1. A bear walks one mile south, one mile east, and one mile north, only to find itself where it started. Another bear, more energetic than the first, walks two miles south, two miles east, and two miles north, only to find itself where it started. However, the bears are not white and did not start at the north pole. At most how many miles apart, to the nearest .001 mile, are the two bears' starting points?
Answer: 3.477
Solution: Say the first bear walks a mile south, an integer $n>0$ times around the south pole, and then a mile north. The middle leg of the first bear's journey is a circle of circumference $1 / n$ around the south pole, and therefore about $\frac{1}{2 n \pi}$ miles north of the south pole. (This is not exact even if we assume the Earth is perfectly spherical, but it is correct to about a micron.) Adding this to the mile that the bear walked south/north, we find that it started about $1+\frac{1}{2 n \pi}$ miles from the south pole. Similarly, the second bear started about $2+\frac{2}{2 m \pi}$ miles from the south pole for some integer $m>0$, so they must have started at most
$$
3+\frac{1}{2 n \pi}+\frac{2}{2 m \pi} \leq 3+\frac{3}{2 \pi} \approx 3.477
$$
miles apart.