How to Explain Patterns with Math
Learn how math can help you explain the origin of seemingly inexplicable numerical patterns.
In the last article we talked about two tips for multiplying by eleven. At the end of the article I left you hanging right on the cusp of unraveling a mystery that relates these two tips to explaining the origin of an otherwise inexplicable pattern that emerges from a mouthful of a multiplication problem. Well, I’m happy to tell you that your wait is over. Because today’s article is all about using math to explain the origin of this “mystery” pattern…which we’ll soon see really isn’t a mystery at all.
Recap of the Mystery Problem
So what was that “mystery” multiplication problem/tongue-twister? The problem I’m talking about is
111,111,111 x 111,111,111 = 12,345,678,987,654,321
That’s right…the answer to the problem is a big long number that starts with 1, counts up to 9, and then back down to 1 again. That’s definitely pretty wild and the pattern might surprise you. But by the time we’re finished today we’ll change that surprise into enlightenment.
How to Explain Patterns in Math
So what’s the best way to go about trying to understand the origins of a pattern like this that you discover in math? Well, when investigating complex patterns, it usually helps to first try and understand a simpler version of the problem. Often this kind of “small scale” investigation will lead you to insights that help you understand the bigger pattern…all while doing a lot less work. In other words, you don’t have to walk the entire Great Wall of China to come to the conclusion that it’s a great wall. A few miles will probably do the trick.
Hunting Down the Mystery Pattern
Okay, so what does that mean for our particular problem? Well, instead of starting with the full-blown mouthful of a problem, let’s try something simpler and build up from there. And by simpler, I mean let’s look at a similar problem, but with fewer digits. Namely, let’s start with the problem 11 x 11. We can use the trick for multiplying by 11 that we learned in the last article to quickly find the answer. The trick tells us to simply add the digits of the other number (which also happens to be 11 in this case) and then stick the result between the original digits. In other words, 11 x 11 = 121. Well look at that—the answer to this problem is a tiny version of our larger mystery problem—it starts with 1, then counts up to a higher number (in this case 2), and then counts back down to 1 again.
Finally Demystifying the Mystery
Pretty interesting, right? Now let’s try a slightly more complicated problem that follows the same pattern: 111 x 111. We can’t rely on our trick for multiplying by 11 in this case, but we can use the same bit of reasoning that allowed us to figure out the 11s trick in the first place. Namely, we can use the distributive property to quickly figure out the answer to this problem. The first thing to notice is that we can write 111 = 100 + 10 + 1. Which means that the problem 111 x 111 is equivalent to the problem (100 + 10 + 1) x 111, and using the distributive property, this becomes
100 x 111 + 10 x 111 + 111
Remember that when we multiply something by 100, all we’re really doing is adding two zeros to the end of it. Similarly, when we multiply something by 10, all we’re doing is adding one one zero to the end of it. Which means that the problem we’re solving is equal to 11100 + 1110 + 111. Now, I want you to think of adding these 3 numbers up the old fashioned way stacked one on top of the other, like this:
When you see it written that way, it’s immediately obvious that the answer absolutely must start with 1, then count up to the number of digits in the number you’re multiplying, and finally count back down again to 1. If we do the same thing for the even more complex example
11,111 x 11,111
Again, the pattern that we once were mystified by is now completely demystified! And you can keep on multiplying longer and longer strings of 1s, and you’ll keep on getting a similar pattern—all the way up to the problem that we started with. Pretty cool, right?
Beauty, Knowledge, Mystery?
[[AdMiddle]I’ll end with a quick word about my perspective on one aspect of this problem. It’s funny because a lot of people actually complain about being told why something like this works. They feel that explaining the things they otherwise thought magical somehow kills off its beauty and intrigue. But I actually believe the exact opposite. I’m never disappointed by the reveal and I always think that the reality of the explanation is way more interesting and beautiful than the mystery ever could be.
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, and Happy New Year, math fans!