In a circular swimming pool, I start in the middle and swim in any direction at speed s. You stand outside the pool on the perimeter. How much faster do you have to run, in terms of s, to ensure you can tag me whenever I reach the edge?

It is not so difficult to come up with a strategy that would allow the swimmer to get away if the ratio (pursuer's speed) / (swimmer's speed) = 4 (or, say, ).As for finding the minimum ratio ensuring the pursuer catches the swimmer, I think it is a difficult problem to solve, let alone explain here without detailed drawings, etc. I think the answer would be something like the root of the equation.I would say,

This problem is the generalization of the "circle pool" brainteaser posted in this forum in 2004. There it was noted that a strategy exists to escape from the pool if runner is only 4x faster than the swimmer, I believe. At some speed of course the swimmer cannot reach the edge in time no matter how he/she swims.

QuoteOriginally posted by: HedgefundguyThis problem is the generalization of the "circle pool" brainteaser posted in this forum in 2004. There it was noted that a strategy exists to escape from the pool if runner is only 4x faster than the swimmer, I believe. At some speed of course the swimmer cannot reach the edge in time no matter how he/she swims.May I ask some questions:1. Can the swimmer change the swimming direction at anytime once he/she starts swimming?2. How smart is the runner? Can the runner change the chasing direction once the game starts?

Swimmer can change direction at any moment; runner can also wait, move, or change direction as needed to beat the swimmer to the edge.

It seems logical to suggest, (but this could clearly be wrong) that the optimal thing for the swimmer to do is to always swim directly away from the runner, i.e. along the constantly changing chord. Is this right?

To me it seems logical that the swimmer should maneuver himself into a right position relative to the runner and then make a dash for the edge. At least, that is how the escape strategy for speed ratio = 4 works.

QuoteOriginally posted by: Msccube2. How smart is the runner? Can the runner change the chasing direction once the game starts?QuoteOriginally posted by: HedgefundguySwimmer can change direction at any moment; runner can also wait, move, or change direction as needed to beat the swimmer to the edge.This question may not have any definite answer. It may be a question of competition on the IQ of the swimmer and the runner rather than on their speed. The runner can wait, move or change direction to beat the swimmer to the edge. Whether the runner can always catch the swimmer even though he/she can run fast enough depends on how he/she chooses the running direction.Suppose the max speed of the swimmer and runner are s and v respectively. If the position of the swimmer is inside the circle of radius of r=(s*R)/v,where R is the radius of the pool, the angular speed of the swimmer relative to the center of the pool is always larger than that of the runner.Therefore, inside this circle, the swimmer can always get away from the runner in angular sense. That means, the swimmer can swim to a position180 degrees out of phase from the runner, which is the safest postion for the swimmer. However, the angular speed of the swimmer isalways slower than that of the runner outside this circle. Therefore the runner may be able to get closer to the swimmer in angular sense.Suppose the swimmer starts at the center and swims toward to the edge. He/she always adjusts his/her positions such that it is 180 degreesout of phase of the runner until he/she reaches the circle. At the circle, he/she knows that he/she will loose the angular speed advantage. Let ABbe the line joining the position of the swimmer and the runner once the swimmer reaches the circle and is 180 degree out of phase of the runner.He/she starts to swim in the ziz-zag patterns. He/she swims at a very small angle relative to the perpendicular to AB. Suppose the runner is a programmed "killer robot". The runner will choose the chasingdirection according to the intended landing position of the swimmer. As the swimmer swims in ziz-zag manner. The runner will run in the clockwiseand anti-clockwise direction repeatedly. At the end, the runner still stay at the original position on the average while the swimmer getscloser to the edge. The swimmer can always escape from the runner whatever how fast the runner runs. What happens if the runner is a human being and is not a robot. It depends on when the runner discovers the swimmer is playing a gameto him/her. The later the runner discovers that the "true" direction of the swimmer is toward the edge rather than in the pretended directionsof that ziz-zag pattern, the closer toward the edge the swimmer is. This timming will determine the speed required for the runner to catch the swimmer.Hope can explain my idea without any diagram.

This question does indeed have a definite answer - given earlier in this thread. The precise statement: if speed ratio x < A (the root of the equation cited in the earlier post), there exists an escape strategy for the swimmer. If x > A, there exists a catching strategy for the runner.

Hey Guys, I don't agree that the chaser in the perimeter should travel 4 times faster than the swimmer:The best strategy for the swimmer is to always swimm away from the chaser in a spiral way. Let's supose that the swimmer is just one inch away from the border and swimming with a velocity close to scape. In that case he is practically swimming parallel to the chaser. The only way not to be chased is with a slightly faster speed.That means s = v + delta, where v is the velocity of the chaser.

What exactly you don't agree with? Can you possibly come up with some precise statement?

I agree with Msccube's explanation of the optimal strategy up to the point that the swimmer reaches the edge of the radius r circle. The zigzagging tactic would not work as described as the runner now has the angular advantage [Correction: "...angular speed advantage"]. As soon as the swimmer moves in one direction, the runner will move to intercept and reduce the angle to less than 180 degrees. Subsequently, the swimmer will not be able to open the angular distance unless they swim back inside the circle.

ans: 1+piThe best strategy for the swimmer(S with max velocity vs) is to maintain the line connecting him and the chaser(C with max velovity vc) pass through the center as long as possible and then head radially for the edge. To maintain the colinearity of the 3 points (S,center,C), the angular speed of S should be equal to the angular speed of C which is (vc/R) ; R=radius of pool. This should occur with a non-zero radial speed (=vs_r), else there is no point going purely in circles without gaining any radial advantage. And since vs is max possible speed of S, vs_r = (vs^2 - (vc/R*r)^2)^0.5 ; r=instataneous distance of S from centervs_r >= 0 ----> r <= R/y ; y=(vc/vs)For maximum success S will follow the collinear strategy till r=R/y after which he will head straight to the edge which is at a distance (R-R/y). Simultaneously, C will start closing down on the angluar seperation (which had been pi till now) to cover a distance (pi)R. To tag S, time taken to reach the intended spot on the edge should be <= time taken for S.(pi)R/vc <= (R-R/y)/vs -----> y >= (1+pi)@Msccube: You are right in saying that it depends on their intelligence but assuming their best intelligence (which is equivalent to saying, S and C know each other's strategy very well), one should be able to come up with a solution. Atleast thats what I think the question meant.

Would you by any chance be interested in proving the assertion you make? This would be difficult to do, of course, since your answer is incorrect. The correct answer has been posted earlier in this thread.