Alan Turing was able to break the Enigma due to a combination of factors.

  1. Three Polish mathematicians: Jerzy Rozycki, Henryk Zygalski, and Marion Rejewski were able to reverse-engineer the Enigma and also take advantage of the way Germans used indicators (used in the beginning of the message to show certain settings) to break it. Later the Germans would change their indicator system. The Polish also developed an early version of the bombe which they called bomba. It was made up of Enigma machines.
  2. Certain mechanical features of the Enigma, designed to make it harder to break actually accomplished the opposite. 1) A letter could not represent itself — e.g. an A could never be an A. This made it possible to float possible readings called cribs. If an A in the crib matched up with an A it had to be wrong. 2) The rotors of the Enigma moved at different speeds. The bottom line of this is that if your crib was short enough it would only apply to the fast rotor and you could reduce the odds down substantiallly.
  3. Turing’s insights included 1) recognizing certain repetitive looping patterns in the way the Enigma works 2) starting with a false assumption as a way of ultimately coming up with an answer 3) vastly improving the bombe over the bomba also with the critical help of Gordon Welchman’s diagonal board, 4) coming up with a variety of methods to process data, including paying attention to frequent words like EINS and superimposing data to see certain patterns and more.
  4. Turing and other codebreakers were also helped by 1) regularly repeated weather reports, which could serve as constants 2) situations in which when a new code was instituted, the same message could be repeated in both the new and the old codes 3) carelessness on the part of German operators who chose non-random settings, used names of girlfriends, used dirty words.
  5. For the Naval enigma, it was also necessary to have special bigram tables which you could not figure on your own. Codebreakers like Turing benefitted greatly when German U-Boats were captured and the boarding parties took hold of these tables.

To read more about this in depth, there is no finer book than A Battle of Wits by Stephen Budansky. In this volume you also can read about the great achievements of American Codebreakers, including the incredible story of what codebreaking achieved for the Battle of Midway. There is an especially clever thing that the codebreakers did to prove their point that you will enjoy reading about.

Two important things to note: The book shows that, contrary to others, the attacks  on neither Pearl Harbor nor Coventry were known to codebreakers — at least not enough details to be sure. FDR and Churchill DID NOT let them happen for larger political goals. It was important not to let the Germans know they had been broken and ironically, one of the reasons they felt it could not be broken was because the Germans in fact easily broke the British codes!

So don’t listen to what certain movies say. The real story of how Alan broke the Enigma is much more interesting and beautiful and is very fun mathematics.

READ MORE ABOUT ALAN

Mathematical Knitting with Alan

Alan’s Sponge Cake

In the Navy

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