But That's so Huge?

If you count the number of squares in a $1 \times 1$ square grid it is equal to $1$ , which is equal to $1^2$ .

If you count the number of squares in a $2 \times 2$ square grid it is equal to $5$ , which is equal to $1^2 + 2^2$ .

If you count the number of squares in a $3 \times 3$ square grid it is equal to $15$ , which is equal to $1^2 + 2^2 + 3^2$ .

If we follow this sequence we see that the number of squares in a $n \times n$ grid of squares is equal to:

$1^2 + 2^2 + 3^2 + 4^2 +\ldots + n^2$ = $\displaystyle \sum_{i=1}^n i^2$ = $\huge\frac {n(n+1)(2n+1)}{6}$ .

• Therefore the number of squares in a $1024 \times 1024$ grid is:

$1^2 + 2^2 + 3^2 + 4^2 +\ldots + 1024^2$ = $\displaystyle \sum_{i=1}^{1024} i^2$ = $\huge\frac {1024(1024+1)(2(1024)+1))}{6}$ .

= $\huge\frac{1024 \times 1025 \times 2049}{6}$

= $\LARGE\boxed{\color{#20A900}{358438400}}$ .

We must count the number of $1\times1$ , $2\times2$ , $\cdots$ , and $1024\times1024$ squares. That's a lot of squares, so we should try to come up with a generalization to count the number of squares in an $n\times n$ grid.

Start by counting the number of $1\times1$ squares. There are $n$ columns of width $1$ and within each of these columns, there are $n$ different rows of width $1$ , so the number of $1\times1$ squares is $n\times n=n^{2}$ .

Then count the number of $2\times2$ squares. There are $n-1$ columns of width $2$ and within each of these, there are also $n-1$ rows of width $2$ . Each $2\times2$ square is made from one of the $n-1$ columns of width $2$ and one of the $n-1$ rows of width $2$ . So, there are $(n-1)\times (n-1)=(n-1)^{2}$ squares of width $2$ .

Continuing the pattern, you have that the number of $(n-1)\times(n-1)$ squares is $(n-(n-2))\times (n-(n-2))=2^{2}$ , and the number of $n\times n$ squares is $1^{2}$ .

Summing up the number of squares, you get: $n^{2}+(n-1)^{2}+\cdots+2^{2}+1^{2}= \sum_{k=1}^{n}k^{2}.$ In this case, $n=1024$ , so I computed $\sum_{k=1}^{1024}k^{2}$ with Wolframalpha and got $\boxed{358.438.400}$ .