A traveller starts out from the Earth's equator, heading exactly northeast. Undeterred by mountains, oceans and political boundaries, he continues on a northeasterly heading until he can go no further.

Where does he end up?

How far did he go?

How many times did he circumnavigate the earth? (For these purposes, this means travel through 360 degrees of longitude.)

(Now that the solution is public, I'll feel free tom comment.)

Part of what I like about this puzzle is the tension between the fact that the length of the path is finite, and the fact that this finite path circles around the pole an infinite number of times.

Yes, it is true that for every √2 units we travel we go 1 unit north. This defines the length of the path.

Near the north pole, the path approximates a logarithmic spiral: each rotation around the pole reduces the distance by a factor of about 535 (I've only done this numerically; I don't know the exact mathematical expression).

The other thing I like about this one is the fact that, while the mathematical number of rotations is infinite, by the end of the third rotation we're only about 3 inches from the pole. (I like the disparity between mathematically infinite, but 3 for all practical purposes.)

Posted by Jim Lyon on 2002-09-27 13:22:16