So, I'm at this boring lecture, and I make this great discovery, namely that 1 plus 8 equals 9. Oh, so you already knew that? But did you also know that 1 and 8 are perfect cubes and 9 is a perfect square?

1 = 1 x 1 x 1 = 1³
8 = 2 x 2 x 2 = 2³
9 = 3 x 3 = 3²

Now I'm wondering, if the sum of the first two perfect cubes equals a perfect square, what does 1³ + 2³ + 3³ equal? It turns out that it also equals a perfect square--

1³ + 2³ + 3³ = 6²

It begins to look like I'm onto a pattern. No doubt Euclid and his pals knew all about this stuff, but still it's new to me and that makes it a discovery any way you look at it. I fish a scrap of paper out of my pocket and start adding cubes of the numbers from 1 to 4, 5 and 6, and lo and behold, I get more perfect squares--

1³ + 2³ + 3³ + 4³ = 10²
1³ + 2³ + 3³ + 4³ + 5³ = 15²
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = 21²

All this time, the speaker is going on about how the kids in Israel did better than U.S. kids on an international math test, which is pretty important, I guess, but I'm not listening too closely because I'm figuring there's got to be a formula here.

Let's see, the sum of the cubes from 1 to 2 is 3 squared; the sum of the cubes from 1 to 3 is 6 squared. Take a look at my calculations below and then in the table.

(1+ 2) = 3
(1 + 2 + 3) = 6
(1 + 2 + 3 + 4) = 10
(1 + 2 + 3 + 4 + 5) = 15
(1 + 2 + 3 + 4 + 5 + 6) = 21

I'm thinking, "If only I had a spreadsheet I could check this out all the way." But I've left my computer in my hotel room. Still, I'm pretty sure it's going to work, now the question is, can I prove it?

So I try making a picture with imaginary tiles. In an obvious representation, I can show my first "discovery" that 1 plus 8 equals 9.

Shown this way, one interesting fact that I discover is that the tiles that make up the odd-shaped piece that represents 8 can be rearranged. Try moving the tiles around in your head. What happens to the odd-shaped piece? You've got it! The tiles can be rearranged into a perfect 2 x 2 x 2 cube.

The question is, does this relationship continue?If you go from this square in the sequence to the next, will the tiles form cubes?

Let's see how things work out with the 6 X 6 square.

Imagine moving the tiles around in your head again. If that's hard, use blocks or something else stackable. What shapes do you get? Voila! The same number of tiles in your 6 x 6 square can be rearranged into cubes, just as our table shows.

The next square in the series is the 10 X 10 and yet again the tiles break up nicely into cubes:

It works!

This kind of thing keeps happening as long as the numbers are integers. Try it out with the 15 x 15 and the 21 x 21 squares. You'll see that the pieces come apart neatly to rearrange into the required number of cubes.

Like magic!

The lecture comes to an end and the U.S. kids are still hopelessly behind, but I don't really notice--I'm too excited about cubes and squares.

Paul Horwitz⁵ is the Director of the Modeling Center at The Concord Consortium. He is also Senior Scientist for the GenScope⁶ and BioLogica™ projects.

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