∫ A littler math puzzle

I was still working on coming up with a puzzle for my friend’s neighbor and I came up with this little guy, which I think is easier than the previous one, and also easier to explain. I haven’t decided if I’m going to use it yet, but either way, it’s a nice little puzzle so I wanted to share it with someone. Lucky you!

So here’s the puzzle: Jill walks into her math classroom at the end of the school day to find Jack sitting sullenly at one of the desks. “What’s the matter?” she asks.

“I’m gonna fail my math class,” says Jack. “The teacher told me I could do this extra credit problem for him to get my grade up to a C-, but now I’ll never be able to solve it and it’s not my fault.”

“What do you mean?” asks Jill, sympathetically.

“I mean, he told me to start with 1 and then start adding the odd numbers together in order on my calculator, you know, like 1+3+5+7 = 16, and so on. And then he said ‘ If you tell me what the first ten-digit number you can make just by adding together odd numbers in order is I’ll round your grade up to a 70.0%.'”

“So what’s the problem?” asks Jill, feeling annoyed that all she gets in this dialogue are questions.

“The problem is, I sat here for an hour adding together odd numbers on my calculator, but then I realized that my calculator can only display eight digits at a time anyway. So I’ll never be able to figure it out!”

“Don’t be silly!” said Jill, finally using a declarative. “I know what the first ten-digit number you can make by adding up consecutive odd numbers is.”

And the question, of course, is “What is it?” Answer below the fold.

The answer is “one billion.”

That’s right, one billion, as in 1,000,000,000. In particular, its what you get by adding up the first 10,000 odd numbers in order. So how did I know that?

It’s simple if you already know the trick, a little harder if you don’t. If you don’t, you might have to start adding up some strings of odd numbers to see if there’s a pattern. You’ll notice that there is:

1 + 3 = 4 = 22

1 + 3 + 5 = 9 = 32

1 + 3 + 5 + 7 = 16 = 42

1 + 3 + 5 + 7 + 9 = 25 = 52

In particular, if you add together the first n odd numbers, you get n2. So every time I add together a string of consecutive odd numbers (starting at 1) the answer will be a perfect square. And thus, I just have to figure out what the smallest nine-digit perfect square is. Well, that’s easy. The ten digit numbers start with the number 1,000,000,000, which is itself a perfect square. Namely, it’s 10,0002, so I also know that poor Jack would have had to add together 10,000 odd numbers to get the answer, even if his calculator could display ten-digit numbers. Cool!

Incidentally, I’ve got a nice “proof” of the pattern described above, namely that adding together the first n odd numbers gives you n2. Now, if I wanted to be formal about it, the simplest proof is probably just to write out the sum and use a well-known formula for the sum of a bunch of consecutive (and not necessarily odd) numbers. But that would look like this:

And that’s not cool. So here’s a slicker “proof.” Instead of using numbers, I’m going to use dots. So If I want to find out what, say, 1 + 3 + 5 + 7 is, I just draw one dot, then three dots, then five dots, then seven dots, and count the total number of dots:

In addition to looking like the service bars on my cell phone, using dots this way gives me a nice way to see each of the numbers separately. However, it doesn’t make it particularly easy to count them. So to make it easier, I’m going to “bend” the purple row down:

That’s easy enough, right? And as you can see,  it’s going to make a nice little 2 x 2 square. Since it’s working so well, I might as well keep going, and bend the next two down:

This gives an end result that looks like this:

And what do you know? It’s a 4 x 4 square, made up of the first four odd numbers. And as usual, I guess all I’ve done is “prove” this for the first four odd numbers. But you can imagine continuing the picture in your head, and you’ll see that it will work for any set of consecutive odd numbers that starts with one, even if there are 10,000 of them.

By the way, images like the last one are sometimes called “wordless proofs,” and they’re one of my favorite mathematical things.

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About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

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