Prince Rupert's cube

In geometry, Prince Rupert's cube is the largest cube that can pass through a hole cut through a unit cube without splitting it into separate pieces. Its side length is approximately 1.06, 6% larger than the side length 1 of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution.

Prince Rupert's cube is named after Prince Rupert of the Rhine, who asked whether a cube could be passed through a hole made in another cube of the same size without splitting the cube into two pieces. A positive answer was given by John Wallis. Approximately 100 years later, Pieter Nieuwland found the largest possible cube that can pass through a hole in a unit cube.

Many other convex polyhedra, including all five Platonic solids, have been shown to have the Rupert property: a copy of the polyhedron, of the same or larger size, can be passed through a hole in the polyhedron. It was unknown whether this was true for all convex polyhedra until late 2025, when it was disproven by researchers Jakob Steininger and Sergey Yurkevich after developing the "noperthedron" shape.