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linestein
United States210 Posts
A guard walks around a pool three times faster than you swim. You are in the pool swimming. Can you escape? The pool is a square, and you are swimming in the pool. I thought this was a great puzzle when I finally figured it out. I hope my answer is right!    | ||
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Alejandrisha
United States6566 Posts
edit: upon further thought i was thinking the guard could counter this by hugging a corner while you are in the middle of the pool. but the guard gets to the opposite corner just a tad slower than you do if you both go to the opposite corner at the same time. so i stand by my answer. curious what you came up with | ||
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Harris1st
Germany7414 Posts
On September 06 2019 10:19 Alejandrisha wrote: i believe so. if you start an infinitesimally small distance from the middle and then go the opposite direction, you will travel ~.5000001 the length of one side of the pool and the guard has to travel 4 times that distance to stop you I believe you are wrong. I'm to tired to do higher maths right now so I used number ^^ + Show Spoiler + Worst case: You are in the middle, guard is in a corner Pool is 4x4 Your distance (going to the opposite corner) is 4/2 ² + 4/2 ² = Root of 8 Guards distance is 4+4 =8 Root of 8 *3 > 8 | ||
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Alejandrisha
United States6566 Posts
On September 06 2019 15:08 Harris1st wrote: I believe you are wrong. I'm to tired to do higher maths right now so I used number ^^ + Show Spoiler + Worst case: You are in the middle, guard is in a corner Pool is 4x4 Your distance (going to the opposite corner) is 4/2 ² + 4/2 ² = Root of 8 Guards distance is 4+4 =8 Root of 8 *3 > 8 ah true. in the corner guard case i was using root 2 when i shouldn't have been. should have used side length of 2 rather than 1. thanks! | ||
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calh
537 Posts
On September 06 2019 10:19 Alejandrisha wrote: i believe so. if you start an infinitesimally small distance from the middle and then go the opposite direction, you will travel ~.5000001 the length of one side of the pool and the guard has to travel 4 times that distance to stop you edit: upon further thought i was thinking the guard could counter this by hugging a corner while you are in the middle of the pool. but the guard gets to the opposite corner just a tad slower than you do if you both go to the opposite corner at the same time. so i stand by my answer. curious what you came up with That is essentially correct. Assuming you start in the center: - If the guard is not at any corner, you swim to the opposite side to him and escape. - If he is at one corner, you swim to the opposite corner slowly while watching him. If he never moves, you escape. If he commits to one side, you swim to the opposite side and escape (basically reduces to the previous case with extra advantage for you). | ||
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Harris1st
Germany7414 Posts
On September 06 2019 15:33 calh wrote: That is essentially correct. Assuming you start in the center: - If the guard is not at any corner, you swim to the opposite side to him and escape. - If he is at one corner, you swim to the opposite corner slowly while watching him. If he never moves, you escape. If he commits to one side, you swim to the opposite side and escape (basically reduces to the previous case with extra advantage for you). You are absolutely right. With changing directions mid-swim you can easily evade the guard | ||
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Ej_
47656 Posts
There are 2 "boundary" scenarios: -the guard's in the corner (1) -the guard's in the exact middle of a pool side (2) Let t_s be the time it takes the swimmer to reach the side and t_g the time it takes the guard to reach the same spot. Let v be the speed of the swimmer and a the lenght of the pool side. For (2), the solution is trivial, as the desired spot of the swimmer is directly across the guard and t_s=a/2v; t_g=2/3v, then t_s<t_g For (1), the swimmer has to change their trajectory in relation to the guard (if a solution exists at all) Let us examine then: Let (0,0) denote the middle of a perpendicular coordinate system OXOY, let us set (0,0) in the center of the pool. Let's assume that the guard is initially in (a/2, - a/2) (bottom right). Then, the swimmer's initial trajectory is y=-x (towards top left). The guard will follow, traversing x=a/2. The swimmer should then change their trajectory to such that the guard has to travel the longest possible path, let's denote that as A. Let's denote swimmer's path as B. A solution exists only if: t_s<t_g => B/v<A/3v => B<A/3 (*). For B>2a, the guard would have to walk past a point of where the optimal strategy would be to walk around and approach the corner from the other side. Therefore, he would need to be in (a/2, -a/2+x) such that x<z and the swimmer's final trajectory point is (-a/2, a/2-z). Therefore, the swimmer's optimal trajectory is the initial y=-x, but after certain time, assuming y=a/2-z. That time is equal the time that the guard walks the distance of x (boundary case for x=z), therefore it's equal to z/3v. In z/3v, the swimmer would have swam a distance of z/3 along y=-x, so his position would be (-z/3/sqrt2, z/3/srtq2). So, the above solution exists if (B=a/2-z/3/sqrt2 and A=a-z+a+a/2-z/3/sqrt2) (1) and (B=a/2-z/3/sqrt2 and A=z+a+a/2+z/3/sqrt2) (2) Both satisfy (*). Let's check then: For (1) and (*) z<3sqrt2/14 * a Satisfies the solution For (2) and (*) We arrive at a/z<(3+4/3/sqrt2)/4<1 Which is bollocks and even if the proof above had been without errors (and I'm sure some are there), the swimmer wouldn't ne safe using a non-continously changing trajectiry :D Let me look for a simpler solution | ||
