## The mathematics of the seven messengers

In my previous article I referred to the short story *The Seven Messengers* by Dino Buzzati, of which I have written a translation. Here is a quantitative analysis. I will also refer the reader to a very nice article published in 2009 on this topic: *The Seven Messengers and the “Buzzati Sequence”* by Giorgio D’Abramo from the National Institute of Astrophysics in Rome. It is available here on arXiv. I discovered ita few years ago after working out my own “sequence” and had a short and pleasant correspondence with Dr. D’Abramo. You can compare our respective derivations, which I think are equivalent. Here is mine.

Although Buzzati gives absolute values (40 leagues per day), all that matters is the ratio *m* between the messengers’ and caravan’s speeds (*m* > 1). The relevant measures of time are:

- The messenger-day, which take as
*unit of time*. - The caravan-day, which is
*m*times a messenger-day.

If as* unit of distance* we take ground covered in one day by a messenger, then time is equal to distance.

So if T_{n} is the time when a messenger rejoins the caravan after his n-th trip back home, we have

T_{n} + T_{n+1 } = *m* (T_{n+1} – T_{n}) [1]

Justification of [1]: both sides measure the time from when the messenger leaves (for the n-th time) to when he next rejoins the caravan. Note that the messenger goes back for his n+1-st trip on the very day he completes the n-th one. On the left we have the time/distance covered by the messenger (T_{n} to go home, plus T_{n+1} to catch up). On the right, T_{n+1} – T_{n} is the time/distance covered by the caravan in caravan units, which we multiply by *m* to get messenger-days.

The equality can be rewritten

T_{n+1} = (*m* + 1) / (*m* – 1) T_{n}

yielding a geometric progression

_{ }T_{n} = *K*^{n} T_{0} [2]

where T_{0} is when the messenger leaves for his first trip, and the constant *K* is (*m* + 1) / (*m* – 1).

The Prince, who is as bad at horses as he (unlike Buzzati) is at math, had initially expected *m* = 2. Then *K* is 3 / 1, that is to say, 3. In that case the progression [2] would have been T_{n} = 3^{n} T_{0}. Even then, he would have found the result disappointing: while the first messenger returns the first time after three days, the third messenger, for example, returns the fifth time after about almost 1000 days (3^{5} is 243, to be multiplied by 4), i.e. close to a year, and the last messenger returns for the sixth time after 16 years ( 3^{6} × 8 /365).

The way things actually happen in the the story, the Prince determines after a while that *m* = 3/2 (the messengers go faster by half than the caravan), so *K* is 5. (In the text: *Soon enough, I realized that it sufficed to multiply by five the number of days passed so far to know when the messenger would be back with us*.) The unit travel times (*K*^{n}) of messengers are as follows, giving return times if multiplied by two for the first messenger (since he first leaves on the second day), three for the second messenger) and so on:

(1) 5 days: as stated in the story, the first return is after 10 days for Alexander, 15 for Bartholomew, 20 for Cameron…

(2) 25 days: Alexander returns for the second time after almost one month.

(3) 4 months

(4) Close to two years (20 months)

(5) 8 years and a half

(6) 43 years

(7) 214 years

(8) Millennium

Buzzati was a journalist by trade; I do not know what mathematical education he had, but find his ingenuity and mastery impressive.

(By the way, there might be a good programming exercise here, with a graphical interface showing the caravan and the messengers going about their (opposite) business, and controls to vary the parameters and see what happens.)

Another point on which the Prince is delusional is his suspicion that he would have fared better by selecting more than 7 messengers, a number he now finds “*ridiculously low*”. It would have cost him more money but not helped him much, since the number of messengers only affects the initial value in the geometric progression: T_{0} in [2]. What truly matters is the exponential multiplier *K*^{n}, where the constant *K* — defined as (*m* + 1) / (*m* – 1) — is always greater than 1, inexorably making the T_{n} values take off to dazzling heights by the law of compound interest (the delight of investors and curse of borrowers).

Obviously, as *m* goes to infinity that constant *K* = (*m* + 1) / (*m* – 1) approaches its limit 1. Concretely, what messenger speed would it take for the Prince’s scheme to work to his satisfaction? The story indicates that the caravan covers 40 leagues a day; that is about 160 kilometers (see here). Ambitious but feasible (8 hours a day excluding the inevitable stops, horses on trot); in any case, I would trust Buzzati, not just because people in the 1930s had a much more direct informal understanding of horse-based travel than we do, but mostly because of his own incredible attention to details. So they are going at about 20 kilometers per hour. Now assume that for the messengers, instead of horses that only go 50% faster than the caravan, he has secured a small fleet of Cessna-style individual planes. They might fly at 180 km/h. That’s *m* = 9, nine times faster. Hence now *K* = 10 / 8 = 1.25. So we only lose 25% on each return trip; planes or no planes, the law of compound interest takes its revenge on the prince all the same, only a bit later.