Want to know a secret? Ok, here it is:
24 31 34 51 15 43 15 13 42 15 44 13 34 14 15 43
Did you get it?
Ok, here's a hint: the graphic above is the key.
Yes! That's right, the secret message is, "I love secret codes!" It's true, I do. I have a vivid memory as a child of writing some secret message and hiding it inside the elephant statue / side table in the living room (the hollow leg made a great hiding place!) -- what's funny is years later, when we moved, the secret message was still in there! There's something innately human about our desire to keep and share secrets. In today's blog, I talk about how we can build on that human tendency and create a wonderfully rich mathematics lesson that encompasses cryptography (the study of secure communication, including secret codes & ciphers), history, and algebra.
First, let's talk about the graphic above. It's a particular kind of grid used for encoding and decoding secret messages (ciphers). You probably noticed it is a 5x5 square and that each letter in the alphabet corresponds to a two-digit code: for example, the letter D corresponds to 14 and the letter P corresponds to 35. This particular kind of square grid was first used by an ancient Greek historian and scholar named Polybius and thus it gets its name, a Polybius square. It is a type of substitution cipher because letters in the original message are substituted for something else (in this case, two-digit numbers).
I love these types of ciphers because they offer a fun way for students to practice both substitution (a core concept in algebra) and coordinate pairs (another core concept in algebra). The Polybius square works similar to the way a Cartesian coordinate system does: the two-digit number for each letter is given by the row number followed by the column number. Just like graphing in algebra class, for everyone to get the same answer, we must all agree on the order we will use to designate each letter: (row, column) is not the same as (column, row). It is nice to be able to tell students that just like Rene Descartes decided we would write the x-coordinate first, then the y-coordinate when we give a point's location on a graph, Polybius decided that his cipher would work by substituting each letter by the number of the row followed by the number of the column for its location in the grid. In case you're wondering, he only needed a 5x5 grid because there are only 24 letters in the Greek alphabet (which left a grid location for either a space or a type of punctuation like a period). For us, we have choices: we can combine two letters that are easily determined by context in a word (like I've done by assigning both i and j to 24) or we could create a larger square (6x6 would offer us a few extra spots for punctuation and a space) or we could create a rectangular cipher.
An interesting historical note is that Polybius probably did not construct his grid for the purpose of hiding secrets but rather to aid in long distance communication via telegraphy. He likely envisioned holding up lighted torches to signal messages, sort of like an early version of Morse code. In fact, the Polybius square was even used as an easier-to-learn tapping type of code employed by American soldiers imprisoned during the Vietnam War.
Though it may seem difficult at first, messages encoded using a Polybius square cipher are not very secure. You may have noticed, even in the short message I began the blog with that the number 15 occurred quite a lot, signifying to anyone familiar with our alphabet (or a huge fan of Wheel of Fortune), that it likely represents a letter that occurs often in English (like an E, S, or T). The cool thing about this type of cipher, especially if you use it as an activity in a math class, is that you can challenge your students to think of ways to make this type of cipher harder to crack. Likely they will think of strategies like mixing up the alphabet in the grid or even using something other than two sets of numbers (like a number and a color):
In addition, the message itself can be made more challenging by writing it differently:
24 31 34 51 15 43 15 13 42 15 44 13 34 14 15 43 (original)
24313451154315134215441334141543 (compressed)
243134 51 154315 134 21 5 4413 341 4 1543 (misleading spacing)
There are many ways you can use this in class. One of my favorites is to challenge students to encode messages which are then "intercepted" and must be decoded in order to be read. I've even been known to include a secret message which, if correctly deciphered, might earn bonus points on the assignment or quiz for the crypto-enthused student.
Ready for your own secret code challenge? It just so happens that a secret code just made the news this week and the code-breakers are asking for help! It was found on a pigeon leg from a bird long-dead in the chimney of a house in Surrey and the still-secret message dates back to WWII. True story. Here it is:
Intrigued? You can read more about it from an article in the New York Times and watch a video from the BBC where they discuss it as well. Go ahead, crack it!
The Solver Blog
Author: Dr. Diana S. Perdue