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Ej_
47656 Posts
The actual solution is also much much eaaier than what I initially did and boiled down to basic trig in my case :D | ||
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Harris1st
Germany7414 Posts
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raynpelikoneet
Finland43270 Posts
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Harris1st
Germany7414 Posts
On September 06 2019 21:09 raynpelikoneet wrote: Depends on how fast can the guard swim. He is obviously afraid of water  | ||
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brickrd
United States4894 Posts
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Nakajin
Canada8991 Posts
On September 06 2019 21:09 raynpelikoneet wrote: Depends on how fast can the guard swim. Also depend how fast you can run once your outside of the water. | ||
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Clazziquai10
Singapore1949 Posts
Assuming the swimmer starts at the centre and the guard starts at one of the corners and you try to escape by swimming towards the opposite corner, you can escape by making a juke towards the opposite edge that the guard is running along. ![[image loading]](https://i.imgur.com/hVq22pz.png) As long as you make the juke anytime between the start of your 45-degree diagonal swim and when you are (1 - 2 / (3 Sqrt(2) - 2) ) X from the edge of the pool, you're safe | ||
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linestein
United States210 Posts
here were what i thought of as the two bounding cases. you swim to the middle of the pool and then either A) ![[image loading]](https://i.imgur.com/jD8eNoW.png) or B) ![[image loading]](https://i.imgur.com/sZ7vVhG.png) now obviously in case A you escape but in case B you don't. e.g. 1.41421356237*3 = 4.2426 which is greater than 4. so finally i would refer to clazziquai's solution where you make a little turn to gain ground | ||
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Chairman Ray
United States11903 Posts
For the problem "What is the fastest speed the guard can walk such that the swimmer is still able to escape", here is the optimal solution: ![[image loading]](https://i.imgur.com/QH2kimb.png) The swimmer should start in middle and guard should start in corner. The swimmer should always swim towards to point on the edge opposite of the guard such that the guard is indifferent whether to go forward or turn around, with a slight adjustment towards the point on the edge closest to the swimmer. Using this solution, the swimmer should swim in a curve like in the picture. Calculating that curve involves some calculus that I have forgotten since college. For the problem "What is the shortest time that the swimmer can escape in", here is the optimal solution: ![[image loading]](https://i.imgur.com/xbeSRKZ.png) You also need some tricky calculus to get the exact curve and end point, but the optimal path for the swimmer is to travel below the indifference curve from the previous solution, such that the guard will walk one direction, and then turn around so that the swimmer reaches the edge just before the guard reaches it. For the original problem, using guess and check, all I did was have the swimmer swim towards the opposite corner, and then head straight left after traveling 1/4 of the way. The swimmer makes it. ![[image loading]](https://i.imgur.com/kXh5T9n.png) | ||
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Alejandrisha
United States6566 Posts
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Alejandrisha
United States6566 Posts
On September 06 2019 21:09 raynpelikoneet wrote: Depends on how fast can the guard swim. he's a cat person. and i don't mean he prefers cats to dogs i mean he can't go in water | ||
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fluidrone
France1478 Posts
+ Show Spoiler + | ||
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Chef
10810 Posts
I think the best strategy in this case is to try to catch the guard running to get to you, splash water in his path, and hope he slips and splits his skull or otherwise really hurts himself. Then you've got a shot to find some clothes and skedaddle. Alternatively maybe you can quickly pull his leg and get him to fall into the pool and buy yourself some time that way, that is probably easier to pull off but less likely to incapacitate him. Also risks getting hit in the head with a baton when you go to do it, depends how vicious the guard is. The mathematics are kind of irrelevant to the problem, you don't give yourself much of a head start even in the best case. | ||
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DarkPlasmaBall
United States46475 Posts
That being said, calling a relatively difficult SAT math question "High IQ" did make me smile | ||
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Strangerinastrange1
1 Post
Now if we replace the round pool with a square one with side length 2 the guard has to run 4 to get to the exact opposite side. Because the guard has to run farther you can still hold him at max distance while swimming a circle with radius 1/3. The maximum distance you would have to swim is to the corner, starting with a head start of 1/3;sothe distance is d=(sqr(2)-1/3) which is about 1,0809. As the guard has to run 4 he will take 4/3 time which is longer than you take. I really liked this problem, it was deeper than I thought at first glance. | ||
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linestein
United States210 Posts
On September 08 2019 11:50 DarkPlasmaBall wrote: "Can you escape?" is a weird and and ambiguous question, because that implies that you (the swimmer) have to actually get out of the water after reaching the edge of the pool. It would make much more sense to talk about racing the person outside of the pool to the edge of the pool or something like that, because there's no information regarding how fast you can lift yourself (or walk if there are steps) out of the pool, etc. That being said, calling a relatively difficult SAT math question "High IQ" did make me smile yes, IQ 190 questions are regularly in the "easy section" for graduate student homework. | ||
