This week in my Advanced Pre-Algebra classes, we learned about The German Enigma Machine and Alan Turing’s efforts in WW2 to break the Enigma code.
I started by explaining the workings of the Enigma machine (you can see more details in the video below). The analogy I used was to a Cryptogram, but where every letter has its own unique coding system. I explained the rotors and the reflectors (leaving out the most complicated parts of the machine like plugboards, etc.) It was enough to give the impact of the Enigma Machine on making an (essentially) unbreakable code. I tried to impress upon the students that mathematicians not much older than them (late teens/early twenties) were given the task to break this code, and the stakes were incredibly high.
We watched a 5-minute clip of The Imitation Game where Benedict Cumberbatch (playing Alan Turing) explains how many combinations there are to the machine’s settings and how long it would take a 10-man team to break the code (20 million years!!) I told the students that this impossible task (2o million years’ worth of work in 18 hours, every single day) was in fact accomplished! I asked them to brainstorm how it could possibly have been accomplished. While they were thinking about that question, we built an Enigma machine!
Soda Cans (1 per student), Scotch Tape, Scissors, a copy of this handout (Word Doc): enigma-settings
This makes a replica (fully operational) enigma machine. The strips of paper in the Word Doc above are scaled to perfectly fit around a soda can and rotate around the smooth surface. They can be removed and replaced the same way as in an original Enigma machine. There’s no place to put the reflector on our version, but I usually just have the students set the can on top of the part of the paper that has the reflectors, so as to demonstrate the fact that it “reflects” the signal back up through the rotors in the opposite direction.
I then gave them this message:
I II III/ R X B / B / RP GT MQ PZ WV
The first three roman numerals indicate the rotors (we are using Rotor 1, Rotor 2, and Rotor 3 in that order. So the “hoops” of paper will slide over the can in this way:
The next three letters (R, X, B) give the starting letters for Rotors 1, 2, and 3 (respectively). So you should line those letters up with the “A” on each input. It should look like this:
The next single letter indicates the Reflector (“B”) that we will use for this coding.
The coded message is the next 10 letters.
For the first R: Find R on the first Input and see what it gets paired with on Rotor 1 (immediately below the R). Then find that letter on the second Input and see what it gets paired with on Rotor 2. Then find that letter on the third input and see what it gets paired with on Rotor 3. Find what that letter is paired with on Reflector B. Then work your way back through the Enigma Machine in reverse order, finding that letter on Rotor 3 and seeing what it’s paired with on the Input, etc. until finding the final letter on the first Input.
In our setting, the first letter R gets coded to B, by going through this progression:
R -> Y / Y -> R / R -> W / W – > V / V – > L / L – > C / C – > B
Then Rotor 1 clicks one space in a Clockwise direction, so your new “home row” is ACAXAB. Then you start all over again with the next letter in the code (P)
Watch the video below for more detail about how to code. My students are breaking the code for extra credit (it’s not required – so far about 10 of 32 students in my Advanced P-A classes have successfully broken the 10-letter code) and some are creating codes with their Enigma and sending them to each other. A fun lesson to be sure, and might seem sort of disconnected to any traditional standards. But this is one that gives them a sense of math’s place in history (not just the history of math but global, historical events), gets them excited about a sub-category of mathematics that is sometimes hard to access (coding and code-breaking), a glimpse into the inequity found in many parts of our not-too-distant past (the story of Alan Turing’s suicide at a relatively early age is tragic and unbelievably sad), and a dash of unsubstantiated and unfounded conspiracy theory (the Apple logo).
See the video with slides here: http://cloud.swivl.com/v/e4a4128c685460cef4fbe2fa26392450
Or watch JUST the video here: