As a fourth-grader in 1977, I had a love-hate relationship with my Addison-Wesley textbook. Its contents overflowed with arithmetic problems, but every so often an entertaining brainteaser appeared to break the monotony of drill practice. These puzzles were clearly marked: Each appeared in a box set aside from the main text and featured a bespectacled fish to introduce the challenge.
The back pages of the textbook featured still more drill, but below each set of problems came a mysterious code. Here is an example:
There was no fish or other indication to suggest that these codes were there for my consumption, but I didn’t care. I needed to satisfy my curiosity and uncover their meaning. I soon realized that the codes were a way to represent the answer to each addition problem with letters in place of the digits. If the answer to 420 + 189 was given as HDA, for example, then H = 6, D = 0, and A = 9.
I now had powerful technique for breezing through each set of drill problems without solving each and every one. By focusing on just enough problems to determine the values of the letters A through J, I could complete the rest by applying the cracked code.
Unfortunately, my teacher didn’t share my enthusiasm for thwarting the computations and chided me to get back on course. To this day, I still wonder about the intent of the curriculum authors. Were the codes an Easter egg intended for students to discover and use to their advantage? I’d like to think so.
Several months ago as part of the Dynamic Number project, I decided to create my own Sketchpad game that would focus on codes. The result is similar to my Addison-Wesley textbook, but with a key difference: Now, both the sums and addends are encoded with letters so that no digits appear at all. Take a look at the example below.
A grid displays the letters from A to J, each of which represents a digit from 0 to 9. Red counters sit on B and I, and their sum, AD (where A is the tens digit and D is the ones digit) appears below the grid. By dragging the counters onto different letter pairs, students’ goal is to determine the numerical values of all ten letters. Download the sketch and spend some time with the code, paying attention to the strategies you use to decrypt it.
Unlike the codes from my elementary textbook where the mathematical connection between the codes and the addition problems was minimal, these Sketchpad codes require students to think about the properties of addition in order to decipher them. Knowing, for example, that zero is the identity element for addition can help students to pinpoint which letter represents zero.
When you’ve solved the addition code, move on to the next page of the Sketchpad document. You’ll find a new code to decipher that replaces addition with multiplication.
Share with us in the comments section the strategies you devise for solving the codes. Unlike my fourth-grade teacher, I promise that I won’t disapprove of the time you spend with these puzzles!