 N-queens problem

Place $N$ queens on an $N\times N$ chessboard so that none share a row, column or diagonal. One constraint satisfaction problem encoding uses a column variable $C_r\in \{1,\dots ,N\}$ for each row, with $C_r\ne C_s$ and $|C_r-C_s|\ne |r-s|$ whenever $r\ne s$. The pigeonhole principle explains why one queen per row loses no solutions. Prescribed occupied cells give the queens completion problem.