As posted elsewhere…

Riddle:

You’ve been framed in a murder of a prominent politician. Since your country doesn’t have the death penalty, you’ve been sentenced to spend the rest of your life in prison, but not just any prison, but the latest tax-dollar-saving addition to the department of correction’s arsenal – the circular prison 4000. Now for the riddle:
The prison is very simple: you, the prisoner, are confined to the area of a circle. The perimeter of the circle is patrolled by a vicious guard dog (the dog can walk only on the perimeter of the circle). The dog can run up to 4 times faster than you and it’s smart – it will always try to catch you. If you manage to reach the perimeter and the dog isn’t there – you immediately escape and get a chance to clear your name.
The question is (of course) how the hell do you escape?

click More for answer…

as suggested, the correct solution is a spiral.
You begin spiraling out of the center, always keeping the dog behind you (you can do this since your spiral radius is less that a quarter of the dog’s radius). You keep doing that until your reach the radius on which you and the dog have the same lap time, then you run directly to the fence (remember, the dog is still behind you).

The math is very basic – if the prison radius is R. Then as long as your spiral perimeter is smaller than R/4, you can set the pace since it takes you less time to complete a lap than it takes the dog. When you reach R/4, your lap times are equal since the dog covers 4 times as much distance but runs 4 times faster). At that point, you run directly to the fence. You need to cover 3R/4, the dog needs to cover pi*R (half the circle). The distance the dog has to cover is more than 4 time the distance you have to cover and therefore it doesn’t make it.

(crazy…)

from: http://news.ycombinator.com/news