Abstract

It is a 300-year-old counterintuitive observation of Prince Rupert of the Rhine that in the cube a straight tunnel can be cut through which a second congruent cube can be passed. A hundred years later P. Nieuwland strengthened Rupert’s problem and asked for the largest aspect ratio so that a larger homothetic copy of the same body can be passed through it. We show that cubes and, in fact all, rectangular boxes have Rupert’s passages in every direction that is not parallel to the faces. In the case of the cube it was assumed without proof that the solution of the Nieuwland’s problem is a tunnel perpendicular to the largest square contained by the cube. We prove that this assumption is unwarranted not only for the cube, but also for all other rectangular boxes.

摘要

莱茵河的鲁珀特王子有 300 年历史的违反直觉的观察，在立方体中可以切出一条笔直的隧道，第二个全等立方体可以通过该隧道。一百年后，尼乌兰（P. Nieuwland）加强了鲁珀特（Rupert）的问题，并要求最大的长宽比，以便可以通过同一物体的更大的同质复制品。我们展示了立方体和实际上所有的矩形盒子在每个不平行于面的方向上都有鲁珀特通道。在立方体的情况下，没有证据就假设 Nieuwland 问题的解决方案是一条垂直于立方体所包含的最大正方形的隧道。我们证明了这个假设不仅对立方体是没有根据的，而且对所有其他矩形框也是没有根据的。