A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and for the purposes of the problem, it is assumed that there is no time lost in refueling, either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant ground speed and rate of fuel consumption and that all planes return safely to their island base?
Three airplanes are quite sufficient to ensure the flight of one plane around the world. There are many ways this can be done, but the following seems to be the most efficient. It uses only five tanks of fuel and allows the pilots of two planes sufficient time for a cup of coffee and a sandwich before refueling at the base. And there is a pleasing symmetry in the procedure. Planes A, B and C take off together. After going one eighth of the distance, C transfers one fourth tank to A and one fourth to B. This leaves C with a tank one fourth full; just enough to get back home. Planes A and B continue another eighth of the way, then B transfers one fourth tank to A. B now has half a tank left, which is sufficient to get it back to the base, where it arrives with an empty tank. Plane A, with a full tank, continues until it runs out of fuel one fourth of the way from the base. It is met by C, which has been refueled at the base. C transfers one fourth tank to A, and both planes head for home. The two planes run out of fuel an eighth of the way from the base, where they are met by refueled plane B. Plane B transfers one fourth tank to each of the other two planes. The three planes now have just enough fuel to reach the base with empty tanks.
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A version of this puzzle originally appeared in the August 1958 issue of Scientific American.
Editor’s Note (4/7/25): The puzzle solution was edited after posting to provide a process with even fewer planes, as described in the November 1958 issue of Scientific American.