EGMO 2023/4: Another problem featuring Turbo the snail

追加: 2026-05-06

[00:00:00]Okay, welcome back. So, this is um this is problem four from the 2023 European girls. Um you're given uh how how do you draw a snail like I just do that or something? Uh and okay, here's a shell. Have some eyes. Hey, that that turned out a lot better than I thought it was going to. Anyways, there's a snail. And then you're given a bunch of C eyes and then you want to um the snail either goes this way or that way in each one and wants to not cover the entire circle in slime I guess or something like that. Um anyways, so um okay, so if if I if the CIS get more than one half, that's where things get things are really dicey, right? Because if C are more than half actually do you actually lose immediate do you lose right away if CI is greater than a half? Um yes because what happens is you No actually is that true.

[00:01:50]Okay. So I agree if like if if for all ICI is no actually wait why is that true?

[00:01:59]So if our all ICI is little little CI is strictly less than a half um then so anywhere standing wherever a turbo has gone before No I don't actually no it's it's no wait I'm confused again um wait I don't think you lose if if CI is exactly 1/2. Then you just keep covering the same arc back and forth, right?

[00:02:33]Yeah.

[00:02:35]So, so the set of points that's visited at some in some place or other is always going to be like some arc. And you don't you I don't even know for sure that you know I will um be at the end point of the ark but wherever you visited okay so wherever I'm standing if there is if CI if the CI are less than one half that means that the move I'm forced to make doesn't cover the entire circle but that doesn't actually no actually it code. So the the you're you're always in the interval of points you've already visited say after the first move. Maybe you're in an end point, maybe you aren't. But if all the CIS are less than a half, then the set of places you can go is like okay. So I don't know is there is there a case where something like this happens? Well, actually it doesn't matter. Like if you're if you're in the interval and there's unvisited points um then there's unvisited points in one of the what am I trying to say there there's some unvisited points. Okay. So you can always make a move that doesn't cause you to die. Uh yeah let's just that should just be true. Hang on.

[00:04:06]So if CI is if CI if if capital C is less than equal to half I think you're just okay because yeah there is an unvisited point somewhere so you can ensure there's still an unvisited point by making sure you don't hit it. Um I think I want to hold a proposition that the set of visited points is always a closed interval which it is. Um and then so if there's an omitted point actually there's a omitted it's not just the oh actually if CI is less than a half doesn't matter never mind if whatever it's just fine.

[00:04:39]So now if capital C is strictly greater than a half I mean I I guess that's why they want a strict equality inequality in the house.

[00:04:52]If if that's true I feel like I should just be able to I have to make sure I do this correctly.

[00:05:04]Um the claim should be that if C is let's say oops [ __ ] if C is greater is equal to 1/2 plus epsilon and the epsilon will probably matter here I think.

[00:05:26]No I lost the green color. What what was the color before? Was it this one? It see a half plus epsilon. No, that's not the right color. You know, let's just use the darker green. It looks nicer epsilon.

[00:05:54]Well, let's say C is less. Okay, let me write it this way. Suppose C is less than C is greater than 1/2 plus epsilon just so I can actually set ceilings.

[00:06:01]Okay. So I think the point is that I should just be able to um I want to show that I can always in let's increase the interval over say two moves. So suppose I have some slime or whatever. Um, so there there's some slime and then my snail is sitting somewhere.

[00:06:28]Um, we'll get we'll give this now a spiral on it show. Um, Like I want to show that I in a in some sequence of moves I can cause the length of the interval. So, um there's an antipode and I get let me I'm going to put a two in front of this. All right, I'm do I want a two? Never mind. No, no, that's let's not have a two. So, I have an antipode and I can I can make turbo go um in any like epsilon epsilon.

[00:07:47]So, I'm fine if Oh my god. I think I need to think two moves ahead instead of just one move ahead. Um Yeah. So, this picture is not amazing.

[00:08:30]Um, I mean, this this one's just stupid because obviously you can make the length increase. So, I really need to draw a picture where like the covered points are they're at least like up to here or something. Um or no, sorry, that that's not good. Um, okay. So, suppose I'm in a scenario. I think the scenario that's blocking me is if I have this entire thing filled and then like I don't know how to draw this picture.

[00:09:10]Uh I feel like I want like two different epsilons.

[00:09:18]Well, let's just do the case where the entire thing is covered first. I don't think the other I can I can always have the epsilons.

[00:09:25]Um, do I have the wrong answer? It could be 2/3.

[00:09:41]Okay. I mean, 2/3 1/3, you don't even need an infinite sequence for that, right? Like, if it's 2/3, like 2/3, 1/3 forces you to recenter and then the other 2/3 finishes it, right?

