Five serial numbers. Now guess how many they built.

A factory stamps its output 1, 2, 3, and on. You capture a handful and read the plates. The largest number you have seen is your obvious guess — and it is always too small, by an amount you can work out.

What you're seeing

The fleet is hidden. Its vehicles are stamped 1, 2, 3, and so on up to some number you do not know. Each capture picks one of them at random and shows you its plate.

Your instinct is to answer with the largest plate you have seen. That guess is not merely uncertain, it is wrong in a direction. To be right it would need the very last vehicle ever built to be one of the few you happened to capture, and it usually is not. So the largest plate you have seen sits somewhere below the top, and there is a gap.

Here is the trick. Your captures also cut the range below the maximum into gaps, and on average every gap is the same size — including the one above your maximum, the one you cannot see. With k captures the visible part is chopped into k pieces, so the invisible piece is about one kth of what you can see. Add that back and the answer stops being systematically low.

Press Do it 4,000 times and the lower panel makes the point without any arguing. The cyan pile is the largest-seen guess: it sits entirely to the left of the truth, because it can never overshoot. The amber pile is the corrected guess, and it straddles the line. Not more accurate on any single try — it is wider, and about half the time it overshoots. What it is, is centred. Repeated over many months of intelligence reports, that is the difference between a number that drifts and a number that lands.

Set the captures to one and watch the correction double your single plate. It looks absurd, and it is right: one draw from a uniform pile sits halfway up on average, so twice it is the pile. This was not a thought experiment. Allied statisticians read the serial numbers on captured German gearboxes and wheels and put tank production at 246 a month for June 1940 to September 1942. Conventional intelligence said about 1,400. The German records recovered after the war said 245.

The rule, exactly. Serial numbers run 1 to N. Capture k of them, all different, uniformly at random, and let m be the largest you see. The expected value of m is k(N+1)/(k+1) — below N, always. Adding back the average missing gap gives N̂ = m + m/k − 1 whose expectation is exactly N, for every N and every k, with variance (N−k)(N+1)/(k(k+2)). The companion proof (improve/verify/110-german-tank.js, 96 checks) does not simulate these: it enumerates every one of the C(N,k) possible captures at nine settings and takes the true expectation, so "unbiased" is checked to a billionth rather than to sampling noise. Six negative controls:

The naive maximum is low by exactly (N−k)/(k+1).
Two plausible near-misses, m+k−1 and m(1+1/(k+1))−1, are biased by exactly computable amounts.
Capturing the same vehicle twice breaks the estimator by twenty standard errors.
Serial numbers that do not start at 1 break it by the offset.
The alternative 2×(mean)−1 is also exactly unbiased, and always has larger variance. So being unbiased is not on its own a reason to pick an estimator.
A deliberately rigged sampler is rejected by the same uniformity test that clears the real one.

What this needs and real life often refuses: numbering that starts at 1, runs without gaps, and is not scrambled — and captures that do not favour new vehicles over old.