Question 194
Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)\gt f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)\gt f\left(x_{2}+1, y_{2}\right)$ and $f\left(x_{1}, y_{1}+1\right)\gt f\left(x_{2}, y_{2}+1\right)$.
Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y\lt 100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$.
Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y\lt 100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$.