It’s not common for smart people to wake with a start when their subconscious solves a complex problem or stumbles across an incredible discovery. I like to imagine that this is how Jonas Salk figured out the key to the first successful polio vaccine: jolting upright from a deep sleep, wearing a big floppy nightcap, and just screaming I SOLVED THE POLIO PUZZLE! at the top of his lungs.
Fortunately, we lesser mortals aren’t exempt from such revelations. For me, it was a warm September evening when I woke in a cold sweat upon realizing that three ternary digits (or “trits”) can be used to represent exactly 27 different numbers and that – hold the phone – our alphabet (plus a space) is represented by 27 different characters!
Of course, I quickly came to find out that puzzle writers have been using ternary numbers to represent the English alphabet for years for this very reason, but my discovery sure seemed profound in the moment. Don’t take this away from me!
Modern humans are most familiar with decimal numbers – numbers represented in base ten – that use the digits 0 through 9. The digits of a decimal number from right to left represent the ones, tens, hundreds, thousands, etc… More generally, these are the powers of 10:
(1 = 100, 10 = 101, 100 = 102, 1000 = 103, etc…)
On the other hand, a ternary number is represented in base three, and only uses three digits: 0, 1, and 2. Its digits – from right to left – represent the ones, threes, nines, twenty-sevens, etc… In other words, powers of 3:
(1 = 30, 3 = 31, 9 = 32, 27 = 33, etc…)
Let’s work through a quick conversion from a ternary number to a decimal number. For this example, we’ll consider the ternary number 1201.
Based on the above, 1201 (ternary) would be equal to (1 × 33) + (2 × 32) + (0 × 31) + (1 × 30) = 27 + 18 + 0 + 1 = 46 (decimal)
In creating this puzzle I wanted to be able to take the commonly-used decimal number to letter exchange (A = 1, B = 2, etc…) and swap in each decimal number for its three-digit ternary equivalent (padded with leading zeros if necessary to reach three digits). For example, the letter G can be represented by the decimal number 7 which reads as 021 when converted to ternary. Here’s the complete list:
I don’t blame you if you glazed over that last section, but it’s time to tune back in because here’s where things start to get interesting. The ternary digits 0, 1, and 2 all carry inherent meaning, so I thought “Why not use some sort of code to represent the digits without necessarily revealing them?” Since my plaintext is a nine-letter word, I also thought it would be cool to arrange the letters – or rather, their numeric representations – in a 3×3 grid. Ultimately I decided to represent each number with three colored squares (chosen from green, orange, and purple); each square would represent a single ternary digit. Behold:
The solver’s task is now to figure out the correspondence between the color and the digits 0, 1, and 2. Fortunately, I included this convenient key, right next to the grid:
Dammit Sam, that’s not convenient! Is a Cucurbita supposed to be purple? Also…what’s a Cucurbita?
I’ll discuss how to get from here to there in Of Trits and Tangrams – Part II.