"Yeah. I understand the mechanics of it, shithead. I just don't understand how this is any less retarded than what I'm suggesting." - Kiley; Housebound.

A bear hunter walks one mile south from his camp (starting point).
He sights a bear and walks one mile east towards it.
It runs away and he follows it 1 mile north and ends up at his camp (starting point).

What could his starting point(s) have been ?
Hmm...

Progress might have been all right once, but it has gone on too long -- Ogden Nash

Consider a circle 'C' with center as South Pole, that has a circumference of 1 mile. From any point on this circle walk a mile due north. This is one starting point, say 'S' for the hunter. All he has to do is walk down to the unit circle down south, goes round it for a mile, whereupon, he lands up where he began on the unit circle (bcos the circumference is 1) and then he walks back UP north to the point 'S' (starting point).

However any point on 'C' is a potential candidate. Since there are infinitely many points on the circle 'C', there are infinitely many such points like 'S'. The set of all these points is a circle whose radius is 1 + (radius of C).

The radius of 'C' is 1/2*pi (since circumference is 1).

Hence the solution is the set of all points on the (outer) circle with radius (1 + 1/2*pi) centered at the South Pole.

But that is not all. One can draw another circle C' smaller than C, whose circumference is 1/2 mile.
Now the hunter walks around twice to make it a mile. (Ask: Would he know if he is going round in circles ?)
Thus the circle around South Pole, with radius (1 + 1/2*2*pi) is also a solution.

One could likewise construct circles C'', etc with circumferences of 1/3, 1/4 ... of a mile.
Thus, any point on a circle centered at the South Pole whose radius is (1 + 1/2*k*pi) k = 1,2,.. is a solution.

That and the North Pole.

Hope you liked it.

Onto the next puzzle.

Progress might have been all right once, but it has gone on too long -- Ogden Nash