How The Allies Used Mathematics To Win World War II

Aadhya Khatri - Jul 16, 2019


How The Allies Used Mathematics To Win World War II

When World War II is mentioned, what first comes to mind is probably explosions and guns. However, the war against the Nazi was not won by weapons alone

When World War II is mentioned, what first comes to mind is probably explosions, soldiers, and guns. However, the war against the Nazis was not won by weapons alone; lots of the work happened behind the scenes with the help of mathematics.

Calculate The Number Of German Tanks

In World War II, there was no doubt that German tanks were more capable and superior than those of the Allies. With that in mind, the big question was the number of tanks was under production so that the Allies can counter the threat. This might sound like a matter of intelligence, but in reality, mathematicians were the ones who came up with the answer; and they made use of reverse-engineered serial numbers.

german-tank-war-world-II-mathematics
In World War II, there was no doubt that German tanks were more capable and superior than those of the Allies

However, the Allies first attempted the traditional intelligence gathering, including decoding, spying, intercepting transmissions, and even interrogating enemies.

They did get the answer, and it was that from June 1940 to September 1942, German military-industrial complex produced around 1,400 tanks per month. They had the numbers, but the figures did not sound right at all.

The Axis utilized 1,200 tanks in the Battle of Stalingrad, which lasted for about eight months and resulted in the death of nearly two million people. Given this information, the number of 1,400 tanks was way too high.

Battle-of-Stalingrad-math
Battle of Stalingrad lasted for about eight months and resulted in the death of nearly two million people

When intelligence gathering did not yield the desired outcome, the Allies turned to mathematics, and they found a vital clue, serial numbers.

They found out that each tank they had captured bear a unique serial number. After observation and calculation, they concluded that the numbers signified the order of tank production. This data was vital for the Allies to create a mathematical model to come up with the German capacity of tank production. The model estimated that during the time from the summer of 1940 to fall of 1942, around 255 tanks were produced each month.

Using the serial number was a wise decision as after the war, it turned out that the German produced around 256 tanks per month, just one more than the estimation.

Here is the math they used:

Suppose one is an Allied intelligence analyst during World War II, and one has some serial numbers of captured German tanks. Further, assume that the tanks are numbered sequentially from 1 to N. How does one estimate the total number of tanks?

For point estimation (estimating a single value for the total), the minimum-variance unbiased estimator (MVUE, or UMVU estimator) is given by:

where m is the largest serial number observed (sample maximum) and k is the number of tanks observed (sample size). Note that once a serial number has been observed, it is no longer in the pool and will not be observed again.

This has a variance of:

{\displaystyle \operatorname {var} \left({\widehat {N}}\right)={\frac {1}{k}}{\frac {(N-k)(N+1)}{(k+2)}}\approx {\frac {N^{2}}{k^{2}}}{\text{ for small samples }}k\ll N,}

When the Allies knew the number of tanks, they could determine the scale of the force needed to defeat them.

Cracked Nazi Codes And Cyphers

The Enigma Machine was a famous encryption device to send encrypted messages. What the machine made were seemingly impossible to cipher, which changed at least once a day and had around 159 million million million settings.

Enigma-Machine-math
The Enigma Machine was a famous encryption device to send encrypted messages

To understand the messages, the British knew that they needed the most brilliant minds. That was why they gathered the best mathematicians, cryptologists, and linguists for the job. To keep the deciphering process secret, the experts were stationed in huts scattered around the Bletchley estate.

Here was what Winston Churchill said about it:

“I told you to leave no stone unturned to get staff, but I had no idea you had taken me so literally.” 

It is believed that the war was reduced by two years as a result of the deciphering work at Bletchley Park. And that was how mathematics had helped the Allies win World War II.

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