Whether it’s military tanks, roaming wildlife, or busy cutlery in a restaurant, counting moving objects can be quite challenging. Thankfully, there exists a method that enables you to estimate the total number of items without having to count every single one.

The capture-recapture technique works by sampling. For instance, you allow some animals to roam, then collect a subset. After marking the individuals, they are returned to the population. Later, you can capture another group and count how many of them are marked.

If your first capture involves 50 marked animals, and you find that half of the second group are marked, you can deduce that approximately half of the total population is marked. Therefore, the entire population can be estimated to be around 100.

During World War II, Allied statisticians aimed to estimate the number of tanks manufactured by the German forces. Instead of releasing captured tanks, they labeled tank components with serial numbers. By recording the serial numbers of both captured and destroyed tanks, they could estimate total production under the assumption of uniform distribution. If the highest serial number recorded is l and n is the number of captured tanks, then the total tank count can be estimated as l + L/n.

For example, if the maximum serial number logged is 80, you might estimate the full range to be around 80/4 = 20, resulting in an overall estimate of about 100 tanks. This problem is commonly referred to as the German tank problem in statistics.

One of my favorite stories about estimating populations comes from a friend’s teacher. The class was tasked with estimating the number of forks in the cafeteria.

The students “captured” several forks, marking each with a spot of nail polish before releasing them back. A week later, they recaptured a sample and used it to estimate the total fork count.

Researcher executed a similar study 20 years ago. Concerned about missing teaspoons in their lab, they marked and released a number of spoons, tracked their movements, and published their findings. The outcome proved effective, prompting the notorious return of five misplaced teaspoons by the culprit in the building.

Katie Steckles is a mathematician, educator, YouTuber, and author based in Manchester, UK. She also serves as an advisor to Brent Wister, a puzzle column for New Scientist. Follow her on Twitter @stecks.

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