Square up to this

Consider a $n$ by $m$ board where $m= n+r$ .

The number of $1$ by $1$ squares is $n*m$ .

The number of $2$ by $2$ squares is $(n-1)*(m-1)$

The number of $3$ by $3$ squares is $(n-2)*(m-2)$ and so on.

Thus the number of squares is the summation of $n*(n+r)$ that is summation of $n^2 +nr$ where n has value from $1$ to $n$ .

Summation of $n^2= \dfrac{n(n+1)(2n+1)}{6}$

Summation of $rn = \dfrac{rn(n+1)}{2}$

The equation simplifies to $\dfrac{n(n+1)(2n+1+3r)}{6}$

Substitute $n=100$ and $r=101$ , The number of squares is $848400$

The number of squares of side length 1 unit in a $n$ by $m$ board will be : $n*m$

The number of squares of side length 2 unit will be : $(n-1)*(m-1)$

The number of squares of side length 3 unit will be : $(n-2)*(m-2)$

So total number of square will be : $\sum_{i=1}^n (n-i)*(m-i)$

(Here I assume $n$ to less than $m$ )