Game of Cat and Mouse - Numberphile

I'd like to set your puzzle the puzzle is about a cat a mouse and as is traditional in cat and mouse situations one is chasing the other the mouse is super quick compared to a cat even though the cat is pretty quick but in this case to keep interesting the mouse is without realizing run into a pond there's this whiskers and little tail and if you go anywhere in the pond because the cat worst mouse it's got to be small isn't it like I could draw it big and then you'd be like worst pond ever compared to the mouse anyway to scale then here is the cat good enough No how's that it's gotta have ears all right yes it's not bad that's better so for the sake of the puzzle the cat is gonna sit on the edge but it can cruise around the edge waiting for the mouse to leave the pond that's like stalking the air exactly now I think it's believable that even though the mouse could run quicker than the cat on land in water it's going to slow down it but it can swim we're not gonna we'll do the modeling situation you know Steve some things like the mouse it's not going to drown it can swim that assuming the cat doesn't actually go in the water although it probably could and swipe it from the edge all of that stuff let's just assume it sensibly for puzzle world and let's assume the cat can run four times quicker than the mouse can swim in fact that's the only numerical detail I'm going to give the cat can walk or run four times quicker than the mouse can swim and that's it so the simple question now is can the mouse escape [Music] it's not a Finn sports video without a computer animation bit of a geogebra file with some nice brown paper to represent our pond as ever we've got a cat who runs around the edge luckily he looks in the right direction I'm very proud of that animation I'm very glad that everyone else is appreciating that the mouse can go anywhere he likes if he goes the other side of the cat does check where he's going but let's assume he starts at the center and there's a couple of things to sort out first is that one tactic the mouse could do is the obvious one of run away in this case swim away but you could just say like why don't you as a mouse just swim away from the cat and what I mean from that is like if you imagine a line from the cat to the mouse wherever that is imagine that line and the mouse swims in the opposite direction and I can show you that happening actually so I'm going to click this button up here is the away tactic and the reason I'm showing you is that it doesn't work then you see the mouse is following this little arrow which is moving around depending where the case but the count is so quick that you can always kind of get around to head the mouse off even though it looks like he was going to make it the cat camera was real quick and we get these actually nice pans are turning up which are typical of these curves that pursuit they're usually kind of curvy they're not always easy to model mathematically and these ones that you know they're always getting symmetrical but there's a little bit of randomness you can see the cat wobbles when he gets in line cuz he's he's constantly trying to move but he doesn't need to move because he's already as close as he can get to the mouse but it's pretty obvious after watching this for a few seconds this is going to repeat even if it's not gonna be exactly the same path the mouse looks like he's doomed so the runaway tactic or swim awake tactic doesn't work which means if you've been thinking about that on your little pause where you were just thinking about it you should probably stop thinking about it because it's not going to get there all right well I think it's a pretty sensible tactic and we it'll be a stupid thing not to try it first and it's difficult to sort out in on paper with mathematics whether it's gonna work so the nice thing to do quite often for mathematicians even physicists do this sometimes is to model it on a computer and realized very quickly doesn't work I'm gonna suggest a hint now though that I think the mouse can't escape and therefore it's instructed to think if this isn't working how else could he possibly do it and get away okay so again if you want to have a think about it how think about Mouse has not gone away yet the obvious tactic of running away or swimming away doesn't work and one other tactic which a lot of people tell me with this puzzle that might be sensible is instead of running away from the cat maybe you should run towards the point which is farthest away from the cat so I think that's obviously in this case the pond edge point opposite the cat no no it sounds like that's the place you want to be if you can arrange it because the cat's not anywhere near to eat you of course that the opposite point tactic so I'm going to show you what happens if you try a mouse doing the opposite point tactic here we go you see the green arrows I'm heading towards the point opposite the cat which is chugging away around because the cat is quick and it keeps chasing the mouse and this angle is trying to it's gonna