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Harris1st
Germany7414 Posts
On September 08 2019 02:13 Chef wrote: Does this question assume that once I'm out of the pool I'm faster than the guard? I think I'm probably expending more energy just by swimming or treading water, so I'm already going to have a tough time. Plus even if I can swim at that speed, it takes time to get out of the pool, more than enough to catch up for the guard unless I've got the strength of a dolphin to leap out in one go. I think the best strategy in this case is to try to catch the guard running to get to you, splash water in his path, and hope he slips and splits his skull or otherwise really hurts himself. Then you've got a shot to find some clothes and skedaddle. Alternatively maybe you can quickly pull his leg and get him to fall into the pool and buy yourself some time that way, that is probably easier to pull off but less likely to incapacitate him. Also risks getting hit in the head with a baton when you go to do it, depends how vicious the guard is. The mathematics are kind of irrelevant to the problem, you don't give yourself much of a head start even in the best case. The moment, when a math problem leads to murder :D | ||
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FFGenerations
7088 Posts
don't know why i'm imagining everyone in the pool naked but i am | ||
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FlaShFTW
United States10566 Posts
On September 09 2019 19:36 FFGenerations wrote: can you turn back and repeat the sequence multiple times to slowly gain a greater edge? that way you could get ahead far enough to account for getting out of the pool and grabbing a towel to put over your dinger don't know why i'm imagining everyone in the pool naked but i am these are the real questions no one is asking, and thus you should be 200IQ. | ||
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Acrofales
Spain18446 Posts
On September 06 2019 07:56 linestein wrote: The renowned universal genius, Rick Rosner, who has a verified IQ of 192 posed this question to his twitter followers: A guard walks around a pool three times faster than you swim. You are in the pool swimming. Can you escape? The pool is a square, and you are swimming in the pool. I thought this was a great puzzle when I finally figured it out. I hope my answer is right! Seems to me that if you start in the middle and swim to the opposite side of the pool from the guard you will get there before him as long as it is on one of the sides. If he is in the corner and you swim towards the opposite corner, he will get there before you, but he has to go around one side or the other. So when he picks a side, swim straight towards the edge so he would have to walk past the corner you were heading to to get there. | ||
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Acrofales
Spain18446 Posts
On September 06 2019 15:33 calh wrote: That is essentially correct. Assuming you start in the center: - If the guard is not at any corner, you swim to the opposite side to him and escape. - If he is at one corner, you swim to the opposite corner slowly while watching him. If he never moves, you escape. If he commits to one side, you swim to the opposite side and escape (basically reduces to the previous case with extra advantage for you). Great, we reached the same answer. IQ 192 confirmed!  | ||
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Dingodile
4139 Posts
Albert Einstein created an intelligence test for us all, said that 98% of people can't solve it (he said that ~80years ago). Today I am sure ~20% of all 1980+ born people can solve it within 30 mins. | ||
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Alejandrisha
United States6566 Posts
On September 13 2019 04:37 Dingodile wrote: https://iqtests4kids.com/einstein-IQ-test.html Albert Einstein created an intelligence test for us all, said that 98% of people can't solve it (he said that ~80years ago). Today I am sure ~20% of all 1980+ born people can solve it within 30 mins. i heard somewhere that the average iq has drifted from 100 to 115 since the tests were originally made because of the proliferation of information. have no sources tho | ||
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Acrofales
Spain18446 Posts
On September 13 2019 04:37 Dingodile wrote: https://iqtests4kids.com/einstein-IQ-test.html Albert Einstein created an intelligence test for us all, said that 98% of people can't solve it (he said that ~80years ago). Today I am sure ~20% of all 1980+ born people can solve it within 30 mins. Took me more than 30 mins, but I don't have an adequate notebook, so spent quite some time drawing lines and tracing them because they weren't adequately straight Still, top 2% according to Einstein! + Show Spoiler + For real? You sure? + Show Spoiler + Really sure? + Show Spoiler +
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SkrollK
France580 Posts
Took me ~40 mins. But I honestly don't think only "top 2%" on earth can solve this lmao. Probably loads more !! Still, it was a nice test | ||
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ninazerg
United States7291 Posts
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FFGenerations
7088 Posts
who do they think they are | ||
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Harris1st
Germany7414 Posts
