Von Neumann was often considered by colleagues to be the smartest person they'd ever met. There's an old story that goes something like this:
Von Neumann was asked a mathematical puzzle: two bicyclists start 20 miles apart and head towards each other at a steady speed of 10 mph each. At the same time, a fly that travels at a steady 15 mph starts from the front wheel of the southbound bicycle and heads to the front wheel of the northbound one, then immediately turns around and heads to the wheel of the southbound one, and continues until the bicycles meet and he is crushed. What total distance did the fly travel?
The trick is that it seems you'd need to compute an infinite series, figuring out the distance traveled on the first journey, then the second, then the third and sum them all. In fact, you can just realize that two bicycles starting 20 miles away, each moving at 10 miles an hour, will meet in 1 hour; if the fly's traveling at 15 mph constantly during that time, it travels 15 miles.
Von Neumann provided the solution immediately. The man who posed it to him said "Oh, you must have heard the trick already!" "What trick?" said von Neumann. "I just summed the geometric series!"