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Permutations

The number of arrangements when order matters

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Practice Permutations
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Estimated12 minsto complete
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Top Secret Codes

Credit: CIA
Source: http://commons.wikimedia.org/wiki/File:Enigma-Machine.jpg
License: CC BY-NC 3.0

Did you know that mathematicians played a crucial role in World War II? Permutations were at the heart of top-secret messaging systems in Nazi Germany, and it took a team of the most talented number crunchers to finally crack the code.

Why It Matters

During World War II, the German military used encryption, or coding, devices called Enigma machines to send secret messages. Enigma machines were like typewriters with a complex system of electrical circuits that matched the letter you typed with a completely different letter. Reading encrypted messages was much harder than just figuring out the alphabet pairs and translating. A series of rotors, or gears, inside the machine turned each time a key was pressed, changing the circuit paths so that letters paired differently each time depending on the position of the rotors. A plug board also allowed the coder to pair any combination of letters to swap when one was typed.

Credit: Greg Goebel
Source: http://commons.wikimedia.org/wiki/File:Four-rotor-enigma.jpg
License: CC BY-NC 3.0

This system of plugs, rotors, and circuits made for an incredible amount of possible settings on the machine, and the Germans believed no enemy would have time to test all these permutations without knowing the exact setting used to encrypt the message. The video below estimates that the average coder had hundreds of quintillions of possible settings to choose from. During the war, a team of mathematicians convened at Bletchley Park, a code-breaking center in England, where they found a critical flaw in the coding system that would help them crack it. The flaw was that no letter ever coded as itself.

See for yourself: http://www.youtube.com/watch?v=G2_Q9FoD-oQ

Explore More

Watch the following video to learn more about how the Allied forces managed to crack the Enigma code.

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