QuoteOriginally posted by: MsccubeI do not have the time to go through all the maths given in this Paper,It seems to me that this paper has not considered the ziz-zag strategy. It just claims that the best strategy is for the lady to run in this direction .......It does more than "just claims the best strategy." If anything, it refers to a book dealing with the subject in general and this specific problem in particular.P.S. I am somewhat curious if you understand the concept of "strategy", at least as it is used in mathematical problems.

QuoteOriginally posted by: sevvostP.S. I am somewhat curious if you understand the concept of "strategy", at least as it is used in mathematical problems.Well, I have posted the escape Strategies in my posts already very early on this thread!What you have posted on this thread are only a formula and a link to a paper. If you claim that it is the best strategy and the value of x = 4.6 is the answer, would you please show your proof to us? If the formula that you have posted come from this paper, you should cite this reference in your early post if it is not your work.

The paper points to a diferent source where, hopefully, a detailed explanation and a proof is provided. But I do not have the time to pursue it now. While calculating the "relative" velocity of S w.r.t. C, the paper scales down the chaser's velocity by a factor r/R. I dont understand why. May be a detailed explanation is given in the book. But even if one accepts that, the strategy involves swimming in such a direction that the resultant vector (the vector difference, vs - vc*r/R) is orthogonal to vc (remember, vc*r/R is always greater than vs outside the critical circle). This is clearly not possible (else, vs should be the hypotenuse and hence the longest of the three). May be the book it points to provides a correct/clearer explanation.QuoteOriginally posted by: MsccubeHi The Theorist,An interesting fact: supposing that the runner and the swimmer know each other very well and there is no secert between them,the runner knows the max speed of the swimmer and the final landing point of swimmer whatever the course the swimmer takes.It means that it does not matter how the swimmer swims, the runner knows what the swimmer is thinking and his intended finallanding point such that the runner can choose the shorest path to reach that point. It also assumes that the swimmer knowsthat the runner has known everything.If it is the case, the answer should be x = 1+pi since the swimmer must take the shortest path once reaching the circle of radius r.(That is swimming in the radial direction toward the edge.) The runner will run only 180 degrees.Msccube [Edit] : I am sorry that there is a typo. I mean x = 1 + pi , not x = 4.3 . My apology.Such a case will make the problem very bland, doesnt it? If C knows S's landing point at the start itself, then the max he has to run is diagonally across which gives pi for the minimum ratio (if S takes minimum time which is swimming radially from the center). It is interesting only if we consider both of them revising their strategy continuously depending on the other's position (and we can assume the other knows of it immediately). Now I understand that my 1+pi solution fails during zig-zag (to whatever extent) tactics. Its tough to take this problem off our head, isnt it?

This is getting silly.QuoteOriginally posted by: MsccubeWhat you have posted on this thread are only a formula and a link to a paper. This statement is false. QuoteIf you claim that it is the best strategy and the value of x = 4.6 is the answer...I never claimed that "x = 4.6" is the answer.Quotewould you please show your proof to us?I think I have answered this already, haven't I? If I have time, I might, although, frankly, I have doubts as to why I should bother.QuoteIf the formula that you have posted come from this paper...It does not come from this paper.Quote...you should cite this reference in your early post if it is not your work.Oh, really? Thank you for telling me, I did not know that forum posts had to follow the rigorous standards of academic papers.

I think the zig-zag tactic of S, as suggested by msccube, is non-trivial. It forces C to pick a direction and run (and not base his direction on the other, a privilege that only S has). Else, he will stay put like a buridan's donkey while S zig-zags with a non-zero angle to the radial vector and escapes.Consider the swimmer(S) standing at a point outside the critical circle of radius R/y. Now, C picks a direction and starts running. Once C picks a direction and runs, the best strategy for S is to move in the 'same' direction as C (means, if C runs clockwise, S should swim clockwise). But, this direction should constantly make an angle p<90deg with the instataneous radial vector. Else, if p=90deg, and if S is on the critical circle, he will forever remain on the critical circle and if S is outside the critical circle, C will catch up in sometime (since C's angular velocity is greater than S's when outside the circle). Hence p<90.At a particular instant,The radial component of swimmer's(S's) velocity, vs_r = vs*cos(p). The angular velocity of S, vs_a = (vs*sin(p)) / (R/y+t*vs_r), where t=time elapsed from the moment S leaves the critical circleRelative angular velocity of C w.r.t. S, vrel_a = vc/R - vs_a = vs*y/R - vs_a (remember, it is positive)In the time,T, it takes for S to cover the radial distance R, C should catch up with S. i.e. cover a relative angular distance of pi from the moment S leaves the critical circle.T = R(1-1/y)/(vs*cos(p))Integrating vrel_a w.r.t time 't' over the limit (0 to T) and equating it to pi, we get the following result,(y-1)*sec(p) - tan(p)*ln(y) - pi = 0Remember, the assumption we have made so far is that p is a constant over the path. But whatever angle S chooses, C will keep running at a constant (angular) speed which S knows he will do even before starting out? So, what is the motivation for S to change the angle as he swims? I cant think of any The above equation gives a value for y for all possible p between 0 and 90deg (above this, S spirals inward and below this, S swims towards the direction of C). Now, select the maximum y for the answer by equating dy/dp=0, select the optimum path angle 'p' and substitute back to get the minimum ratio required for C to catch S.Kindly comment.ps: my earlier solution of (1+pi) is a special case of p=0 as the above equation shows but is not the maximum value.

