Pizza Puzzle

Ever obsess on something trivial when you know you have a ton of work due about an hour from yesterday? Here’s my obsession for today:

In the last Math Mojo Monthly Newsletter (look up at the top in the left menu bar of this post to make sure you get it. It’s free, of course) I posed this puzzle:

What is pizza = 307.72” ?

I just wrote the answer for this week’s edition of the Math Mojo Monthly (hurry and sign up for it now, so you’ll get the issue when it comes out, or else you’ll miss it – nudge, nudge!)

I got some interesting answers back, but the two most representative ones were:

The circumference of a circle whose diameter is 49 ” will be 307.72″, taking pi=3.14 So that is the size of the pizza — a 49″ pizza !!!

To which I wrote back:

Good try. How do you figure, though? Pi times diameter wouldn’t give you 307.72, would it?

Hint: Remember, this is a puzzle. There is a puzzle element at work here. What would make the question clever?

The reply came:

Oops. I meant to write 307.72/(2pi) = 49. So Area = 2pi*r² means a 7″ pizza. That’s your pizza.

Which was the same as the other representative answer, namely:

307.72 is the surface area in square inches of a pizza or circle with a radius of 7″

To which I replied:

Keep trying. You’re not too far off. Once you get it, you may get the “aha!” moment that the puzzle is actually after.

Both were noble attempts, but a little checking would have shown the errors. Plug those figures into the formula for a diameter of a circle, and you won’t get 302.72 . Each of those pizzas would have been pretty far off from the diameter of a standard pie, though. That could also have been a clue that they were off.

Here’s the reason I wanted to write this post,though. The first set of answers came from a mathematician. This gentleman is leaps and bounds ahead of me in his understanding of mathematics (I’m not a mathematician, just a math dilettante). He got the answer wrong. My point is: So, big deal? He was also the first one to write in. He had the interest and the effort going for him.

Too often schools surpress interest and discourage effort by their rush to judge students by testing them. It’s that false “accountability” platitude.

So I wrote back to the gentleman and we had a nice dialogue. I hope he communicates on this blog often, because his insights would certainly benefit readers.

This is what I wrote back to him, concerning his answers. I think it may be motivating and encouraging to some people, so I’m reproducing it here:

One of the things I am most concerned with about math are common mistakes, and why people make them.

Too often students feel intimidated about making mistakes. I think standardized testing is what kills this spontaneity and curiosity. I could be wrong, but I think that’s the way to bet.

One of the missions of Math Mojo is to relieve some of the fears and stress of students.

I think a great way to do that is to show them that anyone can make simple mistakes, and that there is no shame in that. The shame is simply not trying. There is no shame in trying to making honest mistakes.

I am frequently a great example of this. I make plenty of mistakes on my website. It has a great advantage for me, which I did not expect. Every time I make a mistake, I can count on more comments to that blog post than usual! I don’t have to make the mistakes on purpose — because I make enough without trying.

A problem I have when posting puzzles, is that many people, especially young people, assume that I know what I am talking about, and I am the guy who makes up the puzzles, and I am some kind of genius. I think they assumed that I could solve these puzzles each time I see them.

Nothing could be further from the truth. I found a version of this puzzle on the Internet. The answer was right there with the question, so I didn’t actually have to solve it. I just found it very clever and interesting.

To tell you the truth, I don’t think I could have solved it myself. But I certainly would have tried, and probably wasted a whole weekend doing it. Of course it wouldn’t really have been wasted, it would have been fun, and I’m sure I would have learned many things along the way.

Which gets me to the point of why I’m telling you this: I think you may have made another honest mistake.

As you know, the formula for the circumference of a circle is pi times diameter. It could also be pi times two times the radius (because twice the radius is the same as the diameter). C = pi*d, or  C = pi*2r.

The formula or the area of a circle is pi times the radius squared. A = pi * r².

I think the confusion stems from the fact that 2r looks like r². But they are not the same. One is two times the radius, and the other is the square of the radius.

In second answer, you  said, “ Area = 2pi*r²“.  There is an extra, unneccesary 2 in that equation. I think the extra 2 might be due to the above explanation.

It is a totally common mistake, that even a mathematician can occasionally make. I would like to let my readership know that there is no reason to feel that someone must be “bad at math” to make such mistakes. Clearly, if a mathematician can have a foggy day, then so can others. It doesn’t mean you’re “bad at math.”

When I explain the solution to the puzzle in the next Monthly, I think you will find it very entertaining. I’ll bet you will try it on some of your mathematician friends in the future.

I really appreciate our dialogue, and I thank you for playing the puzzle in the Math Mojo Monthly. I hope you keep working on it, because I really appreciate talking with an actual mathematician occasionally. It keeps me on my toes.

This all brings me back, once again, to the value of puzzles. Puzzles are a great place to be able to make mistakes. You don’t get graded on them (unless you have a teacher who’s making a great pedagogical mistake). Puzzles are a safe place to practice and  improve your intellectual ability.

Another benefit of puzzles is that they are not always straightforward. They test and hone your ability to think laterally, creatively. This wasn’t just a simple equation. It was a challenge to think differently. It could have been solved without doing any equations at all if you thought like a puzzler instead of a mathematician.

The mathematician might only be looking at the numbers, where a puzzler may have been looking from a wider perspective, and seen the letters, and recognized right away that the “pi” in “pizza” might stand for π

Maybe I’m giving away too much here, but you might think about what the double z could stand for. And don’t forget the a.

I won’t be giving away prizes for anyone who solves this (I offered one in the last Chronicles, but nobody won). But if you have an complete answer, send it in before the end of the weekend and I’ll give you an “honorable mention” in the upcoming issue of  The Math Mojo Monthly newsletter. (Wow, it just doesn’t get any better than that!)