On September 16 2019 17:59 ninazerg wrote: If you grab the life-guard by the ankle, he will probably slip and fall, breaking his hip. You will then be able to escape pretty freely, but then suddenly realize that the life-guard isn't there to keep you in the water, but to help you in case you drown. Oh no, what have we done? That guy is just doing his job! You are like the fifth person who wants to inujure or kill the poor guard I just realized there is a shitton of really cruel people on TL... | ||
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Vivax
22414 Posts
My random lazy guess without math and expecting the guard to walk towards where I'm swimming. I'd swim to a corner, swim back to the middle towards the opposing corner leading the guard there and then back to the initial corner. | ||
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Alejandrisha
United States6566 Posts
On September 16 2019 21:57 Harris1st wrote: That guy is just doing his job! You are like the fifth person who wants to inujure or kill the poor guard I just realized there is a shitton of really cruel people on TL... lol right? what's up with killing the guard. just swim good. | ||
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WombaT
Northern Ireland27415 Posts
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Vivax
22414 Posts
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Acrofales
Spain18446 Posts
On September 17 2019 08:09 Vivax wrote: The pool thing might be the solution. Why am I being guarded while in a pool? Because it's my bodyguard, dummy! How do I escape? I don't, I just climb out. It's a hotel pool and you're there after hours. E: more surprising is that it's square! | ||
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Grumbels
Netherlands7032 Posts
On September 08 2019 11:50 DarkPlasmaBall wrote: "Can you escape?" is a weird and and ambiguous question, because that implies that you (the swimmer) have to actually get out of the water after reaching the edge of the pool. It would make much more sense to talk about racing the person outside of the pool to the edge of the pool or something like that, because there's no information regarding how fast you can lift yourself (or walk if there are steps) out of the pool, etc. That being said, calling a relatively difficult SAT math question "High IQ" did make me smile It's a difficult question, because you have to first reason about what your optimal starting point is, what the guard's optimal starting point is, whether you're changing direction mid-stream and so on. And even if you find an escape using one scenario you have to prove it's an optimal scenario of some sort. | ||
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Chef
10810 Posts
On September 16 2019 21:57 Harris1st wrote: That guy is just doing his job! You are like the fifth person who wants to inujure or kill the poor guard I just realized there is a shitton of really cruel people on TL... In justification of the scenario I imagine that I am an old Jewish man trying to relax in a pool in burgeoning Nazi Germany. The 'guard' is a member of the Nazi Youth, not wanting to get his stupid uniform wet and patrolling around the pool shouting slurs at me and telling me how he's going to beat me up when I come out of the pool. I'm not necessarily a violent person or wanting to really injure him, but as a matter of self-preservation I know his threats are real and I'm not going to feel that guilty if he hurts himself when he slips. To me the question of the op was one of thinking outside conventions and picking out the actually important details. If it was just a math problem, why have the story about the pool? Most people seem to have decided the context and actual problem are not the important part of the question. In order for the mathematical solution to be realistic, it would have to be more like a video game, where one player or the computer is the guard and can cover a small amount of the border, and the other is a little dot inside the border that moves at a third of the speed. But that's not the same question, even if it is an interesting question. | ||
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FFGenerations
7088 Posts
that's the scenario i was imagining | ||
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Harris1st
Germany7414 Posts
No blood, breaking of limbs or killing involved | ||
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Kitai
United States886 Posts
On September 13 2019 21:33 Acrofales wrote: Took me more than 30 mins, but I don't have an adequate notebook, so spent quite some time drawing lines and tracing them because they weren't adequately straight Still, top 2% according to Einstein! + Show Spoiler + For real? You sure? + Show Spoiler + Really sure? + Show Spoiler +
That was a fun one. Took me less than 30 mins, but I cheated and used Excel to color-code matching clues  . Here's a picture of a few failed attempts with my successful solution at the bottom:+ Show Spoiler + ![[image loading]](https://i.imgur.com/F5D5R6B.jpg) | ||
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Grumbels
Netherlands7032 Posts
On September 20 2019 01:24 Kitai wrote: That was a fun one. Took me less than 30 mins, but I cheated and used Excel to color-code matching clues . Here's a picture of a few failed attempts with my successful solution at the bottom:+ Show Spoiler + I tried it as well, took me like an hour using notepad.  But in my defense, at some point I got stuck trying to do "what-if" scenarios wasting a lot of time. By the way, I did a class on logic solvers for university, and these problems are very easy to define using almost pure boolean logic (with natural numbers extension). For instance, here you would need 5 arrays for each of the labels a house can have, then you would use a pre-set function demanding that the array is bound 1..5 and that every value is unique. Then you can say that e.g. Pet[0] = Color[3], and if there is a unique solution it will find it in less than a second. See here: https://rise4fun.com/Z3 | ||
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