The Imitation Game: The Enigma Machine

The Enigma machine was developed at the end of World War I by a German engineer, named Arthur Scherbius, and was most famously used to encode messages within the German military before and during World War II. (Mental floss).

All german messages were crypted using enigma and send via radio which was very easily accessible. In the past , code breakers were linguistic specialist but this code was ultimately cracked by mathematicians. (Enigma Machine Labelled)
I had first heard about the Enigma code while studying history but it had interested me after watching the Imitation Game recently. In the movie , they showed how this ‘unbreakable’ code with “million million million”

The code settings were given to the generals and told them the individual settings for each day of the month. Even if the Allies had managed to get the paper with the codes , after a month they would no longer be able to decrypt the messages. “Key Sheets were extremely closely guarded and were printed in soluble ink. If it ever looked as though a Key Sheet might be captured by the Allies, German soldiers would dip it in water and wash off all the information” (Plus magazine)

(Plus magazine)
How enigma was solved
Initial Idea – Rejewski’s Theorem
It was the poles who first started solving enigma. Rejewski used a mathematical theorem—that two permutations are conjugate if and only if they have the same cycle structure—that one mathematics professor has since described as "the theorem that won World War II." (Solving Enigmas Wiring)

The radio operators would code the first 6 letters in the global settings of the settings for the rest of the message. The first six letters would be a repitition of three letters
e.g. XYZXYZ and the coded message would be ABCDEF
As the code was a repitition of itself Rejewski deduced that
X was A and three letters later X was

However , if a letter could not be encrypted to itself it would dramatically reduce the number of possibilities. This was the flaw in Enigma.

But if a letter could not be attached to itself , these are permutations where a number cannot stay in its original place called derangements. In order to express this simply we use letters instead of letters and just three letters.

For normal permutations the number of possibilities would be 3! = 6

The list of permutations are tabulated as shown abc acb bac bca cab cba

The number of possibilities are reduced to just 2 as the bolded words all have letters in its original position . This had reduced the number of possible arrangments in enigma which allowed the code to be broken within a realistic time.

For n=3 the number of derrangements notated as !n is only 2
To calculate the number of derangements of enigma . let n=the number of letters=26

And D=Derangement of

Since no number can return to its original place, the number of derangements is given as the subfactorial . !n is the notation of derrangements for n