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?
I am very confused. How can a bear walk south, north, east and return? Shouldn't the bear should be $1$ mile away from the starting point?
For those curious, the solution which I have trouble comprehending is item 1 from the IXth Annual Harvard-MIT Mathematics Tournament, 2006 (PDF link via amazonaws.com) Reproduced here:
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 $\frac{1}{2n\pi}$ 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 $1 + \frac{1}{2n\pi}$ miles from the south pole. Similarly, the second bear started about $2 + \frac{2}{2m\pi}$ miles from the south pole for some integer $m > 0$, so they must have started at most
$$3+ \frac{1}{2n\pi} + \frac{2}{2m\pi} \leq 3+ \frac{3}{2\pi} \approx 3.477$$
miles apart.
Best Answer
You are probably thinking that the first bear's path should look like this:
One mile south, then one mile east, then one mile north. This indeed is how it would look if the bear started near the equator, where the lines of latitude and longitude resemble a rectangular grid.
But the solution gives the clue that the bear starts near the south pole. The lines of latitude and longitude there look more like this:
The south pole is at the center of the circles, and each circle is a line of latitude. North is always directly away from the south pole, so it is toward the edge of the map no matter where you are. And east is clockwise around the pole.
So the bear walks one mile directly toward the pole, walks one mile in a circle around the pole, and then walks one mile directly away.
The complete solution for the two bears looks like this:
The second bear walks twice as far toward the pole and walks around a circle twice as large. To get the maximum distance between the starting points of the bears, we start them on opposite lines of longitude. So to travel in the shortest path from the first bear's starting point to the other, you go one mile toward the pole, then the radius of the smaller circle to reach the pole, then the radius of the larger circle to reach that circle, then two more miles to reach the second bear's starting point.
If the distances were one thousand and two thousand miles then spherical geometry might come into play, but on this scale (assuming the Earth is about $4000$ miles in radius) we can get a good enough approximation by assuming everything is in one flat plane.