Question 24
A move consists of replacing a given pair $(a,b)$ of integers by one of the pairs: $(a+1,b-1)$, $(a+1,b+1)$, $(a-1,b-1)$, $(a-1,b+1)$. Initially, we start with the pair $(100,100)$. Let $S$ be the number of pairs $(x,y)$, where $1\le x\le100$ and $1\le y\le100$, that can be obtained after a finite number of moves. What is the remainder when $S$ is divided by 2023?