After our 3 frequently asked questions about math puzzles episode last week, math fan Cynthia wrote to tell me about one of her favorite puzzles. As luck would have it, Cynthia’s puzzle is based upon one of the same ideas that—as we’ll soon find out—makes our as-yet-unexplained third-and-final puzzle from last time tick.

What’s the tie-in between the two? As we’ll see, they’re both based upon some pretty amazing properties of the mysterious and sometimes seemingly magical number 9. How does it all work? And what makes the number 9 so “magical?” Those are exactly the questions we’ll be answering over the next few weeks!

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A Mathemagical Trick

Before we get to those amazing properties of the number 9, I want to start by telling you about the mathemagical trick that math fan Cynthia sent me. This is definitely one that you’ll want to play along with. Here’s how it goes:

• Start by thinking of a number, any number.
• Now, multiply that number by 9.
• If the result is a multi-digit number, add its digits together to come up with a new number.
• If that new number is still a multi-digit number, add its digits together to come up with yet another new number. Continue doing this until you end up with a 1-digit number.
• Once you have a 1-digit number, subtract 5 from it.
• Now, using the standard numbering of the English alphabet (where 1 is A, 2 is B, and so on), find the letter corresponding to your number.
• Next, think of a European country that begins with that letter.
• Then take the last letter of that country and think of an animal that begins with that letter.
• Finally, take the last letter of that animal and think of a color that begins with that letter.

Okay, now—oh, wait a minute—you do know that there aren’t any orange kangaroos in Denmark, right?

Ha! Is that what you came up with? If not, you’re probably thinking I’m crazy right now. But I’m betting that “orange, kangaroos, and Denmark” are exactly what a bunch of you did come up with. (Just for fun, I’m curious to get some—admittedly totally unscientific—statistics to find out how well this trick really works. So please take a quick minute and send me an email letting me know whether you came up with “orange, kangaroos, and Denmark” or something else.)

So, how does this work? How could I know what words you came up with? To find that out, we first need to recap our third-and-final and as-yet-unsolved puzzle from last time.

Mysterious 9s Puzzle

As you’ll recall, math fan Natalie asked a really great question last week. Natalie wrote:

“What I want to know is why, no matter what number you use, if you [add its digits together, subtract this from the original number, and then repeatedly sum the digits of the resulting numbers], the answer is always 9? Take the number 3,568 for example:

• Add those digits together: 3 + 5 + 6 + 8 = 22
• Subtract 22 from your original number: 3,568 – 22 = 3,546
• Add those digits together: 3 + 5 + 4 + 6 = 18
• Add those digits together: 1 + 8 = 9

I come up with 9 no matter what I do. I just want to know WHY!?!”

As I think you’ll agree, this is certainly a very strange and very cool pattern that Natalie has noticed. And, as we’ll soon see, it’s partially based on the very same idea that gave us all of those orange kangaroos.

But before we get to the connection between the two puzzles, let’s look more closely at the first part of Natalie’s mysterious number 9 puzzle……

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How to Get a Multiple of 9

Subtracting the digit sum of a number from itself always gives a multiple of 9.

The first step is to find what’s called the digit sum—which is just a fancy way of saying the number you get when you add up all the digits of the original number. We’re then told to subtract this digit sum from the original number.

The question to think about is: What happens when we do this? To see, let’s imagine we’ve started with some 2-digit number which we’ll write as xy—where x is the number in the 10s place and y is the number in the 1s place. For example, x would be 7 and y would be 5 for the number 75.

The first thing to notice here is that the number xy can also be written 10x + y. The next thing to notice is that the digit sum of the number xy is just x + y. Which—and here comes the sneaky part—means that subtracting the digit sum of a number from that number always gives (10x + y) – (x + y) = 10x + y – x – y = 9x.

Big deal? Yes, this is actually a huge deal. It says that subtracting the digit sum of a number from itself always gives a multiple of 9. And this turns out to be key for both of our puzzles.

A Common Thread

We’ve now seen that the first few steps of Natalie’s puzzle have turned our number into a multiple of 9. And if we look back at Cynthias “orange kangaroo” puzzle, we can also see that the first few steps there were also designed to have us end up with a number that’s a multiple of 9 (since multiplying any number by 9—as we were instructed to do—always gives a multiple of 9).

And that, my friends, is precisely where the paths of these two very different looking puzzles cross. Because, believe it or not, now that we’ve figured out that both puzzles start by turning our random number into a multiple of that amazing number 9, we’re just a short hop away from discovering out how both puzzles work.

But, unfortunately, we’re all out of time for today. Which means that the exciting conclusion will have to wait until next time.

Wrap Up

Okay, that’s all the math we have time for today.

Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

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!

Number 9 image from Shutterstock.