous Neil, humongous Neil. Some you might have issues with. Deep Neil, petite Neil. Oh, that's fine.

Low Neil, skinny Neil. That's a different dimension. Long Neil. I take my Neil's long. Long Neil is very funny. Wide Neil. Yeah. Long kneel. I take my kneels long. Long kneel is very funny. Wide kneel. Yeah. Yeah. I mean, I put the word large into word hyposasaurus. Now some of these are conjoined words, but it's got a dash in between. I'm going to say no to dashes.

Oh, you purist. Because it's meant to be one word. But even so, it's over 500 words. That's just for large. And are they all legit? Give us a couple of neels. Big Neel, Colossal Neel, Enormous Neel, Giant Neel, Gigantic Neel, Huge Neel, Great Neel, Massive Neel, Monumental Neel, Humongous Neel, Sizable Neel, Grand Neel, Immense Neel, Super Neel, Astronomical Neel, Cosmic Ne cosmic Neil, Elephantine Neil, gargantuan Neil, ginormous Neil, Titanic Neil is fun. Titanic Neil.

Serious Neil. Serious Neil. Now that is a serious amount of Neil. Unwilledly Neil, prolific Neil, stately Neil. Great. Phenomenal Neil. Phenomenal Neil. Phenomenal Neil, definitely. Kingly Neil, cracking Neil or phenomenal Neil phenomenal definitely

Kingly Neil cracking Neil increased Neil I'm gonna just allow that one lengthy Neil, which is a bit like good friends with long Neil. Yeah exactly Well, so economy Neil If you can't afford Gant you a Neil. Yeah. Um, theatrical Neil.

Oh yeah. Then we're getting back to issues we've had in previous problems where it kind of comes into the eye of the beholder as to what's big. What do you think is a reasonable, opulent Neil? Unfathomable Neil. Neil beyond the comprehension of the human mind. Noticeable Neil. Which also suggests that the other director is a noticeable Neil.

Like I'm just looking at the big ones. Let's have a look at small ones. Give us smalls. For every Neil there is an equal and opposite Neil. Neil is such a great name for this by the way. This must be good for people listening at home backers laughing too much to even read out the Neil names to me. Oh I love these.

Okay. Yep. Toy like Neil. Toy love these. Okay. Yep. Toy like Neel. Toy like Neel. Aww. That is just gonna take a moment and we will resume the Neels. Portable Neel. Good There is also pocket meal. Pocket meal. Shrimpy meal. Bantam meal. Some of these are unavoidably an insulting name for a meal. Reduced meal. Reduced meal.

It's like a recipe. Yeah. Oh gosh. Concise meal. Peanut meal. Oh gosh, concise kneel is us. Peanut kneel. Peanut kneel. Travel kneel. Now, we wouldn't be able to use this one because it's two words, but one of the options in

here is not big. Not big? I feel like you put the word not in front of all the big words. Embryonic kneel. Embryonic kneel. Lesser kneel. Embryonic kneel. Lesser kneel. Oh. Yeah. Lower kneel.

Neat kneel. Trim kneel. Dainty kneel. Cute kneel. Aw, that's all kneels. Handy kneel. It's a convenient kneel. Yes.

Dinky kneel. Dinky kneel. Narrow kneel. I feel like there's no limit. Yeah. Um, mere kneel. Mere kneel. Yeah. Um, mere nil. Mere nil. He's but a mere nil.

I mean, look, there is a limit in that at some point... There's a finite number of words. There is a finite number of words. Yes. How many of those you're happy to class as a size-based word is also limited? Yeah, but there's at least hundreds. Yeah. I would say... class as a size based word is also limited. Yeah, but there's at least hundreds. Yeah, I would say- I'd say not thousands.

Yeah, but I would say at least a thousand, both going upwards and downwards. Yes, up and down. Agreed, agreed. So I think you could get many, many generations of Neil out of this if you could agree on the scale of these words. Well, I would say, first of all, going back to Matthew Smart here, we do need a ranking because they want ever decreasing sizes. So we'd have to solve that problem, but we have a mechanism by which we can do that. They also do say these are work colleagues. Now, we're pretty confident push comes to shove.

We could whip up a thousand kneels. We could whip up a thousand meals. It now comes down to how big a workplace does Matthew Smart work in? Because if they got fewer than a thousand colleagues. They won't need that. We're done. Yeah. If they've got, they work somewhere bigger than that, we might have issues. Yeah.

So I think we might have to throw this back. So, I think we might have to throw this back. So, I can't yet give you an ever decreasing series of dings, but I feel like we go back to Matthew and say, look, at a pinch we could find and rank a thousand nils. Yeah. How many do you need? Otherwise, you're looking at just big nil, bigger nil, bigger nil. Some kind of algorithmic way. Yeah, big nil, bigger nil, bigger nil, bigger Neil, bigger Neil, bigger Neil, bigger Neil, etc. Deflated Neil.

Deflated Neil. As opposed to inflated Neil. I'm sorry, these are very fun. So I think you have brought us a colossal solution, but we'll throw it back to Matthew to find out if it's adequate. Yeah, you can ask Epic Neil. We've reached any other birdsness. I think we decided on birdsness.

Yeah. Well, in the meantime, I've looked it up. Oh, but Beak and Bill. Yeah. The, in the meantime, I've looked it up. Oh. Beak and bill. Yeah. The difference? Yep. There is none. What? They are cinnamon. They are cinnamon. You can tell I'm a bit hungry, can't you? It's been a long day.

They are synonymous. Oh, yep. With each other. So all birds have a bill. Or a beak. But most people have chosen to use beak for the more pointy types and bill for the flatter rounder types. But that's not a strict definition. No, it's not.

