探索“魔方”宇宙
Exploring the "Rubik's Magic" universe
摘要：
通过对Rubik's Magic谜题使用两种不同的不变量，一种是度量型的，另一种是拓扑型的，我们可以极大地减少谜题的可构造构型的论域。然而，找到一组实际可构建的形状仍然是一项具有挑战性的任务，我们通过首先将目标形状减少到特定配置来解决：八维3D形状，所有瓦片平行于一个坐标平面;以及平面“面朝上”形状，所有瓦片（考虑无限小的宽度）位于一个公共平面中并且没有重叠的连续瓦片。仍然有很多有趣的配置不属于这两个集合中的任何一个。可构造的配置集（那些可以通过从初始情况操纵未修饰的难题获得的配置集）是具有消失不变量的配置集的子集。我们实际上能够构建所有八角形的形状与消失的不变量和大多数的平面“面朝上”的配置。特别重要的是拓扑不变量，其中我们最近发现提到的一份文件由汤姆Verhoeff。
英文摘要：
By using two different invariants for the Rubik's Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing the target shapes to specific configurations: the octominoid 3D shapes, with all tiles parallel to one coordinate plane; and the planar "face-up" shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles. There are still plenty of interesting configurations that do not belong to either of these two collections. The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants. We were able to actually construct all octominoid shapes with vanishing invariants and most of the planar "face-up" configurations. Particularly important is the topological invariant, of which we recently found mention in a paper by Tom Verhoeff.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
