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A cardinality constraint restricts how many members of a specified collection are selected or satisfy a property. For Boolean variables, exactly $k$ selected entries gives the displayed sum; upper and lower bounds give at-most and at-least variants. Constraint propagation can maintain the number already selected and the number still available. A requirement involving many variables is a global constraint, even when it has a compact arithmetic description.

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.

Let $Q=\{1,\dots ,n{\}}^2$ be the board squares and let $\mathrm{Attack}(p,q)$ mean that the absolute coordinate differences are $(1,2)$ or $(2,1)$.

One variable per knight. Use $X_1,\dots ,X_k$, with each constraint domain equal to $Q$. For every pair $i<j$, requireEvery satisfying assignment places $k$ distinct nonattacking knights, and any such placement gives an assignment after labelling its knights. This has only $k$ variables and $O(k^2)$ binary constraints, but domains of size $n^2$. Relabelling the knights gives $k!$ assignments for the same unlabelled board, so the representation contains considerable redundancy.

One Boolean variable per square. Use $Y_q\in \{0,1\}$ for every $q\in Q$. Impose the cardinality constraintand for every knight-attack pair $p,q$ impose $Y_p+Y_q\le 1$. There are $n^2$ variables, each with a two-element constraint domain, and only $O(n^2)$ local attack restrictions, because each square attacks at most eight others. The single counting constraint is a global constraint, not a binary constraint.

The square encoding avoids knight-label redundancy and has sparse local attack relations; the knight encoding uses fewer variables when $k$ is small and makes single-knight moves natural. Both are exact representations of the nonattacking knights problem. Neither may impose one knight per row: several knights can legally share a row.