Grid of overlapping squares

Question

I have a grid made up of overlapping $3\times 3$ squares like so:

The numbers on the grid indicate the number of overlapping squares. Given that we know the maximum number of overlapping squares ($9$ at the middle), and the size of the squares ($3\times 3$), is there a simple way to calculate the rest of the number of overlaps?

e.g. I know the maximum number of overlaps is $9$ at point $(2,2)$ and the square size is $3\times 3$ . So given point $(3,2)$ how can I calculate that there are $6$ overlaps at that point?

Answer 1

The maximum number of overlaps is always $k^{2}$ (where $k$ is the small square's side length) (assuming the larger square is always of side length $2k-1$).

The number of overlaps at any other point $(i,j)$ is:

$\min(i,2k-i) \cdot \min(j,2k-j)$

As a pointer towards a proof consider the first row and the first column of the larger square which must always be the sequence $(1,2,\ldots,k,\ldots,2,1)$. Then look at why the other rows and columns are the products of their intersection with the edge row and column.

Answer 2

If you are just considering $3\times 3$ squares then the number of overlapping squares at the $(i,j)$ is the number of $1\times 1$ squares (including itself) which are internal neighbours. i.e. neighbouring squares which are not on the edge.