We say that a polytope has the Rupert property if we can make a hole large enough in to permit another copy of to pass through. In this article, we show that among the 13 Archimedean solids, 8 have this property, namely, the cuboctahedron, the truncated octahedron, the truncated cube, the rhombicuboctahedron, the icosidodecahedron, the truncated cuboctahedron, the truncated icosahedron, and the truncated dodecahedron.

Summary It is well known that a hole can be cut in a cube large enough to permit a second cube of equal size to pass through, a result attributed to Prince Rupert of the Rhine by J. Wallis more than three centuries ago. C. Scriba showed nearly 50 years ago that the tetrahedron and the octahedron have this same property. Somewhat surprisingly, the remaining two platonic solids, the dodecahedron and the icosahedron, also have this property: each can be passed through a suitable tunnel in another of the same size and kind. We supply the details.

Gallai in 1966 raised the question about the existence of graphs with the property that every vertex is missed by some longest path. This property will be called Gallai’s property. In this paper we present some graphs embeddable into Archimedean tiling graphs, with both connectivity 1 and 2, satisfying Gallai’s property.