Say I have an outer square like this, which is 14m x 14m. Each inner square is separated by a space of 3m, and one square is 1m$^2$. I of course have the image up, but how would I mathematically figure out that 16 squares fit inside?
Context: I'm approximating the number of trees within a specific area (Imagine looking down on the trees from a top, 2d view. To do that I broke the area up into squares (as shown), and now have to find how many trees fit in a single square.
Let $x$ denote the length of the side a small square, $d$ the distance between small squares and $S$ the length of the side of the large square.
In order to get $n^2$ small squares inside the large square, the large square must have side length at least $S\ge nx+(n-1)d$.
In order for $S$ to be not large enough to accommodate $(n+1)^2$ small squares it must be the case that $S<(n+1)x+nd$.
$$\frac{S-x}{x+d}<n\le\frac{S+d}{x+d}=\frac{S-x}{x+d}+1$$ Applying this to your specific case, $x=1, d=3, S=14$ we get
$$\frac{13}{4}<n\le\frac{13}{4}+1$$
So $n=4$ and the number of small squares is $n^2=16$.