The Restless Fly from "The Moscow Puzzles" by Boris Kordemsky

Two cyclists began a training run simultaneously, one starting from Moscow, the other from Simferopol.
When the riders were 180 miles apart, a fly took an interest. Starting on one cyclist's shoulder, the fly flew ahead to meet the other cyclist. On reaching the latter, the fly at once turned back.
The restless fly continued to shuttle back and forth until the pair met; then it settled on the nose of one of the cyclists.
The fly's speed was 30 miles per hour. Each cyclist's speed was 15 miles per hour.
How many miles did the fly travel?

The problem looks more complicated that it actually is. At first, students might say there is not enough information given, since we don't know the distance between the two cities. On the other hand, although it is highly unlikely that a fly would travel from biker to biker until the two meet, students will probably find it funny and will try to figure out the solution. It is not a practical problem, but probably it is both memorable and strange because of the travelling fly. I still remember some physics problems where we had to determine the speed and angle of a brick thrown out the window of a tall building so it would land on the pedestrian's head who was walking by. It was a weird problem but interestig enough for us to figure out the solution to it.
We could extend the problem by having different speeds for the cyclists and maybe having another fly travelling from the opposite direction with a different speed.

If the two cyclists had the same speed, they meet halfway, both travelling 90 miles. Since the fly's speed was twice as much as the cyclists' speed, and flew without stopping, it had to travel twice as much, a total of 180 miles.
If we look at the time spent until the cyclists meet, t=90/15, we get 6 hours. The fly travels between the two cyclists until they meet, for the same amount of time. This means that the distance covered by the fly will be d=30x6, d=180 miles.