Before finishing our “proof” that math is not an awful nerd, we first need to learn how to look for and discover patterns in sequences of numbers.
The objective of the math game we started last time was to figure out what happens when we add up positive odd integers. To solve this puzzle, we started by looking at the sequence of numbers we get when we add up the first 1 positive odd integer (which is just the number 1), then the first 2 positive odd integers (which is 1 + 3 = 4), then the first 3 positive odd integers (which is 1 + 3 + 5 = 9), and so on forever…and ever. Of course, we stopped long before reaching forever (or else we’d still be going), and after just a few more addition problems we found that adding up increasing numbers of positive odd integers gives the sequence: 1, 4, 9, 16, 25, 36, 49, and so on.
But so what? Why did we do this? Of course, our real objective was to “prove” to ourselves, our friends, and the entire world that math is not an awful nerd (which we’ll finish doing next week). But for now I’d like to focus on the fact that the sequence of numbers we found leads to a few interesting questions. Namely, can you find a pattern in the sequence? And if you can find a pattern, can you also figure out why that pattern exists? Which is exactly what we’re going to talk about today.
What Are Patterns in Numbers?
As I mentioned last time, it’s great if you see a sequence of numbers like 1, 4, 9, 16, 25, 36, 49, and so on, and see a pattern right away. But you shouldn’t worry if you don’t. The truth is that it takes time and practice to develop an eye for this sort of thing. So whether you’ve seen a pattern in these numbers or not, let’s take a few minutes to walk through the process that I might take to search for one.
The first thing you might notice about the sequence 1, 4, 9, 16, 25, 36, 49 is that the numbers all keep getting bigger and bigger. Is that a pattern? Sure. But is it an interesting pattern? In other words, can it be used to help figure out what the next number in the sequence has to be—since that’s what we’re really interested in here. In this case, not really. After all, we created the sequence in the first place by adding larger and larger positive integers together. So the sum absolutely has to keep getting larger as well. Which means that while this is a pattern of sorts, it’s not particularly interesting.
The Value of Making Mistakes in Math
Since I’m so picky about patterns, let’s set our sights on finding something more insightful. How about the differences between the numbers? In other words, do you get the same number every time you subtract one number in the sequence from the next? Or maybe you get some other interesting result? Let’s try it out and see. The first few differences between the numbers in the sequence are: 4 – 1 = 3, 9 – 4 = 5, and 16 – 9 = 7. So the differences are not all the same (which seems kind of interesting) and it looks like they’re following a pattern (which also seems pretty interesting). We now need to ask ourselves whether or not this is actually an insightful pattern we’ve discovered? Unfortunately, when we do that, we see that it’s not. In fact, we see that we’ve simply (and kind of sillily) rediscover the sequence of positive odd numbers—3, 5, 7, and so on—that we used to create our list in the first place. Although you could argue that this knowledge does allow you to predict the next number in the sequence, it’s not new information. And that means that it’s also not very insightful.
Which brings us to an important lesson: Math is a process that should be full of silly missteps. There’s no harm in making mistakes, so don’t be afraid to try things out. More often than not the first thing you try will be wrong, as it often is when I’m solving math problems for the astronomical research I do. But trying wrong things is the first step in discovering right things. And since we’ve only tried wrong things so far for our puzzle, we’re still looking for the winning pattern. So what else can we find?
How to Look for Patterns in Numbers
To gain a fresh perspective on our puzzle, let’s try a completely different tact: Imagine you have several groups of blocks. Besides having a solitary block, you also have three other groups of blocks arranged in straight lines: a 4 block strand, a 9 block strand, and a 16 block strand. Do these numbers—1, 4, 9, and 16—look familiar? They should by now since they’re the first four numbers from our sequence. So how are these strands of blocks going to help us find a pattern in our numbers?
Imagine that the blocks in each strand are held together by a rubber band so that you can move them around and change their overall shape. For example, you could take a 4 block strand that’s arranged in a straight line and move 1 block to turn the strand into kind of an “L” shape. And, if you think about it, you’ll see that you could also take the straight 4, 9, and 16 block strands, rearrange them a bit, and fold them each into a perfect square—the 4 block strand will become a 2x2 square, the 9 block strand will become a 3x3 square, and the 16 block strand will become a 4x4 square.
The Mystery of the Perfect Squares
Which means that the first four numbers in our sequence can all be folded into perfect squares. Not only is that “interesting,” I’d say it’s incredibly interesting since most numbers in existence can’t be folded into squares like this. And yet ever single one of ours can be! Does the pattern keep working? In other words, is every number on our list the square of some number? Well, the next number in the list is 25, and 25 is equal to 5x5. So, yes, that one works too. The number after that is 36 which is 6x6, and if you keep going you’ll find that this pattern continues to hold up…forever!
Does any of this strike you as being at all weird? After all, we started by doing nothing more than adding up positive odd numbers. And we somehow ended up creating a list of all the very special numbers known as perfect squares—1, 4, 9, 16, and so on. If you think about it for a minute, this is even weirder because why in the world should adding up odd numbers have anything to do with creating perfect squares? They seem like totally different beasts! Is there something super-deep going on here? Now that we’ve discovered the pattern, we’re finally ready to answer this question and figure out why it exists. But since we’re all out of time for today, that’s going to have to wait until next time as we once and for all debunk the opinion that math is an awful nerd!
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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, math fans!