Algebra High School

Talking Inverses and the Enigma Machine

Here is a challenge which has appeared on my classroom board, in various forms, over the past 10 years:


Can you decode the message?  In 10 years, I have given out zero gift cards….so good luck.  More info on this challenge below.

A trip today to the Franklin Institute science museum in philadelphia reminded me some of cryptography nuggets you can use in math class; in particular, discussion starters for inverses, and code-breaking using matrices.  One of the first artifacts we encountered in the exhibit was the Enigma machine shown below, which I fawned over like a teenage girl at a One Direction concert.


The Engima machine is a coding machine, used primarily during World War II, to both code and decode messages.  Messages were typed using a standard keyboard.  The electric signals from the keyboard passed through a system of rotors and plugs, and lit up a letter, which was recorded.  There were a number of variations of the machine over the war years, and the Allied forces employed many mathematicians, many working through Blechtley Park in London, to intrcept and de-code messages.

Consider this intercepted message:


How would you even start to decode this message?  Does a one-to-one correspondance seem reasonable?  How else can letters be coded?

You can try your hand with some coding using this Enigma Simulator, which shows the coding rotors, inputs and outputs.  But here’s the neat thing about the Enigma machine: the machine is used to both code AND decode messages, using similar procedures, which are outlind here.


So, now you have everything you need to decode my message it seems.  You have a message, and a device.  Oh, but those pesky rotors.  If they aren’t set correctly, then the machine is of little help.  Working through this issue was the task of many of the mathematicians during WWII.  And I want you to be successful!  Set those pesky rotors to R-J-L (my initials), and start typing!  You can also copy and paste the message, but it is far more fun to watch the rotors do their work as you type.

Embedded in all of this crypography history are some neat math discussions:

  • After looking at some messages and their coded outputs, is there a ONE-TO-ONE correspondance here?  For example, does the letter E in a coded message always map to the same decoded output letter?
  • Are there any patterns we can use to help decode the message? Any predictable behavior?
  • A message is coded using a rotor setting.  Then this coded message is typed, using the same rotor settings, and we get back the original message.  The Enigma machine is its own INVERSE!  How exciting is that!  How many ideas or devices do we know of which are their own inverse?

Here are some sites with additional information relating to the Engima:

Exploring the Enigma, from +Plus Magazine.  Good student reading, with guiding questions.

This Numberphile Video has a demonstrations of the gears and plugboard of the Enigma, and some explanation of combinations.

In my next post, we’ll look at Hill’s Cipher, a cryptography application of matrices, and think about my Best Buy challenge!

High School

A New Start for The Blog?

So, the name of this blog is “MathCoachBlog”.  I picked the blog name about a year-and-a-half ago, as I was working in my district’s curriculum office, and hoped to use this forum to share ideas, resources and experiences.  I treasure the opportunities I have had to share with others, the kind feedback people have given me about many of the activities, and the many friends I have made through this blog and twitter.

But here’s the thing: I’m not an academic coach anymore.  After 2 years working with some great people in my office, I have chosen to go back to the classroom.  This was purely my decision, and I am looking forward to implementing many of the ideas and resources I have been encountered in the last 2 years.  I’m thinking of it as a mid-career re-set, and in some ways I am more energized to teach classes than I ever was before.

But the blog….keep the name?  Change the name?  Keep it?  Change it?  We’ll get back to that….

This week, I had to set-up a new classroom for the first time in a long time.  I was in the same room for 13 years before, and had to pack a lot of stuff when I moved into an office.  So, time to dust off the cobwebs, think about what’s important, and do some moving-in.  Here are some elements I like to have in my classroom.  What are some neat things you like to have to create a positive classroom culture?

I like to tell a lot of stories in my class, think about anecdtoes from previous years, and keep in contact with as many former students as I can.  For AP Statistics, the “Wall of 5’s” has always been a topic of conversation, and a goal for many students who strive to eventually be “immortalized”.  It’s a nice hook the AP classes.  Thanks to my colleague Joel, who kept the wall alive the past 2 years, and has now made a duplicate wall for his classroom.

Wall of Fives

Along the same lines, I like to have lots of pictures from previous years around.  Many of these are from our annual Stats Fair, and are great conversation starters.

Stats Fair

The Wall of Badges: more chances to talk about my experiences as a Siemens STEM Fellow, an AP reader, and conference junkie.


Cool art.  Escher works always generate buzz.  Now with 100% more Legoes!


T-shirts from math contests our math club has attended provide just the right amount of geek-pride.


And finally, the oragami art a graduate made for me is the best gift ever, and gets it own spot in the room.  Special appearance by John McClain – a “Secret Santa” gift.


So, about the name of the blog…  Since I announced my return to the classroom, I have had lots of conversations with colleagues in my department, and know I am blessed to work with many fantastic people.  Sometimes we don’t agree on things, and that’s healthy.  And I’m thrilled to be able to implement so many of the great new things I have learned, and continue to share them out to you.  So, I’m no longer a coach in my district, but I think that, in many ways, I way wind up being a more effective coach to my friends online through the sharing of classroom ideas.  So, MATHCOACHBLOG LIVES ON!

Also, it’s sort of a pain to change the name,….so there ‘s that.