Solution. Around the World (#138)
Answer: 3
Solution
We will show how three planes can accomplish the task, and then we will demonstrate that fewer planes won’t suffice.
Let’s number the planes: #1 (which will make the round-the-world trip), #2, and #3 (the support planes). For convenience, we’ll measure time in terms of fuel consumed (assuming uniform fuel consumption).
0) Launch 3 planes in the same direction.
1) After each plane has consumed 1/3 of its fuel (i.e., each has flown 1/6 of the great circle), transfer 1/3 of the tank from #3 to #2 and send #3 back to the base. After this instant refueling, the fuel balances are as follows:
- #1: 2/3
- #2: 1
- #3: 1/3
Note that the remaining fuel in plane #3 is exactly enough for it to return to the base (1/3 tank there and back).
2) Consider the situation when the first plane reaches a quarter of the journey. Since the last step, another 1/6 of the tank has been consumed by each plane, so the balances are:
- #1: 1/2
- #2: 5/6
- #3: 1/6
Refuel the first plane from the second plane to a full tank and turn the second plane back to the base:
- #1: 1
- #2: 1/3
- #3: 1/6
3) Note that in another 1/6 of the tank’s worth of fuel consumption from the last step, plane #3 will arrive at the base. At this point, we refuel it and send it to meet plane #2:
- #1: 5/6
- #2: 1/6
- #3: 1
4) After another 1/6 of the tank’s worth of fuel consumption, planes #2 and #3 meet. At the moment of their meeting, plane #2 is running on fumes (i.e., its tank has just emptied), but we refuel it instantly with the necessary amount of fuel (sufficient for all, since plane #2 is not far from the base):
- #1: 4/6
- #2: 1/6
- #3: 4/6
5) After another 1/6 of the tank’s worth of fuel consumption from the previous step, planes #2 and #3 will return to the base, and plane #1 will have covered exactly half the circumference of the world.
- #1: 1/2
- #2: 1
- #3: 1
Now, we only need to use symmetry. Send both support planes to meet the main plane and transfer 1/2 of the tank at a distance of 1/2 tank from the base with the same sequence of actions.
If we consider having only 2 planes, the most we can do is transfer 1/3 of the tank to the main plane at a distance of 1/3 tank from the shore (see point 1). But this will only cover 1/6 + 1/2 = 2/3 of the journey. Meeting the main plane with just one support plane also won’t work.
It is noteworthy that since all three planes were continuously in the sky, a total of 3 x 2 = 6 tanks of fuel were consumed.
