Kinesys materializes on a tiny island, the only fixed point on Dynamos, a planet where everything else spins, races, and swings nonstop. “At least nothing moves here,” Liz sighs, right before noticing that the island hosts an extremely busy airport.
The local pilots are preparing the Great Circumnavigation: one plane must fly all the way around the planet. There’s a catch, though: a full tank only gets you halfway. Dord shakes his head: “Designing fuel tanks that small is simply irrational.”
M00N already has an idea: “What if they help each other? Together they can do it!” The chief pilot looks at her with interest: sure, but what’s the smallest number of planes it takes?
The Riddle
A group of planes is stationed on a small island on Dynamos. Each plane’s tank holds exactly enough fuel to fly halfway around the planet, but any amount of fuel can be transferred from one plane’s tank to another’s while the planes are in flight. The only source of fuel is on the island, and we assume no time is lost refueling, either in the air or on the ground.
What is the minimum number of planes needed to get one of them all the way around the planet, assuming all the planes fly at the same constant ground speed with the same fuel consumption, and that every plane makes it back to base safe and sound?
Hint
The support planes can take off at different times, and even in the opposite direction, flying out to meet the main plane on the final stretch. It helps to measure everything in eighths of a lap.
Solution
3 planes are enough. Let’s call them A, B, and C; remember that a full tank lasts half a lap, so a quarter tank lasts 1/8 of a lap.
- The three planes take off together. At 1/8 of a lap, each has used a quarter tank. C transfers a quarter tank to A and a quarter tank to B, who are full again; C is left with exactly a quarter tank, just enough to fly back to the island.
- At 1/4 of a lap, A and B again have three-quarters of a tank. B transfers a quarter tank to A, who is full again; B is left with half a tank, enough to get back to the island.
- A, with a full tank, can continue for another half lap, that is, up to 3/4 of the lap, where it arrives with an empty tank.
- Meanwhile C, after refueling on the island, takes off in the opposite direction and meets A exactly at 3/4 of the lap (that is, 1/4 of a lap from the island, on the other side). C has used half a tank and transfers a quarter tank to A: now both have a quarter tank, enough for 1/8 of a lap.
- At the same time B, also refueled, takes off in the opposite direction and meets A and C exactly at 7/8 of the lap. B has used a quarter tank, and gives a quarter tank to each of them: now all three have a quarter tank, just enough for the last 1/8 of a lap.
The three planes reach the island together running on fumes, but safe and sound. The timing works: C gets back to the island when A is at 1/4 of the lap and must take off again when A is halfway around; B gets back when A is halfway around and must take off again when A is at 3/4.
Source: Martin Gardner, Mathematical Puzzles and Diversions (Italian edition: “Enigmi e giochi matematici”)