OlympiadBench / 2012 / Let Z_(≥0) be the set of non-negative integers, and let …

Let Z_(≥0) be the set of non-negative integers, and let f:Z_(≥0)×Z_(≥0)→Z_(≥0) be a bijection such that whenever f(x₁,y₁)>f(x₂,y₂), we have f(x₁+1,y₁)>f(x₂+1,y₂) and f(x₁,y₁+1)>f(x₂,y₂+1).

Let N be the number of pairs of integers (x,y), with 0≤x,y<100, such that f(x,y) is odd. Find the smallest and largest possible value of N.

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 $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right)$ and $f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}, y_{2}+1\right)$.

Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$.