Ian Stewart, in his excellent book Game Set and Math, had a nice little story about Mother Worm's Blanket.
Esentially, Junior Worm is a squiggly little thing exactly one unit long. His mum wants to cover him with a blanket while he sleeps (while he is sleeping he is not moving). Father Worm, being frugul, wants to make the blanket as small as possible and still be able to position it so Junior is completely covered by the blanket when he is sleeping.
The question is, 'What is the smallest blanket that meets this criterion?' Certainly a circular blanket of radius 1 will work, since no point on Junior is more than 1 unit from the tip of his tail. But can you make it smaller?
Indeed you can! But no one knows what the minimal area is. This is a nice little problem because the more you work on it, the more you can see ways to reduce the area. The article gets the blanket down to a semi-circle of radius .5, and implies that the best solution to date was an area of pi/8.