As a child, I used to puzzle over a self-made conundrum, which is similar to the taxi problem I set a while ago. Here it is.
Imagine the eight lines that connect the 16 primary points of the compass (N, NNE, NE, ENE, E, ESE, SE, SSE, S, SSW, SW, WSW, W, WNW, NW and NNW). Each line connects two of these sixteen points. So, for instance, the line from N to S constitutes a single line, as does that connecting SSE and NNW.
Each line has the same length: one kilometre, let's say.
You walk each of the lines in succession, but for each line, you can choose which direction to take. So, for the N/S line, you can either walk north or south for one kilometre. After doing so, you take the NNE/SSW line from your previous end point, again in either direction. Etc.
Here's the conundrum: is it possible to end up where you started. And if not, how close to the start can you end up and how do you do this?
I've just worked out the answer, which I'll post as a comment.
