论文标题
六边形滑动难题的平价特性
Parity Property of Hexagonal Sliding Puzzles
论文作者
论文摘要
我们研究各种形状和各种孔的六角形滑动难题的拼图图。拼图图是一个组合模型，可捕获连续机械拼图的可溶性和复杂性。先前已经研究并解决了与拼图图有关的问题，并解决了有史以来最著名，最不可分割的方形滑动拼图的15个拼图。众所周知，对于诸如15个拼图之类的方形难题，可溶解性取决于将拼图图拆分为两个组件的平等性能。就六角形滑动难题而言，我们获得了更多有趣的奇偶校验特性，这些特性取决于木板的形状以及板上缺失的瓷砖或孔。我们表明，对于带有六角形瓷砖的大型六角形，三角形或平行四边形形状的木板，所有带有三个或更多孔的难题均可解决。对于带有两个或更多孔的难题，我们提供了一个涉及平价特性和瓷砖在板紧密拐角处的含量的可溶性标准。拼图图是用于在不同的基于底塞尔的域上移动的硬图块（六角形或正方形）配置空间的离散模型。了解拼图图的组合可能会导致理解这些配置空间拓扑的某些方面。
We study the puzzle graphs of hexagonal sliding puzzles of various shapes and with various numbers of holes. The puzzle graph is a combinatorial model which captures the solvability and the complexity of sequential mechanical puzzles. Questions relating to the puzzle graph have been previously studied and resolved for the 15 Puzzle which is the most famous, and unsolvable, square sliding puzzle of all time. It is known that for square puzzles such as the 15 Puzzle, solvability depends on a parity property that splits the puzzle graph into two components. In the case of hexagonal sliding puzzles, we get more interesting parity properties that depend on the shape of the boards and on the missing tiles or holes on the board. We show that for large-enough hexagonal, triangular, or parallelogram-shaped boards with hexagonal tiles, all puzzles with three or more holes are solvable. For puzzles with two or more holes, we give a solvability criterion involving both a parity property and the placement of tiles in tight corners of the board. The puzzle graph is a discrete model for the configuration space of hard tiles (hexagons or squares) moving on different tessellation-based domains. Understanding the combinatorics of the puzzle graph could lead to understanding some aspects of the topology of these configuration spaces.
