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Think of a number, add 3, double it, now subtract 4, divide by 2, and finally subtract your original number from the result. You get 1, right? How do I know? And what are some other fun math tricks that you can use to amaze your friends? Keep on reading to find out!

By
Jason Marshall, PhD,
November 30, 2013
Episode #176

Do you know any good math puzzles? If so, please send me a note and let me know what they are—I'm always looking for good puzzles to add to my collection and to share with math fans around the world. If not, today we're going to take a look at 3 fun listener-inspired math puzzles that you can use to kick-off your collection and amaze your friends at future dinner parties.

Ready to be puzzled? And then un-puzzled? Good, because that's exactly what today is all about. Enjoy!.

## Puzzle 1: Think of a Number

Our first puzzle comes from math fan Kristina. She writes:

"Math Dude, can you help me explain how this works??

Think of a number. Add 3. Double the result. Subtract 4. Cut that number in half. Subtract the number you first thought of. Your answer is 1!"

If you're playing along, you'll have found that the number you ended up with is indeed 1. But how in the world can that work all the time no matter what number you start with?

As you'll soon see, the answer is surprisingly simple. There are a ton of different versions of this kind of "think of a number" puzzle out there, and until you realize what's going on, they all look kind of magical. But no magic is required here other than the magic of math itself—which, you might argue, is actually pretty magical. So how does this non-magic trick work?

## Puzzle 1 Solution

Long time math fans may recall talking about a very similar puzzle in our How to Amaze Your Friends with Number Tricks episode. Back then, we found that the key to understanding how all of this works is to know that under the hood we're actually creating an manipulating an algebraic expression.

To see what I mean, let’s use the question mark symbol, “?”, as a variable representing the initial number we think of. When we add 3 to this number, we create the algebraic expression: "?" + 3. When we double this, our expression changes to: 2 x ("?" + 3). When we subtract 4 and then cut this in half, our expression becomes: [2 x ("?" + 3) - 4] / 2.

This looks awfully complicated right now, so let's take a minute to simplify it. In particular, let's use the distributive property to first multiply out the part inside square brackets to get [(2 x "?") + (2 x 3) - 4] / 2 = [(2 x "?") + 2] / 2, and then use the distributive property again to divide everything inside square brackets by 2. This leaves us with the expression: "?" + 1.

The final step of the puzzle has us subtract the original number—which in our case is "?"—from the result. We can now see that when we do that we've rather sneakily been tricked into completely removing our original number from the problem: "?" + 1 - "?" = 1. Which is precisely why the answer must always equal 1—no matter what number we start with!

## Puzzle 2: Beat a Hasty Retreat

Our second puzzle, called "Beat a Hasty Retreat," comes to us from math fan Donald. He writes:

"I recently pulled-out one of my favorite math puzzles for a friend at a pub and had to actually do the math to prove my solution. My question is, am I doing the math right? I know that the answer is right. Here’s the question: You awake one morning, get ready for work, jump in the car and drive to the office. Being a bit of a math geek, you realize that on the way in, your average speed (door-to-door) was exactly 30 MPH. As you're about to kill the engine and head in, you notice the lot is completely empty and you realize that it's Saturday!

So you decide to beat a hasty retreat and, given the relative emptiness of the roads on the way in (and not wanting to spend more of this beautiful day on the road than you absolutely need to), you decide to drive home fast enough to make your round-trip (door-to-door-to-door) average speed exactly 60 MPH.

You set off to drive home via the exact same route you took in. The question is exactly how fast do you need to drive on your trip back home to achieve this goal?"

## Puzzle 2 Solution

This is one of those puzzles that seems like it should have a quick and easy answer. In fact, your initial gut reaction might tell you that you simply have to drive 90 MPH on your way back home to make your average speed for the trip 60 MPH (since the average of 30 and 90 is 60). But is that right?

Well, let's say your commute is 30 miles, then your trip to work at 30 MPH would take 1 hour. If you drive home at 90 MPH, your return trip would take 1/3 hour or 20 minutes. Which means that your average speed for the whole trip would be 60 miles / 1.333 hours — which is less than the 60 MPH that you wanted. So, no, 90 MPH isn't fast enough!

What about traveling 120 MPH on the way home? Well, that return trip would only take 1/4 hour or 15 minutes. Which would make an average speed of 60 miles / 1.25 hours — which, again, is less than the 60 MPH average speed for the entire trip that you wanted.

In fact, if you think about it, you'll see that no matter how fast you drive on the way home, you'll never be able to quite reach an average speed of 60 MPH since the denominator of the distance-over-time fraction is always a number that's greater than 1. Sadly, unless you somehow manage to travel home at infinite speed, you'll never be able to reach an average of 60 MPH.

The task is not possible in the universe we inhabit.

Or, as Donald wrote when he posed the puzzle to me:

"The answer is this: Your goal cannot be accomplished. The task is not possible in the universe we inhabit. At twice the distance and twice the average speed, the round-trip should take exactly the same time as the trip has already taken, leaving you no time to make the return trip."

## Puzzle 3: Mysterious 9s

Our third and final puzzle this week comes from math fan Natalie. She writes:

"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?

Examples:

7,269

• Add those digits together: 7 + 2 + 6 + 9 = 24
• ​Subtract 24 from your original number: 7,269 - 24 ​= 7,245
• ​Add those digits together: 7 + 2 + 4 + 5 = 18
• Add those digits together: 1 + 8 = 9

3,568

• 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!?!"

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

## Puzzle 3 Solution

Well Natalie, that is an excellent question. The pattern you've discovered is very cool, and the reason for its existence is very cool, too. In fact, it's so cool that we're going to devote the entire next episode to it—and to looking at a few additional amazing properties of the mysterious number 9 as well. So be sure to check back next time to find out how it works!

## 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!

FAQ image from Shutterstock.

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