Hi sevvost,I think that everyone can read this thread, and that paper. They will find your answer of x \approx 4.6, and x is the root of that formula which you have posted without proof.Have a nice day. Msccube QuoteOriginally posted by: sevvostThis is getting silly.QuoteOriginally posted by: MsccubeWhat you have posted on this thread are only a formula and a link to a paper. This statement is false. QuoteIf you claim that it is the best strategy and the value of x = 4.6 is the answer...I never claimed that "x = 4.6" is the answer.Quotewould you please show your proof to us?I think I have answered this already, haven't I? If I have time, I might, although, frankly, I have doubts as to why I should bother.QuoteIf the formula that you have posted come from this paper...It does not come from this paper.Quote...you should cite this reference in your early post if it is not your work.Oh, really? Thank you for telling me, I did not know that forum posts had to follow the rigorous standards of academic papers.

Yes, the answer is (but not x = 4.6), so I would be wary of making a bold statement ''everyone can read this thread".Here is another paper treating the problem. This one actually describes an escape strategy (among other things) for the swimmer - see section 4.7, pages 8-9. Curiously, the numerical value it gives seems to be slightly off. The one found in yet another paper looks better (pages 20-22).Probably the best source is the book referenced in the earlier paper.

QuoteOriginally posted by: TheTheoristIntegrating vrel_a w.r.t time 't' over the limit (0 to T) and equating it to pi, we get the following result,(y-1)*sec(p) - tan(p)*ln(y) - pi = 0Hi The Theorist,Would you mind giving out the numercial solution of the above equation?Msccube

QuoteOriginally posted by: sevvostHere is another paper treating the problem. This one actually describes an escape strategy (among other things) for the swimmer - see section 4.7, pages 8-9. most helpful...now if only I could speak German

QuoteOriginally posted by: TheTheoristI think the zig-zag tactic of S, as suggested by msccube, is non-trivial. It forces C to pick a direction and run (and not base his direction on the other, a privilege that only S has). Else, he will stay put like a buridan's donkey while S zig-zags with a non-zero angle to the radial vector and escapes.Consider the swimmer(S) standing at a point outside the critical circle of radius R/y. Now, C picks a direction and starts running. Once C picks a direction and runs, the best strategy for S is to move in the 'same' direction as C (means, if C runs clockwise, S should swim clockwise). But, this direction should constantly make an angle p<90deg with the instataneous radial vector. Else, if p=90deg, and if S is on the critical circle, he will forever remain on the critical circle and if S is outside the critical circle, C will catch up in sometime (since C's angular velocity is greater than S's when outside the circle). Hence p<90.At a particular instant,The radial component of swimmer's(S's) velocity, vs_r = vs*cos(p). The angular velocity of S, vs_a = (vs*sin(p)) / (R/y+t*vs_r), where t=time elapsed from the moment S leaves the critical circleRelative angular velocity of C w.r.t. S, vrel_a = vc/R - vs_a = vs*y/R - vs_a (remember, it is positive)In the time,T, it takes for S to cover the radial distance R, C should catch up with S. i.e. cover a relative angular distance of pi from the moment S leaves the critical circle.T = R(1-1/y)/(vs*cos(p))Integrating vrel_a w.r.t time 't' over the limit (0 to T) and equating it to pi, we get the following result,(y-1)*sec(p) - tan(p)*ln(y) - pi = 0Remember, the assumption we have made so far is that p is a constant over the path. But whatever angle S chooses, C will keep running at a constant (angular) speed which S knows he will do even before starting out? So, what is the motivation for S to change the angle as he swims? I cant think of any The above equation gives a value for y for all possible p between 0 and 90deg (above this, S spirals inward and below this, S swims towards the direction of C). Now, select the maximum y for the answer by equating dy/dp=0, select the optimum path angle 'p' and substitute back to get the minimum ratio required for C to catch S.Kindly comment.ps: my earlier solution of (1+pi) is a special case of p=0 as the above equation shows but is not the maximum value.My above equation does not have a valid answer! I would be happy to see someone point out the flaw in my above argument and come up with a better strategy.I am returning to this problem after almost a year and I remember remaining entangled for couple of days, while thinking about this.

Actually, I have used the concept of economics rather than math to solve this problem.The runner is forced to change his running direction because he does not know the swimmer's max speed.It is the economics asymmetry information flow.When the runner finds out the swimmer's max speed or the swimmer is not careful enough to let him know,the runner can decide the running direction. It is the signaling economics.

If the chaser (C) changes his direction, he is just going to lose time and let the swimmer (S) get closer to the edge (with no real advantage for C) - your zigzag method is an extreme case of this change of direction strategy. So, C has to pick a direction and run even if he does not know S's max speed - this is the best strategy for C. Now, if I can guess C's best strategy, then surely S can guess too! So, he will always swim at his max speed.. else, he is just going to let C gain on him for no reason.

Not necessary.The swimmer has no obligation to swim at his max speed. He can swim at non constant speed to foolthe runner. The runner must take the risk if he does not change direction. That is to bet whether it is the max speed of the swimmer, which he does not know.

I am not sure what you mean by 'S can fool C by swimming at a reduced speed.' And, I am also not sure what S's speed has to do with C's direction. Just for clarity, can you tell me the direction (left/right) that C would choose for different ranges of S's speed and why so?

As I said before, when the swimmer changes his direction in his ziz-zag way, the runner must answer an importantquestion ---- what is his max speed? If the runner continues his running direction, the runner will be required to runa longer distance after the swimmer changes his direction. What happens if the swimmer suddenly swims much faster andthe runner is required to run a longer distance if the runner does not change direction at the same time. Of course, one can argue thatthe runner can gather the historical speed of the swimmer but the historical performance may not be the indicator of thefuture performance. It is the situation of the economics of asymmetry information flow--- the runner does not know the swimmer'smax speed but the swimmer knows his max speed. So the runner must decide whether he should change his direction to run ashorter distance after the swimmer changes his direction.Hope I have explained.