The classic "Bear" problem requires the answer "White", which most people have heard enough times that they do not bother to reason why, The house in this case is clearly at the North Pole, so that going 2 mls S then 2mls E leaves one at a position that is still 2 mls S of the pole. Similar cases close to the pole permit one to travel along an arc of longitude where ones return journey N converges with ones outward journey.

Shooting the bear makes any answer about the S Pole wrong, as there would be no bears there. But once the bear is out of the question (hence "taking a leak") then positions near the S pole are possible. If, after walking 2 miles S the man is on a circle (centred on S pole) of radius 1/(2ð) then walking 2 mls E will take him in a circle once around the pole. So any start point on a circle 2 mls N of this circle is suitable to build a home to fit the question. Similarly, moving closer to the S Pole, we can find positions where the man wals 2 mls E and circles the pole twice. Giving another set of answers, and we can continue inwards for more answers. However, his house cannot be closer than 2 mls from the S pole, as then he would be unable to start by walking 2 mls S.

Hope this clarifies the situation.

Why don't polar bears eat penguins ??