Planes Around the World

Here's a Martin Gardner puzzle which I enjoyed.

On a spherical planet there is a airport located on an island at the equator. There are n planes each of which can hold enough fuel to get half way around the world. The planes are capable of using refilling equipment (a big pipe) to transfer as much fuel as they like midflight. Planes can restock at the airport which takes a trivial amount of time.

Given that every plane has to land safely back at the airport, how many many planes must there be in order to get one plane to circumnavigate the planet by going around the equator?

Answer below:

…

The reason that I like this puzzle so much is because I was so off with my answer that it required an infinite number of planes. In fact we can do it in 3. So it goes. I guessed infinite because there is a similar puzzle involving camels crossing a desert and trying to preserve water, where it becomes a geometric series, each doubling of camels reducing the remaining distance to go by a half. However, the key here is that the planes can fly in either direction around the planet. There is a bit of non-uniqueness in the answer because 3 planes gives a bit of leeway, but here's a graph of a nice symmetrical solution:

In words, the idea is that all three planes set off at the same time and when a quarter of the tanks have gone Plane 1 transfers a quarter of its tank to each of the other two and heads back with just enough fuel to get back home. By this point one eighth of the distance has been travelled. Another eighth of the journey in and both Plane 2 and Plane 3 will have burned through half a tank, but they will have three quarters of a tank left. Plane 2 transfers a quarter over to Plane 3, leaving half a tank to get home and Plane 3 carries on with a full tank, having gone a quarter of the way around the planet.

This tank is enough to carry it another half way around the world which takes it to the three quarter mark. At this point Plane 1 has flown out to meet it after a restock, but flying in the other direction around the world. This second meeting mirrors the initial half of the solution.