School-Safe Puzzle Games

The Girl in the Lake

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We are reaching the end. Here is the 4th logic puzzle:

A girl is at the center of a circular lake of radius R and a man at the perimeter. The man desperately wants to catch the girl. The only way the girl can escape is to reach the perimeter of the lake and run away. Swimming speed of the girl is S. Running speed of girl and man is 4S. What strategy/path should the girl follow to guarantee a safe escape?

29 Comments to “The Girl in the Lake”

  1. kasabubu | Profile

    She has to swim half the way to the shore. Then she should swim in a circle around the center, maintaining the disance to the shore. Swimming a circle at a radius of only 0.5R, she can easily get to the opposite side of the man. Now she can swim to the shore without the man being able to catch up with her.
    The math behind it:
    0.5R/S < 3.14R/4S

  2. hex | PUZZLE MASTER | Profile

    What is the man’s strategy with respect to the girl’s moves?

    Does he always run clockwise?

  3. suineg | PUZZLE MASTER | Profile

    I have some questions Bilbao:
    the girl can run in the water at the same speed the man run in the perimeter?
    the girl can choose between running in the water or swimming even when running is 4 times faster than swimming?

  4. MFox | Profile

    Is the man allowed to enter the water?

  5. MFox | Profile

    If she begins to swim directly away from the man, he will have to commit to running around the lake either clockwise or counterclockwise. The instant he does, she should begin to veer the same way (clockwise or counterclockwise) to whatever point on the shore is exactly opposite the man’s position.

    Presuming she never gets tired, she’ll keep the man chasing her as she gradually spirals closer and closer to shore.

  6. MFox | Profile

    I should clarify that as the man’s position changes, she should continually change her course toward the point on the shore that is opposite him.

  7. hex | PUZZLE MASTER | Profile

    I think that a sure strategy to escape, provided the man is coherent and starts at the North of the lake, is for the girl to:

    1- Always go in a direction diametrically opposite to that of the man. She will go in a circle of diameter R/4 and would have covered a quarter of it while the man would have run a quarter of the lake circle so that he is east of the lake.

    2- At that point, she has to go West in a straight line. She will reach near the West of the lake before the man.

    3- She puts high gear and flees West on foot (or motorcycle).

  8. Mashplum | PUZZLE MASTER | Profile

    Call the man’s starting point 12 o’clock. The girl should swim toward the man until she is (1/4)R from the center. The man will not move. Then she should swim clockwise the circumference of a circle with diameter (1/4)R that intersects the center of the lake. This should cause the man to run clockwise around the lake to keep her from getting to the edge. It will take the girl (1/4)pi*R/S to complete her circle. The man can circle the whole lake in 2*pi*R/4S or (1/2)pi*R/S. So when the girl reaches the center of the lake, the man will be at the 3 o’clock position, and when the girl completes her circle, the man will be at the 6 o’clock position. If the girl then swims straight to 12 o’clock, it will take her (3/4)R/S which is less than the (1/4)pi*R/S it will take the man to return to the same point.

    Of course, if the man figures out he can double-back somewhere between 3 and 6, the plan will not work.

  9. Shawn | PUZZLE GRANDMASTER | Profile

    If the girl swims in a straight line toward a point directly opposite where the man is standing, he will be able to catch her.

    Time for girl to reach shore = R/S

    Time for man to reach the same spot = (2*pi*R/2)/4S = (pi/4)*(R/S)

    Because pi/4 is < 1, the man will arrive at the point first.

    So the girl’s best strategy would be to continually adjust the direction of her swimming so as to always be heading toward a point directly opposite from where the man is at any given time. This will result in a curve emanating out from the center of the lake, and looking like the yin-yang curve that we saw a while ago in another thread.

  10. mashkalji | Profile

    If the man is allowed into the water, then she swims towards him, and then let her chase her, then get out of the water, assuming they both swim at the same speed.

    If the man can’t enter the lake, she first swims towards him, for a minimum distance of more than .0215*R and he won’t move, then swims at the opposite direction away from the man, then he will move to meet her at the other end, and at the moment he is exactly in front of her she changes her direction 180 degrees, and as she gets out of water she escapes away from the man. since their speed on ground is the same he can’t catch her.

