### Brain Teasers

# Swimming to Shore

A man is standing on a rock in the middle of a circular lake of radius 1. There is a tiger on the shore of the lake that can run four times as fast you can swim, however the tiger cannot swim. The tiger is hungry and always attempts to keep the distance between the two of you at a minimum. How can you safely swim to shore?

Assume that as long as you can get to the shore before the tiger gets to you (even if it is only a split-second before), you will be safe.

Assume that as long as you can get to the shore before the tiger gets to you (even if it is only a split-second before), you will be safe.

### Answer

Until you are more than 1/4 of the radius away from the rock you can swim fast enough so that you can keep the tiger the furthest distance possible from you. Regardless of which direction the tiger moves around the circle, you swim the other way, always keeping the rock between you and the tiger. While you are doing this you will be able to move outward, since you will have some energy to spare.Before long you will be 1/4 of the radius away from the center and the tiger will be 180 degrees away. At this point swim straight to the point on the shore furthest from the tiger. You will be able to get there in 3/4 units of time, while it will take the tiger pi/4 =~ 0.7854 units of time.

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## Comments

Tigers swim better than people, just like any other cat, they may not like water but they can swim very nicely and if the tigers life will depend upon swimming, he'll go for it.

I thought swim in a spiral until you are 1/4 radius. A great teaser!

Good one. I went for swimming in a spiral.

I figured since it was the man on the rock, not me, that I was safe already.

I've seen this before, but it's still a good teaser.

It suggests several other interesting and less trivial problems:

1) What's the farthest from the tiger that the man can be when he gets to shore?

2) What's the shortest distance the man can swim and still avoid the tiger?

3) What is the slowest speed for the tiger where the man can't escape?

The issue with #1 is that once the man reaches 1/4 of the radius, the best path to take is to start swimming directly away from the tiger, and as soon as the tiger commits to going in one direction around the lake, adjust the angle towards the shore. There is an optimal angle where four times the added distance to the shore is less than the added distance along the shore.

The issue with #2 is that since there is some extra time with the first solution, and there is some more optimal path than swimming directly away from the tiger, what path give the shortest distance, and thus the shortest time? Is it better to compromise on the initial spiral by letting the tiger get a little closer as the man swims to the 1/4 radius mark? Is it better to simply head for shore once the man reaches a distance where the straight path to the shore beats the tiger? Is it better to head to the shore even soon, using the strategy for #1 above? Some combination?

The issue with number three is simply to determine the answer to #1 and then adjust for the amount of extra time. The is a complication in that as the tiger's speed increses, the distance along the radius before heading to shore decreases, so it's not simply a linear relationship.

Anyway, these questions are quite a bit harder to answer than the teaser, but would add an interesting twist.

I've seen this before, but it's still a good teaser.

It suggests several other interesting and less trivial problems:

1) What's the farthest from the tiger that the man can be when he gets to shore?

2) What's the shortest distance the man can swim and still avoid the tiger?

3) What is the slowest speed for the tiger where the man can't escape?

The issue with #1 is that once the man reaches 1/4 of the radius, the best path to take is to start swimming directly away from the tiger, and as soon as the tiger commits to going in one direction around the lake, adjust the angle towards the shore. There is an optimal angle where four times the added distance to the shore is less than the added distance along the shore.

The issue with #2 is that since there is some extra time with the first solution, and there is some more optimal path than swimming directly away from the tiger, what path give the shortest distance, and thus the shortest time? Is it better to compromise on the initial spiral by letting the tiger get a little closer as the man swims to the 1/4 radius mark? Is it better to simply head for shore once the man reaches a distance where the straight path to the shore beats the tiger? Is it better to head to the shore even soon, using the strategy for #1 above? Some combination?

The issue with number three is simply to determine the answer to #1 and then adjust for the amount of extra time. The is a complication in that as the tiger's speed increses, the distance along the radius before heading to shore decreases, so it's not simply a linear relationship.

Anyway, these questions are quite a bit harder to answer than the teaser, but would add an interesting twist.

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