It's just something that we've kind of started to vaguely use. Right. One day it might be codified and become. Well, cause I wouldn't say that a platypus has a beak. No, it has a bill. I wouldn't say that platypus has a beak. I would say it has a bill. Yeah. And now I want. Because it's a duck pill platypus has a beak. No, it has a bill. I wouldn't say that a platypus has a beak. I would say it has a bill. Yeah.

Hmm. And now I want... Because it's a duck pill platypus. Yeah. Is that just why? Is it... Would you technically... I don't think it's a... Is it a bird? But I feel like a platypus' bill evolved separately to beaks or bills.

Whereas bird beaks and bills have a common ancestor. Technically... Technically... They don't have... Technically, they don't have a bill. No, cause they've evolved a flappy thing on their face. Yeah. Cause it's a, it's not bone in the same way that a bird's beak or bill is technically bone. So there you go. So in conclusion, any other billness is acceptable. Yes.

Because it's duck themed. Or beakness. Beakness works better. Beakness would be equally correct. Any other beakness sounds like business. Beakness is a better pun. Got it. Right.

In any other beakness, also animal related, in episode 089, we discussed if dogs have four legs. Yep. Or two legs and two arms. Because a fantastic listener named Tim was arguing with their partner. Yeah. Their partner was adamant arms and Tim was adamant legs, all legs. Yeah.

Now we made a lot of fun of Tim. We may have come on a bit strong. Tim, good sport. Yeah. We decided, and we've already done this in any other business, so people think this sounds familiar. We decided definitely arms. Dogs have two legs and two arms. Bunch of biology and evolution reasons.

Made a lot of fun of Tim. Tim took it very gracefully, gave us the ding. Yeah. Thanks, Tim. Good sport. Yeah. I would just like the reason we've circled back to this, if I just want to tell everyone, we're done.

Yeah. When people are still using their front legs to type, can they go to the problem posing page at theproblemsquared.com, they choose solution and they keep adding more votes for arms and some people even supplying more reasons. Yeah. Now it's been a long time since we've had to forcefully close a problem. Yep. The, what animal would form the longest line if they all queued up?

We're done. Dogs have arms. Leave Tim alone. Closed. End of problem. Yep. Good. This is, this is very cathartic.

It is, isn't it? We should enter the closed zone more often. Technically, that's what the ding is meant to do. The ding is meant to be- The ding is the closed zone. Yes. That's why some problems don't have dings. Because they're still open.

Yeah. Dinged. Done. Let it go, guys. We appreciate you all getting involved. We do. We love you. Except in these few specific cases where we're done. However. However.

If you're a Patreon supporter, you can message us online. You're paying for the ride to keep going. We might not read it out loud. No, but we will read it and we won't mind. Yeah, yeah, we won't mind. If you're a Patreon supporter, you are welcome to vent. If you're a Patreon.com slash a problem squared, go nuts. Yeah. Speaking of Patreon supporters.

Oh, I just was. We like to thank three of our Patreon supporters at random at the end of each episode by mispronouncing their names. For example. On this episode, we'd like to thank... BN. That's it. That's it. That's it. C Handler.

That was good. Or C Handler's myth. Myth? They are a myth. I've noticed since we started mispronouncing names, people's names are beginning longer. And sometimes easier. I feel like people are going in and just making sure we've got their full name to mess up.

Yeah. And some of them have been a real challenge. Yeah. Apart from BN, these ones are all pretty full. Okay. C. Harls. I'm just doing what you did. Yeah, you are. Yeah. Yeah, but it works. C. Harls. C. you are, yeah. Yeah, but it works. See hows, see a Elkins, see El-Kai-ens, see El-Kal-Kai-ens. What about Char-Les-Kal-kins?

Char- you love moving the space. I do. Char-Les-Ka- do. Charles Lescar Elkins. Lescal. Lescal. Lescal, Kins. As in the family of Lescal. That's the one.

Thank you. Did you know that Show Catherine Kim is a pun on Kith and Kin? What? What does that mean? Kith is friends, Kin is family, friends and family, Kith and Kin. Oh really? Did I know that? Kath and Kim is a play on that. New to me.

Fun fact. It's a little fact for everyone. Just when you think the podcast episode stopped giving. I know. Now just wait forever. No it's not. Yeah everyone pile it on that. That's fresh. I'd also Now just wait forever. No, it's not. Yeah, everyone piling on that. That's fresh. I'd also like to thank Matt Parker. Hooray. Myself, Beck Hill and someone who I wouldn't even put in the category of ducks. I'd like to thank our once was an ugly

duckling and is now a beautiful swan. I don't know if that's a hundred percent compliment. I'm nagging her. I think so. Our producer. Before you worked with us, you were nothing. Yeah, that's right. Lauren Armstrong Carter, who is infinitely patient with us. Correct.

I would like to think infinite Lauren. You could hold infinite Lauren. The biggest of all the Laurens. In bigans, the biggest Lauren. All right. I got my battleship open again. Oh, mine hasn't fallen out. Oh, mine did. I fixed it because I took a photo last time. Oh, it's my turn, is it? Yeah. Okay. We've hit a rhythm where you go first, I go second every time. Unless I throw my move and then you become the second player. Player two. Yeah.

Player one. Yeah. Yeah, player one. That was, I keep getting that wrong. All right. J6. J6 miss. Now I said A1 last time because I'm systematic. J, six, miss.

Now I said A one last time because I'm systematic. Yep. But this time I'm going to say B two. Ooh. Uh, miss. You can probably extrapolate my next, uh, eight moves based on that.. Alright, till next time. It's just them falling out.