Have you ever noticed that the number 9 is kind of amazing? What's that…did I hear you say "NO?" Then prepare yourself to be amazed and keep on reading to learn all about the magic behind the mysterious number 9.
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!
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......