[00:10:49]Honestly, I feel like after the first move, it should just be like half of cuz you there two/3s 1/3 two. Yeah, I think 2/3 1/3 2/3 is just fine. Wait, what if it's So in this case, I mean So, you actually have the most power if your if turbo is at the very edge of the interval. Um then okay I'm going to do something that's uh I I might actually need like more than one moves like what I what I want to do in this position really is like take this move versus this move um like have a move that either pushes you to the edge of the interval because once you're in the edge of the interval you can defin you can force turbo to the center of the interval and then you can grow the interval by some amount bounded away from zero. So, um, actually, in general, I feel like I just need to set up variables.

[00:13:37]So, you make your first move. It's like a half plus epsilon or whatever.

[00:13:41]Um, and then, okay, we're just going to we're going to actually write things down. So, let's say it's on the left half.

[00:13:57]Um, and we'll we'll say x plus y is greater than a half to start because after the first move you're you're in that situation. Um, yeah, cuz I think 2/3 is good because 2/3s is like in in three moves you can win, right? It's like you you're at the edge right away and then you can push the center and then you do the two/3s and it's fine. Um, but I feel like you should be able to do something like that even when C is between 1/2 and 2/3. It's just you have to take more moves to do that. But if let's say X is if you have the length of the interval L is greater than a half and you have an X and Y and then you make a you you specify CI equals Oh, hang on. Wait, I'm getting the quantifiers wrong. Um, sorry. The CI is spec is fixed in advance. You specify the whole CI sequence and then Turbo studies the entire sequence. It's not an interactive game where someone makes a move and then someone specifies a C.

[00:15:03]Okay. Then then the answer might be greater than half. Uh, if you can see the entire sequence in advance, that might actually make things a lot easier.

[00:15:14]Um, sorry. I've been doing the interactive version the whole time. Um, ah crap. Okay, fine. Never mind. Ignore this picture. Um, this picture is not going to do anything. Uh, so if I have the whole if if it's actually the whole CI sequence is specified in advance, then all I need to do is specify which point I want to miss.

[00:15:50]And then sequences are really long. Interactive version is easier. I mean, I'm sure that I don't know.

[00:16:14]Oh, wait. Are we stupid? Maybe we're stupid. If you do half then half plus epsilon [ __ ] was stupid.

[00:16:34]No.

[00:16:36]Okay, so we can do the interactive version anyways. Never mind. Um, I think I should upload this to YouTube.

[00:16:55]I see. You actually make life harder for yourself if I try to use the hole in your Oh.

[00:17:13]God damn it.

[00:17:24]All right. Cool. Good control team.

[00:17:26]Thanks, Jar.

[00:17:39]Hey, now I have to decide whether I want this on YouTube or not.

[00:17:48]All right, fine.

[00:17:51]All right.

[00:18:06]All right.

[00:18:14]Okay. Um, yeah, let me let me type this.

[00:18:16]Although there's not very much to type.

[00:18:19]Um, Oops.

[00:19:10]I move there is at least one point say PI obviously before because CI less than a half turbo can also choose a direction to avoid PI. Okay.

[00:19:42]Suppose 12 plus epsilon.

[00:19:47]So choose epsilon greater than zero such that c is less than half plus epsilon.

[00:19:58]Let CI equals half when I is odd and a half plus epsilon.

[00:20:21]What happened to my Hey, don't they have a llinter that's supposed to remove trailing white space?

[00:20:27]Okay, there we go. I don't know why I didn't fire just a moment ago.

[00:20:32]Okay. I mean, All right. I'm I'm just going to leave it like this because I don't actually want to write out the rest. Um but yeah, this this should actually work.

[00:21:24]I mean because the point is okay fine I'll write a little bit um the first move is clockwise to see this works then second move must be counterclockwise then I mean clockwise.

[00:22:06]Okay. So has visited an interval of length 1/2 of epsilon and is at an end point. Third moves must be counterclockwise and the fifth move must be clockwise.

[00:22:51]No two moves. So no two consecutive moves can be in the same direction. I mean that's the racist way I did. This means Oh, actually if if no two consecutive moves must be in the same direction.

[00:23:06]Uh so we can assume they go clockwise.

[00:23:15]Then at the end of the 2k move for k one turbo is at the end point of a visited interval of length 1/2 + k epsilon which eventually making k large enough finishes the problem I don't know what I'm doing. Yeah it's just if if cis are bigger than half they the moves have to alternate. So if the moves have to alternate then all right we we'll just pretend this um turbo doesn't teleport. What?

[00:23:59]Okay. Anyways uh okay anyways. Yeah anyway that happened. All right. Good job team. Um cool. But that's all for today. Um or at least for this problem. I I'm the next problem might be better. I don't know.

[00:24:25]I agree. It was not obvious we should take all CI greater than one half because it feels like you want to move turbo to the center. I just didn't it it should have occurred to me that as long if you do that then there's only one sequence to analyze because the consecutive moves have to alternate directions.

[00:24:45]I think that's the thing. I think if you notice that consecutive moves have to alternate if the the sum of the CIS is greater than one then then you should be okay because then then you then you can just pick the sequence.

[00:25:05]All right.

[00:25:07]The green screen's not happy today.

[00:25:11]Yeah.