Saturday, August 2, 2008

Answer: Square Wheel
New Problem: To Swim or Not To Swim

Square Wheel
Sorry to be lame, but I'm not going to prove this one. It's a catenary.

To Swim or Not to Swim
There's a lake in the middle of the field, as shown. You want to go from point A to point B as quickly as possible. You can go over land or over water, but you're twice as slow in the water as one the land.

Assume (or prove) the best strategy is to head straight to some spot on the shore, swim straight across, and then head on from the far shore to point B by land. The shape of the lake is such that it doesn't matter where you choose to enter - all paths take the same amount of time. What is the shape of the lake?

Assume: points A and B are equidistant from the center of the lake. The lake is symmetric about a vertical line through its center.

4 comments:

kangway said...

I could do a mathematical proof, but I want to do something clever and talk about this problem reminds me of lenses. Of course, I don't remember much about lenses, so I'll prove it later.

Nikita said...

I think it could be a meniscus shaped lake. Because regardless of where you start swimming from, you swim the same distance and so you also walk the same distance.

Markkimarkkonnen said...

yup kangway, it was intended to be like lenses and whatnot. i don't know much about them either. in fact i did not know the answer to this problem when I wrote it down on the whiteboard at camp.

nikita - i don't understand. why would you swim the same distance regardless of where you start from?

Nikita said...

because that's the way a meniscus is shaped.
doesn't matter, the answer is definitely wrong. quite a bad guess actually. :|