*Photo by jaycoxfilm.*

**Math concepts:** mental calculations, math vocabulary, and anything else you want to include

**Number of players:** any number, but I think it works best with two players who alternate asking questions

**Equipment:** imagination and, if necessary, scratch paper

Many years ago, I read a magazine article by mathematical music critic **Edward Rothstein**, wherein he described a game he invented for his daughter:

**“What number am I? If you add me to myself, you get four.”**

Rather than explaining the rules of the game, let me tell you a story…

## A Long Time Ago, in a Minivan Far, Far Away

I gave Rothstein’s question to then-6yo Niner (pronounced NEE-ner) during a family car trip. The Cowgirl, age 9, wanted in the game. I tried a question with slightly bigger numbers, but she rolled her eyes.

“That’s too *easy*, Mom.”

So I asked her:

**“What number am I? If you take away one fourth of me, then add two, you get 17.”**

The older they get, the harder Mom has to work. For 12yo ComputerGeek, I asked:

**“What number am I? If you multiply me by myself and add one, you get half as many as the number of pennies in a dollar.”**

[To put this story in historical perspective: **Last week**, Niner had to enter **her photos** in the County Fair as an “adult” competitor. **The Cowgirl** and her brother graduated long before I thought of **using blogs** as part of our language arts education, so I invented their online names just now. If they complain, I can always come back and change them.]

### Turn-About Is Fair Play

My question kept ComputerGeek busy for a few minutes. After he figured it out, he came back with:

**“What number am I? If you divide me by two and take away four, then add five, then multiply by three and divide by two and add seven, you get me again.”**

“What?!”

He repeated the question.

“This is really a number?” I said. “You figured out an answer to this?”

He nodded, with the smug grin of a preteen who knows he has Mom skewered.

### Problem Solving Tool #1: Guess and Check

I pulled out a notebook and pen. He gave the series again, and this time I wrote it down. I figured the answer had to be zero or one — those magic numbers that make multiplying easy — but neither worked. I tried 100. No luck.

I heard my son chuckle from the seat behind me.

“Wait,” I said. “Give me a chance.”

### Mom Solves the Problem, But…

The Engineer was driving, but he glanced over at the notebook. “You know,” he said, “you could set that up as an equation.”

No way. ComputerGeek had not needed algebra to figure it out, so neither did I. I tried 10, then 50, then 20. Okay, that narrowed it down. Now I knew it was between 20 and 50, but I had run out of easy numbers.

I nibbled on the end of my pen.

ComputerGeek hummed to himself.

“I’ve got it!” I spun around as far as the seat belt allowed. “The answer is—.”

“Nope.”

“What?!”

### Problem Solving Tool #2: Try Algebra

I sent the numbers through my mind again, then did it out loud. ComputerGeek agreed that my answer would work, but it was not his number. I would have to guess again.

The Engineer protested that there could not be another answer. If a polynomial doesn’t have an *x*^{2} or something similar, there can’t be more than one solution.

The little brat stood his ground, smirking.

### What Do You Do with a Kid Like That?

I conceded. “What’s your number?”

“Infinity! It doesn’t matter what you multiply or take away, it’s always infinity.”

Aha! He was right. Even better — from his point of view — he had stumped the teacher. You won’t find a chance like that in a 6th-grade textbook.

### To the Mathematical Purists

Of course, I know that infinity is **not a number**. You cannot calculate with it. [But you can have **plenty of fun with it**, even so.] But after all, ComputerGeek was only in middle school, so I cut him some slack.

By the way, did anyone bother to figure out the answers to our puzzles? Here is one more:

**The second question above has more than one solution, depending on how you interpret the words. Naturally, the Cowgirl did***not*interpret it the same way I did. Can you find both of our answers?

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Looks like 34 to me, I don’t see how to interpret it another way that I wouldn’t consider wrong. I think the kids will be graduating from I spy to a version of this game soon.

Yep, that’s the answer I got. The question with two interpretations was, “If you take away one fourth of me, then add two, you get 17.” And you or I would probably consider HorseyGirl’s solution wrong, but I find it wise to allow for elementary students’ immature understanding of the English language. If the student can explain her solution, and if it is correct given her interpretation of the way the problem was stated, then I usually let the answer stand.

I get 20. Let’s put this into algebra terms and let X=”I”. So the question is

X – X/4 + 2 = 17

3/4X + 2 = 17

3/4X = 15

X = 15*(4/3)

X = 20

How do you get 34?

34 was my answer to the other problem: “What number am I? If you divide me by two and take away four, then add five, then multiply by three and divide by two and add seven, you get me again.”

The question with an alternate interpretation was: “What number am I? If you take away one fourth of me, then add two, you get 17.” Diz gave the standard version above.

But HorseyGirl interpreted it like this: “What number am I? If you take one fourth of me and throw away the rest, then add two to the part that you kept, you get 17.”