[Guest]

A dog is in the center of a round swimming pool (radius 100 meters). A tiger, who can't swim, is on the edge of the swimming pool and tries to catch the dog. What is the maximum ratio of the tiger's running speed and the dog's swimming speed such that the dog can swim ashore without being caught by the tiger? What strategy (i.e. swimming route) should the dog use?

[Guest]

The name of this "riddle" is incorrect - as this is obviously a mathematical problem, it has nothing to do with wit.

The "wit" part is in thinking of a strategy that would allow the dog to get out of the pool safely. In fact, I remember your answer, and I think it had nothing to do with the question or atleast the "wit" part of the question!

Edited July 8, 2009 by DeeGee

[bonanova]

he stays opposite the tiger as far from the center as possible then swims to the nearest point on the shore.

That is, he maintains his position, the tiger's position and the center on a straight line, then swims to shore along that line.

So long as the tiger's speed ratio does not exceed 1 + pi the dog can escape.

he can escape an even faster tiger.

As before, he gets as far from the center as he can [radius r], keeping the three points [black dots in figure] aligned.

Assume tiger and dog are moving clockwise when this happens, and they are aligned North-South.

The dog then swims straight East [upper red arrow] toward a landing point [red dot] at coordinates {x, y} from the center.

Immediately the tiger is able to close the angle from himself to the dog to less than 180^o, and he continues to pursue clockwise [lower red arrow].

Analysis:

The speed ratio of the tiger is s = s_tiger/s_dog.

Then R=100 and r = 100/s meters.

The landing point {x, y} is {100 cos(a), r}

where the angle a is given by sin(a) = r/100 = 1/s.

The dog travels a distance D = x = 100 cos(a).

The tiger travels a distance T = 100(3 pi/2 - a).

They arrive simultaneously if D/T = 1/s = sin(a).

D/T = cos(a) / (3 pi/2 - a) = sin(a).

Simplifying,

cot(a) + a = 3 pi/2.

Thus a = 12.546^o and s = 1/sin(a) = 4.60333884 ...

Note:

It may seem strange the tiger does not reverse direction to shorten his path to the landing point.

But if he does so, the dog immediately reverses and moves on a new tangent to a different landing point, gaining even more advantage.

The tiger's best strategy, once his angle to the dog is less than 180^o and the dog is outside the radius r, is to continue clockwise pursuit.

[Guest]

he stays opposite the tiger as far from the center as possible then swims to the nearest point on the shore.

That is, he maintains his position, the tiger's position and the center on a straight line, then swims to shore along that line.

So long as the tiger's speed ratio does not exceed 1 + pi the dog can escape.

he can escape an even faster tiger.

As before, he gets as far from the center as he can [radius r], keeping the three points [black dots in figure] aligned.

Assume tiger and dog are moving clockwise when this happens, and they are aligned North-South.

The dog then swims straight East [upper red arrow] toward a landing point [red dot] at coordinates {x, y} from the center.

Immediately the tiger is able to close the angle from himself to the dog to less than 180^o, and he continues to pursue clockwise [lower red arrow].

Analysis:

The speed ratio of the tiger is s = s_tiger/s_dog.

Then R=100 and r = 100/s meters.

The landing point {x, y} is {100 cos(a), r}

where the angle a is given by sin(a) = r/100 = 1/s.

The dog travels a distance D = x = 100 cos(a).

The tiger travels a distance T = 100(3 pi/2 - a).

They arrive simultaneously if D/T = 1/s = sin(a).

D/T = cos(a) / (3 pi/2 - a) = sin(a).

Simplifying,

cot(a) + a = 3 pi/2.

Thus a = 12.546^o and s = 1/sin(a) = 4.60333884 ...

Note:

It may seem strange the tiger does not reverse direction to shorten his path to the landing point.

But if he does so, the dog immediately reverses and moves on a new tangent to a different landing point, gaining even more advantage.

The tiger's best strategy, once his angle to the dog is less than 180^o and the dog is outside the radius r, is to continue clockwise pursuit.

Excellent! A perfect 10!