Is there a formula to evaluate the number of all squares in the $m imes n$ grid? Well, I"m just curious, I"ve seen the question like this somewhere at the university, to solve this they were dividing the grid with $m - 1$ and $n - 1$ lines...I don"t know what"s next.

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Suppose $nge m$.

Number of squares of size 1: $mcdot n$Number of squares of size 2: $(m-1)cdot (n-1)$...Number of squares of size m: $1cdot (n-m+1)$

Result: $$eginalignsum_k=1^m k cdot (n-m+k) & =(n-m)sum_k=1^m k +sum_k=1^m k^2 \& = (n-m) m(m+1)/2 + m(m+1)(2m+1)/6 \& = fracm(m+1) (3n-m+1)6endalign$$


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Rectangles in rectangle$$frac(n^2+n)(m^2+m)4$$

Rectangles in square$$frac(n^2+n)^24$$

Squares in rectangle$$m≥ n-1,frac(n^2+n)2m-frac(n^3-n)6$$

Squares in square$$frac(n^2+n)(2n+1)6$$


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