Have you heard this one?
A man leaves his house, walks on mile due south, then another due west. On the western leg of his walk, he sees a bear in the distance. He then walks one mile due north and arrives back home. What color was the bear?
The answer is of course, white, because his journey is only possible if his house is at the north pole, and therefore the bear must have been a polar bear.
Here’s one to challenge your intuition:
A woman walks one mile north, one mile east, one mile south, and finally one mile west. Under what conditions does she end up back where she started?
The obvious answer is that she will always end up back where she started. If you’ve heard the bear color riddle, you might emend your answer to say that you must be more than a mile south of the pole. You’d still be wrong! The only places where she ends up back where she started are places that are exactly 1/2 mile south of the equator.
OK, one more:
What is the fastest route from San Francisco to Vancouver, assuming that the oceans have been drained and the land leveled to a completely uniform height?
Again, there’s an obvious answer: a straight line. That would only be the fastest route if you could pass through solid rock as quickly as through air. The right answer is the shorter of the two arcs on the great circle that passes through both cities.