## Serial numbers on Lugers were deliberately confusing

Friday, March 29th, 2019

While discussing the German “Luger” pistol, Dunlap brought up a point that surprised me:

Serial numbers on Lugers were deliberately confusing, as the Germans did not like to have people adding up numbers and estimating production figures, so they organized a code-series system, which is no military secret now, but which I have never completely solved.

This surprised me, because I’d read about the Germans failing to do just that with their tanks:

The statisticians had one key piece of information, which was the serial numbers on captured mark V tanks. The statisticians believed that the Germans, being Germans, had logically numbered their tanks in the order in which they were produced. And this deduction turned out to be right. It was enough to enable them to make an estimate of the total number of tanks that had been produced up to any given moment.

The basic idea was that the highest serial number among the captured tanks could be used to calculate the overall total. The German tanks were numbered as follows: 1, 2, 3N, where N was the desired total number of tanks produced. Imagine that they had captured five tanks, with serial numbers 20, 31, 43, 78 and 92. They now had a sample of five, with a maximum serial number of 92. Call the sample size S and the maximum serial number M. After some experimentation with other series, the statisticians reckoned that a good estimator of the number of tanks would probably be provided by the simple equation (M-1)(S+1)/S. In the example given, this translates to (92-1)(5+1)/5, which is equal to 109.2. Therefore the estimate of tanks produced at that time would be 109.

By using this formula, statisticians reportedly estimated that the Germans produced 246 tanks per month between June 1940 and September 1942. At that time, standard intelligence estimates had believed the number was far, far higher, at around 1,400. After the war, the allies captured German production records, showing that the true number of tanks produced in those three years was 245 per month, almost exactly what the statisticians had calculated, and less than one fifth of what standard intelligence had thought likely.

1. Kirk says:

I think what Dunlap is referring to “never having solved” are the details like “who made it”, vs. total numbers produced. The three-letter codes on German small arms, like byf, indicated which plant and company made the weapon, but the full details of which code applied where…? Still somewhat confusing. There’s an entire book out there devoted to this stuff, and the full production code set is still somewhat obfuscated for some items and factories, from what I understand.

Tanks were pretty easy–There were only a couple of factories. Small arms? Holy hell, but everyone with a machine shop was producing parts, which would get assembled at different places, seemingly randomly. Friend of mine was a Mauser rifle collector, and if you talked to him about his collection, it didn’t take too long before your eyes would start glazing over as he went over all the little variations. He had a goal to have every single variant of Mauser rifle ever manufactured and issued by the Germans in WWI and WWII, and I think I remember him saying that would be something on the order of a thousand different rifles…

2. Alistair says:

The tanks is True. AFAIK.

However, less well known, is that you can further estimate confidence intervals for the total N using the second and subsequent highest serials taken.

You generate likelihoods for each of nth serial holding its observed value, for a suspected total series of N. The binomial can be used to do this easily – what is the pdf that n-1 out of N serials will occur in a series of trials with P(success) = (n/N). The subsequent series of likelihoods for each n can then be summed for every value of N to get its overall distribution.

This gives you a much tighter estimate of N; I haven’t seen the trick commented upon yet, but it seems obvious and doesn’t throw away information.