Problem 761

[david-kerr]

This problem is driving me up the walls. I fairly easily got 4.60339314 for the circular pool (which is correct) but the same strategy gives 5.90143057 for the square pool. Can someone please confirm that the figure given in the problem for the square pool is correct.

[brob26]

This problem is driving me up the walls. I fairly easily got 4.60339314 for the circular pool (which is correct) but the same strategy gives 5.90143057 for the square pool. Can someone please confirm that the figure given in the problem for the square pool is correct.

What exactly do you mean by the "same strategy" between the circle and square cases? It'll likely be valuable to figure out why your supposed solution for the square pool does not actually work.

[david-kerr]

I think I have spotted my stupidity. An improved strategy for the runner.

[Kaladin1]

I've had a similar problem, but I got a strategy that appears to work for any v<5.82842 (3+sqrt(8)) for the square pool.

The strategy accounts for any actions by the runner, and I'm really struggling to find any errors in my reasoning. Are we sure that v_square is 5.78859314.

[Kaladin1]

Nevermind, I have seen my mistake.

[sgb27]

I got the circle value, but when I started on the square it got me thinking about the circle one some more. Tell me where I'm going wrong and why the following won't allow the swimmer to escape the circle with a higher runner speed than given:

Imagine the swimmer decides to swim in a square, with a square of some length x centred on the centre of the circle. Now, if the time it takes the swimmer to swim one lap of the square (4x) is less than the time it takes the runner to do the circle (2.pi / V), this gives x<pi/(2V). So if the swimmer starts swimming laps, the runner is going to gradually drop behind, and at some point the swimmer will reach a corner when the runner is diagonally opposite. But at this point, the swimmer is further from the centre [r=sqrt(2)*pi/(4v)] than he would be for the solution given for a circle implies.

[jaap]

So if the swimmer starts swimming laps, the runner is going to gradually drop behind, and at some point the swimmer will reach a corner when the runner is diagonally opposite.

When the moment arrives that the runner is half a laptime behind, why would that be exactly when the swimmer is at a corner? The runner could actually hang back to make sure it definitely does not happen when the swimmer is at a corner.

[sgb27]

Ah yeh good point, the optimum solution for the runner is no longer going to be to just move at full speed in the direction of the swimmer.

[jabah013.307]

Thats SO true!

[neverforget]

True and kind of a spoiler IMHO... Half the fun (frustration?) for me was in realizing why the example values are what they are. I spent many nights totally convinced the dev team had made a mistake.. heh

[thedoctar]

I think the problem should state that the circle is a unit circle, and the square is a unit square, and the hexagon is a unit hexagon. I just calculated the circle value using a unit circle.

[Swistakk]

I think the problem should state that the circle is a unit circle, and the square is a unit square, and the hexagon is a unit hexagon. I just calculated the circle value using a unit circle.

Not necessarily, it's quite clear that the answer stays the same for all circles, all squares and all regular hexagons

[thedoctar]

Ah yes you're correct. Silly me!