  11. michaelc | Profile

    It seems like the girl should swim in a slight spiral in the direction away from the man wihtout putting the math to work at it.

    Another thought is this a couple? Maybe the girl wants to be caught? :)

  12. bilbao | Profile

    The man cannot enter the water (let’s assume he doesn’t swim). So he can run in the perimeter either right or left in immediate response to the girl’s movements.
    In the water the girl can only swim (not run) at a speed of S. If she reaches the perimeter at a point where the man didn’t yet arrive she may run at the same speed as the man (4S), and thus, escape.
    Sorry I wasn’t clearer with the wording!!

  13. Sebastian Roughley | Profile

    I’m not sure why, as in I don’t know the maths, but my gut is telling me that she swim in a spiral from the middle to the perimeter.

    I don’t have any workings out to back it up though, sorry!

  14. bizarette18 | PUZZLE MASTER | Profile

    I think she can make it if she firstly swims more than 21.5% but less than 25% of the way to the side. (If it’s a big lake she can swim a shorter distance so she will have less swimming to do, but further if it’s a small lake to give herself more margin, because you can’t really run out of a lake very fast.) Then she swims in a circle round the centre of the lake until the man is exactly opposite. He’s chasing round the edge but can’t keep up because she’s doing smaller circles. Then she makes a dash for it to the nearest bank and runs away from the man. If she swam as far as she could underwater that might give her a bit more margin too. Before she tries to escape she should also keep going underwater and pop up in unexpected places.

  15. preswilli | Profile

    Depending on the depth of the water she may be able swim at a depth where her movements are not visible to him. If she ends up at a location away from his then she will be able to escape.

  16. suineg | PUZZLE MASTER | Profile

    I am more intuitive than logical so I would say its like an spiral shape, the arc that the new direction the girl should go must be the same as the arc the man describe in the circle even if iit looks like she is going toward the center again, logically in an intial point the man is 2 pi R of the point where the girl is R so I think by intuition that in the spiral you go back to 2 pi R for the man but the girl is at a distance that is shorter than R and so on, round and round jajaj cool.

  17. pele1515 | Profile

    Just thinking very basic here…Assume the lake was fairly small, say small enough to swim across it underwater in one breath…

    She could just go underwater and swim to the edge.

  18. barryd | Profile

    Maybe I am missing this but the girl can swin a speed s and the man can run at speed 4s then if she starts swimming from the center directly away from the man then she will reach the edge in r/s time units. Meanwhile the distance that man will have run to get there will be half of the circumference of the lake i.e. half of 2.pi.r or pi.R so will arrive there in pi.r/(4s) time units. Since pi is 3 point something, he will arrive < r/s time units and be there before she arrives. The marginal distance extra he has to cover by her altering her continually direction away from him as he runs should not be enough to overcome 20% edge in time if she does not alter her course. So she is trapped!

  19. Falwan | Profile

    suppose girl is at center of the circle and man is at (r,0)…

    she heads toward to the point opposite to where he is toward(-r,0) at a speed of s..

    by the time he ia at (0,-r), she changes direction opposite to where he is toward (0,r)..

    and when he is at (-r,0), she also chnges direction to (0,r)..

    and so on, and so forth..;that is she keeps changing directin to a point opposite to his location at the perimeter..

    the girl’s circle will gradually increase and she will reach a point at the perimeter before he interceps her..

    that also implies if he heads the anticlockwise direction..

    hope my English is clear for everybody (including me!).

    many thanks to the Doc. and mr. bilbao

  20. kasabubu | Profile

    Is there a reason why my comment, sent October 21st, 2009 – 12:57 am, is still awaiting moderation? I nailed it, huh?

  21. Mashplum | PUZZLE MASTER | Profile

    Kasabubu, in your calculations, the girl’s circle doesn’t contain pi, so it won’t work. Everyone else, I am not convinced of the spiral solution. The swimmer cannot spiral from the center to the edge and expect to get there before the man. Once she is more that (1/4)R from the center, she will not be able to keep herself diametrically opposite the man. I believe that she could spiral to the (1/4)R limit and then take a straight-line path to the nearest edge.