close the angle and it just means the mouse goes in circles because he's not moving quick enough obviously the puzzle will be different if the mouse was quicker but it wouldn't be as interesting because the mouse could just swim through so I just wanted to point out another sensible tactic to try swim away from the cat towards the opposite edge doesn't work either and this time he just goes around in circles there are two more tactics I want to show you neither of which really help but together they're quite good so let me introduce some vocab which I've just made up which will help me refer to these tactics what I'm going to call the dash tactic which is swim away but not away from the cat or towards the opposite edge but just towards the nearest edge of the pond I feel like sensible tactic you're in a pond they wanna get out you don't swim the long way so on the screen this looks like this so I've got a dash tactic and the moment my mouse is closest to that point over there it's kind of like the radius from the point through the mouse to the edge and if I click on - he just heads towards the a now obviously that wasn't very successful but it depends where the mouse is to start with right if I put the mouse on the other side a long way away from the cat and he's close to the edge then a dash will get him out that doesn't solve our problem but it means there are some places in the pond where if the mouse can get to I know he can get out with a dash tactic second tactic then is not a dash technique this is circling tactic and this doesn't help you get out but it does help you get away so what I mean here is that the mouse can move in a circle and this point here is worse for the mouse then this point despite being the same distance from the edge so I think this one's better because in terms of an angle he's further away from the cat which means measuring this angle just drawn a line on here it's a good indication of whether the mouse is far away like opposite or close and actually I think that kind of summarizes the cat's technique as well the cat will try and minimize this angle and if the mouse has got any sense at all we should try and maximize that angle but it doesn't always work so here's a circling tactic if you are near the edge of the pond it's a problem because there mount the cat is much quicker for example the mouse is just circling but it's clearly moving slower around the angle than the cat and if the cat was actually doing something sensible instead of is going round let's actually make him pursue he will get to the mouse and it will stop and then the circling tactic sump doesn't work anymore because whatever the mouse does the cat can keep up but if I get further away from the edge but I put it over here you can see that the cat catches up but just not quite as quick which might give you a clue that if I'm rightly in the middle it's pretty obvious that even if the cat is really close in the angle the mouse can get away now so the circling tactic means sometimes I can get away from the cat in terms of the angle and sometimes I can't because the cat catches me and those two tactics maybe I maybe it's time for another pause with those two tactics I think there's a glimpse of a hope those two tactics the - tactic and the circling tactic which sorts out whether you can get the cat opposite you or not and let's sort them out one at a time so then the - tactic let's assume the - tactic we are opposite the cat so if the cats here the mouse is on this line the other side of the center but if he is at the center I don't think he can make it and maybe you can think about why the cat has to go around here let's assume some things which includes this radius of the pond which I've never told you I'm going to assume it's one and you can scale up if you want the distance the cat has to go round is half of a circumference so around is 2 pi times 1/2 it's PI meters let's go with meters for the sake of unit and let's assume there's some speeds going on let's seem the cat speed for the sake of any other argument is 4 meters per second that's a quick cat and then house is therefore 1 meter per second and that's the only number we set this up with their cat is 4 times quicker than the mouse can swim and you can see that if the the mask from the center will less says 1 is going to take precisely one second to get out but how long does the cat take to get around well yes this is everyone's like wow which how do you calculate this I think the speed equals distance over time thing is a really basic equation that we really need to use carefully how long will it take the cat to get there time is distance over speed the distance is pi so it's PI divided by speed which is 4 is PI by 4 seconds so it's quicker than a second it's because pi is less than 4 this is less than one second the cat is there waiting first thing when the - tactic is clearly the mouse can't depth from the center so the obvious question is where can he desk from and it's the same calculation right he's got PI by 4 seconds at most to get out so because the cat can get to any