Summing Products

Some days your mind wanders into an interesting puzzle: not necessarily because it’s a difficult puzzle, but because it has familiar result. Then the puzzle becomes, how are the two things linked?

For example, I had cause to add up all of the numbers in the times tables - let’s say the grid from $1\times 1$ to $10\times 10$.

I did the natural thing for a mathematician: I split it into ten arithmetic sequences - the numbers from $1\times 1$ to $10\times 1$ sum to 55; the numbers from $1\times 2$ to $10\times 2$ sum to $55\times 2$, and so on.

Then I have $55\times (1+2+\cdots +10)={55}^2=3025$.

In general, the times table grid from $1\times 1$ to $n\times n$ gives a sum of ${\left(\frac{n(n+1)}{2}\right)}^2$, or $\frac{n^2(n+1{)}^2}{4}$.

But hang on a minute

That’s the sum of the cubes from 1 to $n$! Why on earth should that be? I don’t see any cubes!

After a lot of thought, I found them.

Let’s just look at the products ¹ that have a 10 in them - from $1\times 10$ to $10\times 10$ down to $10\times 1$, nineteen products in all.

Imagine each of them as a rectangle. We have a $10\times 10$ square. We also have $9\times 10$ and $10\times 1$, which we can stick together to make a square. Similarly, $8\times 10$ and $10\times 2$. The eighteen non-square products can be paired up to make nine ten-by-ten squares - added to the $10\times 10$, that gives us a $10\times 10\times 10$ cube.

And we’re left with the times tables up to $9\times 9$, on which we can play the same trick. Each ‘band’ of the times table grid can be folded together to make a cube!

I thought that was a lovely, unexpected connection. I’d be thrilled to hear if you had another way of explaining it - the comments are open!

Footnotes:

1. in our original 10 by 10 grid, although it works generally