    This is a variation of my initial solution. I like this one better because it cannot be countered by the man and it requires less precision on the part of the swimmer.

  22. Falwan | Profile

    @preswilli , @pele1515 :

    very intuitive indeed…

  23. hex | PUZZLE MASTER | Profile

    While my strategy above is valid, another simpler one would be:
    Man is North
    She swims South a little bit less than R/4
    She can now swim in a circle with an angular velocity greater than that of the man. She can thus continue circling until she is in a position diametrically opposite to him.
    She now has slightly more than 3*R/4 to reach the shore while he would need Pi*R (at 4 times her speed) to reach that same point.
    3*R/4 < Pi*R/4

  24. bilbao | Profile

    The accepted solution that I have, corresponds to the explanations given by hex, mashplum and bizarrette18.
    I liked the lateral thinking solutions regarding underwater swimming, preswilli and pele1515.
    You are all very resourceful people!

  25. athamidreza | Profile

    I solve it!
    she must swim on a circle at radius of (R/4)-epsilon ,then after a long time! the man and the girl are in a same line go at center. but distance of girl to edge is (3R+epsilon)/4 so that time to reach opposite side is (3*R+epsilon)/(4*S)<(Pi*R)/(4*S).

    P.S: epsilon must be :0 < epsilon < (Pi-3)

  26. eh00748 | Profile

    Easy. she just swims underwater so the man can’t see her. when she gets to the shore there’s a 1 in 360 chance that the man will be where she surfaces.

  27. Puzzler | Profile

    If the man really wants to catch her, he would shoot her with a tranquilizer dart.

  28. newacct | Profile

    The solution is very simple. She should swim out to radius R/4 while always maintaining that she is on the opposite side of the circle than he is. She can do that because at radius = R/4, she only needs to swim 1/4 as fast as the guy to match his angular movement. At radius < R/4, she needs even less speed in the angular direction to match, and can use the remaining "spare" speed to move in the radial (outward) direction, increasing the radius in the process. (Initially most of the speed could be spent in the radial direction, but as it gets closer to R/4, she needs more speed in the angular and can afford less on the radial. The radial and angular components are at right angles and add up to her speed by the Pythagorean Theorem.)

    At radius R/4, she can swim in a straight line outwards. Since 3R/4S < pi R/4S, she can get to the shore before he can round half the lake.

    On further analysis, the factor of 4 is not magical. In fact, the problem would also work with any factor less than pi+1 = 4.14

    Another interesting point is that many people said that she should swim to a radius of R/4 – epsilon, where epsilon is some arbitrarily small number (i.e. given any small number epsilon, there is a finite time she can reach radius R/4 – epsilon in). This is because it is not obvious that she can actually reach radius R/4 (because as she approaches radius R/4, less and less of her speed can be spent in the radial direction), and R/4 – epsilon is sufficient to solve the problem.

    However, perhaps counterintuitively, she does reach radius exactly R/4 in finite time. To see this, consider that she swims at speed S; she needs to swim at a speed of 4Sr/R in the angular direction to match the man's movements on shore, when she is at radius r. So by the Pythagorean theorem, that leaves her S * sqrt(1 – (4r/R)^2) speed in the radial direction. Let this be the rate of increase of her radius, i.e. dr/dt in calculus. Now we have a differential equation dr/dt = S * sqrt(1 – (4r/R)^2). Long story short, this can be solved with r = R/4 * sin(4S/R * t). As you can see, r = R/4 can be reach when sin(…;) = 1, which happens when the inside reaches pi/2, i.e. t = pi R/8S, i.e. finite time.

  29. Waytooeasy. | Profile

    This one doesn’t seem too tricky, the man can reach the point the girl swims to in pi/4 times the time of the girl. This is about 0.785 of her time. circumference=2piR therefore he must travel piR at a speed of 4s. t= piR/4S gives pi/4 * R/S. The girl must travel R at speed S. t= R/S. Therefore she must be at a distance 0.215R but < 0.25R from the center of the lake and begin swimming in a circle, the man could not move around the perimeter as quickly as she could move around this circle. Therefore she could continually swim until directly opposing the man and swim to safety.

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