point on the circle within ya the farthest away point is PI by 4 seconds away so actually he can only get to the edge when the edge is less than PI by 4 away because he goes 1 meters per second we've set these numbers up to be quite nice but that circle has distance PI by 4 there which means this dis in here is 1 minus PI by 4 and you begin to realize very quickly that my scale historian is terrible we need to know roughly where that line is because outside that line if I'm opposite the cat I'm home and dryer just dash for it initially let's just check you oh you're all estimating us on the video but one minus PI by 4 0.2 one from Center - tactic it's like an event horizon bit yes that's exactly what it is like inside that you're doomed with your - tactics out like that if your opposite the cat you okay so now the question really is can I get opposite the cat outside that event horizon so we actually now you recognize why the circling tactic is important because that's the one that concerns whether we can get opposite attack so let's talk about beer that I call it the TAC I'm at the cat reversing things let's talk about these circling this in some ways is feels harder to sort out at first cuz like running in circles feels difficult but if the cat is running at 4 meters per second and the mouse is at 1 and actually what we care about is how quickly I can get round compared to the cat going round how big a circle can I keep up with the cat if it's 4 times quicker than me I think on moments door is friends of the obvious that if the circle is 4 times smaller I'll be doing the same thing as the cat so actually this circle I've just gone here if that's 4 times smaller than the big one the scale factor of 4 going on the distance round it is 4 times shorter than the big one that's how a scale factors work and so at that point which is a quarter of the size in which case that is 3/4 and the distance in there looking pretty messy thanks to my repeated drawing of mice but it is a 1/4 on that circle I can keep up with the cat in terms of angles which means inside that circle I can outpace it yet in terms of angles so which means I think 1/4 otherwise known as 0.25 meters from the center my circling will work this point these two numbers and their comparison is important and deliberately sort of talking slowly to let the realization stop in this is a nice puzzle where the massive Don is is really lower than GCSE level maths and yet it wasn't obvious to me at all when I first started playing with the cat the mouse but let me show you on the diagram on the screen so now we've established that the there are two event horizons if you like we can put the one here so the - boundary is this line here in fact that was the boundary said was 0.21 away from the center and you can see it's about right and the circling boundary is the outer line there so inside that I can circle okay and outside the - boundary I can - okay can you see what I see that there's a little gap in between which I'm going to officially call the mouse sweet spot if it can get in there then not only can it circle to get away from the cat it can also when it's ready i opposite the cat it can just peg it and it should make it which is weird because just swimming away from the cat won't make it and just swimming towards the opposite bank from the cat won't make it but this tactic according to the math should make it so I set up Joe Dubrow which doesn't know about these boundaries - a tale told the mouse to get in that sweet spot and then when it's opposite the cat or close enough like about hundred 79 degrees or there abouts peg it and I was really quite nervous when I'd done the coding to see if it actually make it because I must as they should make it and no other tactic on the model had worked so I'm going to press play I'm glad Brady that's why you make these videos so I'm gonna start the mouse in the center of the pond because I know that if it's anywhere else I could still swim to the center and solve it from there and let's see what happens I'm gonna click on escape tactic and what I hope the program will tell it to do it it will swim out until it gets into that sweet spot which is that small little region and then it was circle until they recognizes this opposite the cat you'll see the angle go up to about 180 or within some tolerance and then you'll pick it and I've no idea where we'll end up to pick it but if it gets to the edge the exciting payoff will be a bit of red tech saying the mouse has escaped that's what I'm hoping for ready I'm really found the sweet spot very quickly he's only just inside the sweet spot you see the angle is increasing but it's not quickly increasing come on Mouse it's made it I agree that would be a tense moment for the mock-ups but if I assume as I did right in the beginning when the mouse is much quicker on land and the cat I could just imagine a game and their cats like ah but I think the math tells me that wherever the mouse starts whatever the cats doing if it's four times quicker than the master swim it can escape so how fast does the cat