Prince Rupert's problem asks whether a given bounded convex polyhedron has Rupert's property, that is, whether a congruent copy can pass with positive clearance, without turning, through a straight tunnel cut in it (Steininger and Yurkevich 2023). The original cube problem has an affirmative answer, and the largest possible passing cube is Prince Rupert's cube.
Steininger and Yurkevich (2023) proved that the problem is algorithmically decidable for convex polyhedra whose coordinates are algebraic numbers. A positive answer requires only one rigorously verified passage, whereas proving that a convex polyhedron is a nopert polyhedron requires excluding every possible configuration.
The conjecture that every bounded convex polyhedron has Rupert's property was disproved by the noperthedron of Steininger and Yurkevich (2025). Hervay (2026) gives a computer-assisted proof that the small rhombicosidodecahedron is another nopert polyhedron, settling the conjecture of Steininger and Yurkevich (2023).
