Parks: A Doubly Infinite Family of NP-Complete Puzzles and Generalizations of A002464

The Parks Puzzle is a paper-and-pencil puzzle game that is classically played on a square grid with different colored regions (the parks). The player needs to place a certain number of "trees" in each row, column, and park such that none are adjacent, even diagonally. We define a doubly-infinite family of such puzzles, the \((c, r)\)-tree Parks puzzles, where there need be \(c\) trees per column and \(r\) per row. We then prove that for each \(c\) and \(r\) the set of \((c, r)\)-tree puzzles is NP-complete. For each \(c\) and \(r\), there is a sequence of possible board sizes \(m \times n\), and the number of possible puzzle solutions for these board sizes is a doubly-infinite generalization of OEIS sequence A002464, which itself describes the case \(c = r = 1\). This connects the Parks puzzle to chess-based puzzle problems, as the sequence describes the number of ways to place non-attacking kings on a chessboard so that there is exactly one in each column and row (i.e. to place non-attacking dragon kings in shogi). These findings add yet another puzzle to the set of chess puzzles and expands the list of known NP-complete problems described.