Learn how to solve our positively odd brain teaser puzzle about adding up positive odd integers as we prove once and for all that math is most definitely not an awful nerd.
For the past few weeks we’ve been on a noble quest to prove to ourselves once and for all that math is not an awful nerd. And today, we’re going to finish the job! Of course, I use the phrase “awful nerd” here a bit in jest since there’s certainly no shame in being a nerd these days—in fact, a nerd is a pretty hip thing to be! My real point with all of this is that math isn’t just “not awful” (which could still mean it’s just okay), but that it can actually be really great. So, without further ado, let’s find out why.!
Recap: A Positively Odd Puzzle
In the first part of this series, we started to look at a “positively odd” math puzzle where our goal was to figure out what happens when we add up positive odd integers. We started by finding 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. By continuing to add up increasing numbers of positive odd integers (until we got tired and decided to stop), we obtained the sequence of numbers 1, 4, 9, 16, 25, 36, 49, and so on.
Then, in the second part of the series, we asked ourselves if we could see any interesting patterns in this sequence. Why? One answer is that finding patterns in numbers is a great skill to develop because it can help you better understand the world around you and get lots of important jobs (liked a code-breaking math secret agent). And that’s all true. But there’s another—equally valid—answer that I’d like to embrace today: It’s fun! And when we allowed ourselves to have some fun with math last time, we discovered that we can think of the numbers in our sequence 1, 4, 9, 16, 25, and so on as strands of connected blocks, and that each of these blocks can be folded up into beautiful little perfect squares.
Perfect Squares: Mystery Pattern
With that discovery, we arrive at the point in our story where things get very, very interesting. We’ve now found that adding up increasing numbers of positive odd integers gives us larger and larger perfect squares—that is, numbers that are formed when we multiply some number by itself. That’s weird, right? After all, given the infinite number of numbers, why does adding up a sequence of positive odd numbers always produce a very special perfect square? As we’ll soon see, this question has a surprisingly simple, elegant, and to my eyes at least rather beautiful solution. So let’s take a few minutes to discover it.
Perfect Squares: Pattern Explained!
Since they were so helpful when looking for the pattern in our number sequence earlier, let’s see if a few foldable strands of blocks can also help shed some light on the question of why this pattern exists in the first place. Imagine you have foldable strands of 1, 3, 5, 7, 9, and so on connected blocks. As you probably recognize, these numbers of blocks are just the positive odd integers that we added together to build our original sequence. Believe it or not, by doing nothing more than finding a clever way to arrange these strands of blocks, we can show why the incredible relationship we’ve discovered works.
Here’s how: Start by setting the imaginary single block down on an imaginary table (or if you’d like you could do all of this with real blocks or by drawing it on a sheet of paper). Next take the 3 block strand, fold it into an “L” shape, and position it so that it’s wrapped around the single block. What do you have? A perfectly square arrangement of blocks. Hmm, interesting. Now take the 5 block strand, fold it into an “L” shape, and position it so that it’s wrapped around the 4 block group. Again, what do you have? You’ve got another perfectly square arrangement of blocks. And you could keep on doing this with increasingly longer strands of blocks to make increasingly larger perfect squares…forever!