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

TL;DR

It is proved that for each $c and $r$ the set of $(c, r)$-tree puzzles is NP-complete, which 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.

Abstract

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.

Figures (19)

• Figure 1: A 1-tree $6 \times 6$ Parks puzzle.
• Figure 2: A 2-tree $8 \times 8$ Parks puzzle.
• Figure 3: A (2, 1)-tree $8 \times 4$ Parks puzzle (left) and its solution (right).
• Figure 4: A (1, 2)-tree $4 \times 8$ Parks puzzle (top) and its solution (bottom).
• Figure 5: A variable park in a 1-tree puzzle

Theorems & Definitions (4)

• Definition 1
• Theorem 2
• Theorem 3
• proof