need to be for the mouse to never escape Brady I like your style you're doing what all mathematicians want to do in their heart of hearts actually most humans that you're generalizing right the four is critically chosen so that that point to one and 0.25 are really close together and then the right way round if they're the other way around you you can only - from a point where you can't a circle whereas we had a sweet spot and actually I've got a slider on Joe Dubrow so I can change the ratio of the speeds and you can see whether the sweet spot increases or decreases so since you asked let's let's move the mouse back to the center let me turn these too often that is the region that I called the sweet spot the critical region is maybe a more technical name for these sort of things at the moment the ratio is four if I change that to four point one can you see the sweet spots gone really quite small which means if the cat is four point two times quick it is all over somewhere between four point one or four point two looks like is the end of the story if I go the other way you can see the sweet spot is really big this is not a surprising the cats lower the mouse has got a better chance in fact there gets to a point where if the mouse is not quick enough even at three point two in fact if I go between three point one and three point two you can see that something there's it's all sweet spot in the middle and 3.1 at 3.2 do bracket a famous number which I'm sure people have heard of and I've used it's called pi it says to do with actually the speed compared with pi here if it's less than pi it's all sweet for the mouse and in fact it can dash from anywhere and get out so there's a really tiny area and for is this lovely chosen whoever came up with this puzzle I haven't managed to track down who originally came up this puzzle seems to have been around a long time but I hadn't heard it until a couple of years ago when someone told me about this puzzle we were sitting in a bunkhouse in the leg district around fire and he said I got a puzzle for you and just told me about the cabin the mouse and I don't know as bit arrogant perhaps I was like this is gonna be a simple puzzle completely flummoxed me for a week I was thinking about it driving back from the Lake District and when I solved it I was really pleased that it wasn't that I didn't know the maths it's just that it was a nice puzzle and they've chosen the for really well because any other number much bigger than for much more than before and either it's too easy for the master escape or it just always gets caught so that's the cat-and-mouse puzzle a bit of play and a bit of pond so if you like challenging problems like the one we've just shown you but maybe you want slightly better pictures than Ben's Mouse here brilliant might be the place for you brilliant have sponsored today's episode and they have really cool daily challenges like these have a look at them this one about hitting a bowl around a corner appeals to my brain obviously I'm gonna click on this one it's pretty interesting stuff here and look at that that is quite a shot but how could it be played to check all this stuff out go to brilliant org slash numberphile that's not only going to introduce you to the brilliance of brilliant but it also sets you up for 20% of their premium subscription premium means you get access to not just the free stuff that's on the site but all the archives all the puzzles all the courses all the daily challenges that address again brilliant org slash numberphile use the slash numberphile so they know you came from here and check them out 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Channel: Numberphile

Views: 662,966

Rating: 4.9112058 out of 5

Keywords: numberphile, cat, mouse, curves of pursuit, circles, pi

Id: vF_-ob9vseM

Length: 18min 36sec (1116 seconds)

Published: Tue May 28 2019

Reddit Comments

[u/DemIce]

There's a pretty good thread in one of the comments about improving tactics. E.g. the dash should not be a straight line, but diagonal away from the cat's position. This can be optimized for most distance from (= most time until intercept by) cat. The mouse can also enter the 'sweet spot' more optimally by gradually making its way to the sweet spot while maximizing the angle between it and the cat, rather than taking the shortest linear distance to the sweet spot first, which may put it initially closer to the cat.

What other optimizations could be done, and what would be the optimal strategy? I'd love to see them re-visit this after people's input.

[u/Homunculus_I_am_ill]

I'm having such a dejà vu. I feel like I've seen a video on this exact problem before but phrased as a monster and a swimmer, I even recall it being animated with the ski game yeti. I could have sworn it was numberphile, but it could be some other math youtuber.

[u/SupremeRDDT]

Does someone know how to actually prove that you can‘t escape using the first two methods shown? I would guess you could maybe model it using differential equations hmm.

[u/ido4121]

Can't you also use an ellipse maybe?