Ada Pawns

In order to beat AlphaZero, Chef Ada is improving her chess calculation skills. Today, Ada took a big chessboard with $N$ rows (numbered $1$ through $N$) and $N$ columns (numbered $1$ through $N$) and placed some pawns on this chessboard. Let's denote a cell in row $r$ and column $c$ of the chessboard by $(r, c)$. Ada may perform any number of moves in any order. In one move, Ada must take o