Recently I read Seven Brief Lessons on Physics by Carlo Roveli. True to its title, the book is brief, and clocks in at a scant eighty-one pages. Its beauty is in the nearly poetic brevity and the expressive nature of the word choices the author (and translators) use. Unlike massive scientific tomes on particle physics I once attempted to read as a 12-year-old boy, Roveli’s book is written in a way I could have understood even as a young boy.
The book brought me a remembrance of the elasticity of young minds with its references to young physicists like Einstein, and his solving massive problems with simple solutions at a very young age. The book inspired me to write, and this is the result.
When I was a first-grade elementary school student, I recall having a tough time memorizing multiplication tables as students were taught to do at the time. Ever precocious, I wanted there to be some rhyme or reason, a pattern even, to the way we determined the values of these simple mathematical expressions. No, I was told, you just have to add the number you are multiplying by to itself the number of times the multiplier expresses.
For a young, and frankly lazy, Willy Munn, this was an annoyance at best for a number like one, two, and three; and completely unbearable for the silly number nine. You see, not being able to use a calculator or a multiplication table, and being required to show my work meant writing out nine addition problems just to “show my work” and get an acceptable grade.
My young brain started looking for a better way. It wanted to enable the laziness my heart so desired. It wanted to find a way for me to skip writing out these problems and allow me to go back to reading or playing with Lego blocks or watching the Superman movies I taped onto VHS from TV. I WANTED some rule as simple as “1 times anything is the number being multiplied” or “10 times anything is the number being multiplied with a 0 tacked onto the end”. I started to see patterns (we are talking about math after all, which is nothing without patterns). A solution for the ridiculously tedious number nine came first.
Looking at the following multiplication table, I unconsciously looked for patterns I could exploit. What patterns do you see?
I noticed the following in this approximate order:
Reversed Pairs of Numbers as Results
This was the first thing I noticed. 18 and 81, 27 and 72, and 36 and 63 all the results on opposite ends of the table form numeric palindromes. Even 09 and 90 technically work.
Everything Adds Up to Nine
Next, I realized that the first and second digit of every result always add up to 9. This seemed like something exploitable, and I saved the knowledge for later use.
First Digit Is The Number Being Multiplied By Minus One.
Eureka! I had a formula (even if I didn’t know that’s what it was called). I could take any number one through ten and subtract one from it for the first digit. Then I could subtract that number from nine to get the second digit. Simplicity itself!
I took my solution to Mrs. D, my first-grade teacher, for confirmation. I just knew she would be so happy and proud that I had found a better way to do multiplication by nines.
My youthful supposition couldn’t have been farther from the truth.
Mrs. D. assured me that this was not the proper way to do the math and that I must write our all of my answers as she had taught. Needless to say, if young Willy Munn was a thing besides lazy, he was stubborn. I dug in my heels, so to speak. I refused to do this work stating that I would sooner die than do unnecessary work after I had found a better solution. Predictably, Mrs. D marched me right down to the principal’s office and demanded that he deal with my insubordination. She suggested corporal punishment if necessary.
The principal of my elementary school, Mr. F looked at me with kind, wise eyes, mostly gray hair, and just a quirk of a smile on the edge of his lips, and asked me to explain my method to him. I was nervous. I didn’t know what this corporal punishment was, but it sounded serious. With sweaty little palms and a mildly shaky voice, I explained my idea for the number nine. Mr. F asked some thoughtful questions and proved my hypothesis on the blackboard in his office. Then, to the joy of my young heart, he said the words I had so desired to hear from Mrs. D, “Willy, it looks like you’ve come up with a better way to solve this problem. At least for you.” Then the golden words of freedom, “Mrs. D, since this proves out, it shouldn’t be necessary for Willy to write out all of his multiplication solutions should it?” Mrs. D grudgingly agreed with her boss, although I’m fairly sure she held a cold, dark place in her heart for me for the rest of the year. In retrospect, maybe I shouldn’t have challenged her so directly, but I was 6 and thought the world revolved around me.
Mrs. D grudgingly agreed with her boss, although I’m fairly sure she held a cold, dark place in her heart for me for the rest of the year. In retrospect, I shouldn’t have challenged her the way I did, but I was six and thought the world revolved around me.
I went on to “discover” patterns for other simple multiplication problems that saved me a bit of time over the years, and even eventually memorized all the basic times tables. For several years, I got away with doing all of my math problems in my head and never showing my work. This irritated my teachers to no end and probably is what made learning manual Calculus feel like one of the most tedious tasks I’ve ever undertaken.
Many years later, while attending Southern Utah University, I learned additional math practices in my Discrete Mathematics class that used similar patterns and approaches to my early experimentation. Fun realization, that.
The Point (There Is One)
I knew, even as a youth, I had spent more time working on this solution than it would have taken just to write out the problems as I had been asked.
Today, I recognize my solution as less simple from the Mrs. D’s viewpoint. She was giving students one tried and true way of solving all multiplication problems. While it wasn’t easier for me to achieve my work, it was certainly simpler to teach. Just because there is a pattern or formula, doesn’t make a solution better based on that merit alone.
A fractal bacteria colony may BE simple once you understand it, but at first glance, it seems strange and complex.
All anecdotes aside, the pursuit of better, simpler solutions is one that I’ve been chasing from a young age. It is the same pursuit of any scientist including the computer scientist. Let’s not forget this as we write code. Our ultimate goal should be the simpler solution because it will lead to better, more maintainable code. It will save time overall even if there is an opportunity cost up front.
We must remember that what is simpler from our particular point of view may not make sense from another’s perspective. A simple solution is only as good as our colleagues’ ability to grasp it.