Problem: Consider the set of point
S = { (x,y) : x,y are non-negative integers {\le {n}} }.
Find the number of squares that can be formed with vertices belonging to S and sides parallel to the axes.
Solution: S = {(x,y) : x,y are non-negative integers {\le {n}} }
We calculate number of squares by calculating number of |x| squares ,& number of squares number of {{n}* {n}} squares.
Now number of |x| squares = number of choosing one pair of lines with difference 1 parallel to x axis & integer distance x number of choosing one pair of lines to y axis with distance 1 & integer distance from y axis = {{n}*{n}} = {n^2}
Similarly number of {{k}*{k}} squares
= {(n-k+1)^2}
So total number of squares
= {\sum_{k=1}^{n}}{{k}^{2}} = {\frac{n(n+1)(2n+1)}{6}}