# Sunday Puzzle: Air-to-Air Refueling Edition

Need something to wash the taste of Patriots victory out of your mouth? Try the Sunday puzzle.

Much ink has been spilled about the limited range of modern American carrier fighters. It’s one of parvusimperator’s biggest bugaboos. Sometimes though, it’s difficult to get a good sense for the penalties imposed by limited range. Let’s formulate it as a riddle.

You, Colonel Reader, command a fighter wing in Friendly Mideastistan. You have orders to strike a target in Enemy Mideastistan. Your target is six hours away from your airbase. Your fighters only fly for three hours on one tank of fuel.

Some additional parameters: your fighters fly at a constant speed without respect to payload or altitude, and their fuel consumption is also constant. You have exactly one airfield to work with, placed six hours from the target. Planes may not take off from or land anywhere else, nor may a plane run out of fuel in midair. (It looks bad.) Planes may refuel each other; there is no limit on how much fuel a plane may transfer. Takeoffs, landings, and refueling are assumed to be instantaneous.

Question 1: for each plane which drops a payload on the target, how many planes are required for refueling?

Question 2: describe the pattern of refueling which is required to get one plane to the target.

Spoiler:

I first saw this as a question about circumnavigating the globe, so I’m going to talk about 180 and 360 minutes instead of 6 hours.

Two refueling planes are required per strike plane. All three take off at T+0. When they reach T+45, all three have 135 minutes of endurance remaining. The first refueling plane fully refuels the other two, at the cost of 90 minutes of endurance. It has 45 minutes of endurance remaining, and returns home. The other two have 180 minutes of endurance left.

At T+90, the first refueling plane has landed. In the air, the two remaining planes each have 135 minutes of endurance remaining. The second refueling plane fully refuels the strike plane. The refueling plane is left with 90 minutes of endurance, and the strike plane now has a full 180. The refueling plane turns for home.

At T+180, the strike plane drops its payload on the target, with 90 minutes of endurance remaining. The second refueling plane has landed. The first refueling plane, now refueled, takes off.

At T+270, the strike plane and the first refueling plane meet. The strike plane is running on fumes, with 0 minutes of endurance left, and the refueling plane has 90 minutes. It transfers half its fuel to the strike plane, leaving both with 45 minutes of endurance. The second refueling plane takes off.

At T+315, the strike plane and the first refueling plane meet the second refueling plane. The first two are running on fumes. The latter has 135 minutes of endurance remaining. It transfers 45 minutes of fuel to each of the first two planes, leaving all three with 45 minutes of endurance: just enough to get back to home plate.

And this, of course, is the best-case scenario. Our model is simplistic in the extreme; it doesn’t account for the time taken to refuel, the time taken to find the tanker, the difference between fuel consumption based on payload, the ordinary requirement that strikes be flown at a speed and altitude different from those used for best cruise performance, and a myriad of other factors.

Let us consider a real-world example which closely matches our riddle in its setup: the Black Buck raids, flown by the RAF during the Falklands War. The distance between the closest British airfield, Wideawake on Ascension Island, and Port Stanley Airport, in the Falkland Islands, is 6,300 kilometers. Different sources list the Vulcan’s cruising range at between 4,100 and 7,000 kilometers. Even if you choose the worst possible figure, the Vulcan’s endurance is nearly two-thirds the required range, much better than the half in our riddle. Of course, I haven’t been able to find actual range figures; this column does not merit that much investigation.

What it does merit, however, is the answer to our riddle for the real-world case. For each Vulcan strike (they were flown as single-aircraft raids), eleven tankers were required, refueling each other, then refueling the Vulcan six times on the outbound leg and once on the return trip.

Refueling is hard. Not only is it difficult mechanically, not only does it require specialized aircraft (or limited refueling performance, for buddy stores), it also gets you into a vicious cycle in the same vein as rocket design. When you have to carry your fuel, you need more fuel to carry your fuel to where it’s needed, and so on and so forth. The United States has an enormous advantage in that it already operates refueling assets worldwide; much of its fuel is already where it needs to be. This may not always be the case, hence our advocacy for aircraft designs with legs built in.

I hope you enjoyed the inaugural Sunday puzzle. There may be others.