Answers

1. The simplest nonconvex polyhedron with unit-square faces is the 30-face solid formed by attaching a unit cube to each face of a unit cube. Mrs. Pedersen found a way to braid this solid with three strips, each crossing once diagonally over every face of the solid. An infinite family of nonconvex polyhedrons with congruent square faces is obtained by joining any number of these crosses to form a chain.

2. A regular tetrahedron can be colored with four colors only in two ways, each a mirror reflection of the other. The simple formula that applies to all five Platonic solids is to divide the factorial of the number of faces by twice the number of edges. For example, the cube can be colored with six colors in 6!/24 = 30 ways, the octahedron with eight colors in 8!/24 = 168 ways, and so on.

3. A cube can be colored with three colors, each color going on two faces, in six ways: One with all pairs of opposite faces alike, two ways that are mirror images with all like colors on adjacent pairs of faces, and three ways with just one pair of opposite faces alike. Only the first three ways can be plaited with three five-square straight strips in the manner explained.

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