Nonattacking knights problem

Choose $k$ distinct squares of an $n\times n$ chessboard so that no chosen pair has absolute coordinate differences $(1,2)$ or $(2,1)$. One constraint satisfaction problem representation has $k$ square-valued variables, requiring distinctness and pairwise nonattack. Another has one Boolean variable per square, a cardinality constraint fixing their sum to $k$, and one mutual-exclusion constraint for every edge of the knight attack graph. The first has fewer variables when $k$ is small but represents each unlabelled placement up to $k!$ times; the second uses only binary constraint domains and sparse local attack restrictions.

The knight attack graph has one vertex per chessboard square and an edge between squares linked by a knight move. A nonattacking placement is an independent set. Colouring squares by the parity of the coordinate sum makes this a bipartite graph, because every knight move changes that parity. Every vertex has degree at most eight, with fewer attacks near the boundary.