Doing more than Doolittle
When Japan’s attack on Pearl Harbour brought the United States into World War 2, the U.S. was faced with a huge logistical problem. While Britain and Nazi-occupied Europe weren’t very far apart, the distance between the eastern shore of the Pacific and Japan is so large that any offensive across the Pacific would have been almost impossible. Even with the help of Allies such as Britain, the British Raj, and Australia, the challenge was considerable. So how did they bomb Japan?
The first attempt was almost purely symbolic. In the wake of the Pearl Harbour bombing, the Americans sought any means possible to strike back at Japan. The best they could do were the Doolittle raids – medium range B-25 bombers launched from aircraft-carriers sailing as close as they dared to the Japanese mainland. In order to launch from the short deck of the carriers, the bombers had to be lightened, and this constraint, together with the payload, meant that return was impossible. The Doolittle raiders crashed in the China Sea or over China itself. Some were captured by the Japanese, others sheltered by Chinese partisans. It took years for the survivors to return the USA.
Later efforts were also almost comically unsuccessful. B-29 Superfortresses had to fly from India, over the Himalayas, and to Chengdu. There they would deposit fuel, and keep making this trip over and over again until they had accumulated enough fuel in Chengdu in order to fly to Japan to bomb the steel mills there. That was the theory at least. According to American Secretary of Defence Robert McNamara, who was serving in the U.S. Air Force during the war, “We had so little training on this problem of maximizing efficiency, we actually found to get some of the B—29s back instead of offloading fuel, they had to take it on. To make a long story short, it wasn’t worth a damn.”
This problem of maximising efficiency had actually been around since the Middle Ages. One of the earliest descriptions of it appeared in 9th century England, in a book by the scholar Alcuin of York. His Propositiones ad Acuendos Juvenes (Problems to Sharpen the Young), one of the earliest recreational mathematical texts, and likely the first in Latin, included a problem of maximising distance, which goes something like this “A camel can carry a maximum of 1,000 units of grain at a time and consumes 1 unit of grain per mile traveled. You have 3,000 units of grain to transport across 1,000 miles of desert. How can you maximize the amount of grain delivered to the destination?” A version of this with bananas instead of grain is somewhat popular online.
This problem reappeared in the final text of Luca Pacioli (one of the fathers of modern accounting), De viribus quantitatis (Concerning the powers of quantity). It again resurfaced in a more mathematically rigorous form in a paper by D. Gale in 1945: the “problem concerns a jeep which is able to carry enough fuel to travel a distance d, but is required to cross a desert whose distance is greater than d (for example 2d). It is to do this by carrying fuel from its home base and establishing fuel depots at various points along its route so that it can refuel as it moves further out. It is then required to cross the desert on the minimum possible amount of fuel.” The goal is to calculate the maximum distance possible that can be crossed with the fuel available (or the maximum distance when anticipating a return trip). There is also the equivalent problem of minimising fuel consumption with a given distance.
It is easy to see the relevance of this problem to aircraft creating fuel depots at airstrips. It was a growing understanding of this problem, or at least how it is solved, that helped motivate a switch to the island hopping strategy in the Pacific Theatre, which would ultimately prove to be the key to defeating Japan.
The above image visualises the problem with three units of fuel (and thus making three trips). It shows the problem where a return trip must be made. We can see that the amount of fuel left at the first depot is precisely enough to make the return trip from that depot twice, indeed that’s why the first depot is ⅙ into the desert – it’s the maximum distance at which this is possible. Whenever the jeep reaches a depot on the way out it will fill back up to full, and on the way back it will be empty by the time it reaches each depot, filling up just enough to reach the next one. The next depot is slightly further along because with one less trip to make, the maximum distance at which it is possible to leave enough fuel for all future trips increases. By the final trip, no extra depot is created, it just goes as far into the desert as it can. When this trip begins, the first depot will contain ⅓ of a unit of fuel, and the second ½.
This pattern, the harmonic series, will always appear in the final trip, irrespective of the quantity of fuel. It will just start at 1/n, n being the quantity of fuel. The harmonic series is also the solution to the problem: the maximum distance the jeep can go into the desert and return will be equal to the nth harmonic number. The solution to our example is the third harmonic number = ⅓+½+1. Since the harmonic series increases logarithmically (always increasing, but always slowing down), the number of fuel depots needed to cross a desert of a given size increases exponentially. This is one of the many reasons why going through Chengdu to bomb Japan wasn’t working: the low number of fuel depots for the huge distance being travelled meant that they were far from maximum efficiency.
This solution was only found in 1947, by American mathematicians N.J. Fine and L. Alaoglu (who had realised the relevance to WW2, and specifically how it showed why going through Chengdu was failing, although he never published his solution to the Jeep problem). This detailed understanding of the logistical issues helped the Royal Air Force pull off the spectacular and fruitless Operation Black Buck during the Falkland wars, where mid air refuelling was used to attack the Falklands from the British airbase in Ascension island (6300 kilometres away). While reality is of course more complex than the elegance of the solution to the diagramatised Jeep problem, it is nevertheless satisfying to think that a maths problem that belonged to hobbyists has helped win wars and whose insights are still part of modern logistics.
