# Number of squares (Tomato subjective 59)

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
= $${\sum_{k=1}^{n}}{{k}^{2}}$$ = $${\frac{n(n+1)(2n+1)}{6}}$$