The maths hack that can help you count things

Whether enemies in combat, animals in the wild or cutlery in an animated canteen, it is difficult to count objects that move. Fortunately, there is a technique that can estimate the number of something without obliging you to count everyone.

The method of capture -recording consists in obtaining a sample – while waiting for certain animals to walk, for example, then by collecting some – distinctly marking individuals, then by releasing them in the population. After a while, you repeat the process to choose another group of animals and count how many of them are already marked.

If you captured, let’s say, 50 animals at the start and marked them all, then on your reuptake step, you found half of the animals you saw were marked, that tells you something on the whole population. Since half of the sample is marked, this implies that half of the whole population is marked – there must therefore be around 100 individuals. This can give a reasonably precise estimate of a population, without having to find and count each member.

During the Second World War, Allied statisticians wanted to determine the number of tanks that the German army produced. The captured tanks cannot be reissued, but, because the components of the tank are marked with standard numbers, another approach allowed them to make an estimate. They recorded the standard numbers of all the tanks captured or destroyed, working on the hypothesis that they were numbered sequentially and at random. If the biggest serial number in your data is L and the number of tanks captured is nan estimate of the total number of tanks is given by L + L / n.

So, if we had four numbers, the largest of which was 80 years, we could assume that the whole range extends around 80/4 = 20, so there would be around 100 tanks overall. This is known as the problem of the German reservoir in statistics.

One of my favorite population estimation stories was told to me by a teacher friend, who instructed his students to estimate the number of forks at the school canteen – impossible to count because, at any time, a number will be used and others will be in washing.

His class “captured” a set of forks and marked each with a drop of nail polish, then released them in the population. A week later, they resumed another population of samples and used it to estimate the total number of forks.

The researchers carried out a similar experience 20 years ago. A worrying number of teaspoons lacked in their laboratory, they therefore marked a set of spoons before freeing them, studying their movements and publishing the results. It turns out that science is effective: the newspaper’s publication has led to five teaspoons are timidly retained by spoon thieves in the building.

Katie Steckles is a mathematician, lecturer, youtuber and author based in Manchester, in the United Kingdom. She is also an advisor for the New Scientist puzzle column, Brentwister. Follow it @steck

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