How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that \(\lfloor {\frac{n}{2}} \rfloor \) rooks or \(\lfloor {\frac{n}{3}} \rfloor \) queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

保护一个给定的多米诺骨牌的所有正方形需要多少个象棋大王或皇后呢？这个问题是美术馆问题的一个版本，在该问题中，警卫人员可以“看到”新手或女王/王后攻击的平方。我们证明\（\ lfloor {\ frac {n} {2}} \ rfloor \）骗子或\（\ lfloor {\ frac {n} {3}} \ rfloor \）女王/王后足以保护一个与n瓷砖的多米诺骨牌。然后，我们证明，找到保护一个多米诺骨牌所需的最少白嘴鸦或皇后区很困难。这些结果也适用于d维乌鸦和皇后区的d维多维立方体。最后，我们使用二分匹配定理来描述多米诺骨牌上无攻击菜鸟的集合。