# 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 $1\times1$ to $10\times10$.

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 $55 \times (1 + 2 + \dots + 10) = 55^2 = 3025$.

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 ((in our original 10 by 10 grid, although it works generally)) 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!