This problem was proposed by Martin Gardner, and posed to me by
Gary Jackoway (4/2008).

You have an air force base on an island.  With the help of other
planes, you want to fly one plane all the way around the world
without landing. 

For simplicity, imagine the island is on the equator and you want
to fly around the equator, essentially a circle.  All planes fly
at the same speed and carry the same amount of fuel, which is only
enough for a trip HALF way around the world.  All planes may
transfer fuel to each other instantaneously in flight.  What is
the minimum number of planes required so that one plane can
circumnavigate the globe and all other planes return to the island
safely.  No ditching of planes along the way.  Planes may be used more
than once.

Ajit Thyagarajan (3/2010) extended (and solved) the problem to also ask:
What is the minimum number of flights and minimum amount of gas
that the minimum number of planes will use?