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Let L denote the n×n square lattice with wraparound. Call an orientation of L an assignment of a direction to each edge of L, and denote the number of orientations of L such ...

Let F(m,n) be the number of m×n (0,1)-matrices with no adjacent 1s (in either columns or rows). For n=1, 2, ..., F(n,n) is given by 2, 7, 63, 1234, ... (OEIS A006506). The ...

A theorem, also known as Bachet's conjecture, which Bachet inferred from a lack of a necessary condition being stated by Diophantus. It states that every positive integer can ...

The amazing polynomial identity communicated by Euler in a letter to Goldbach on April 12, 1749 (incorrectly given as April 15, 1705--before Euler was born--in Conway and Guy ...

The tetragonal trapezohedron is the dual of the square antiprism. For a square antiprism with unit edge lengths, the edge lengths of the corresponding tetragonal ...

An n-gonal pyramid adjoined to the top of an n-gonal antiprism. In the 3-gonal gyroelongated pyramid, the pyramid and lateral antiprism are coplanar. However, the 4-gonal and ...

A decahedron is a polyhedron having 10 faces. Examples include the 5-trapezohedron, augmented pentagonal prism (Johnson solid J_(52)), augmented tridiminished icosahedron ...

A n-gonal cupola adjoined to a 2n-gonal antiprism.

A polyhedron dissection (or decomposition) is a dissection of one or more polyhedra into other shapes. Two polyhedra can be dissected into each other iff they have equal Dehn ...

A (general) dodecahedron is a polyhedron having 12 faces. Examples include the Bilinski dodecahedron, decagonal prism, elongated square dipyramid (Johnson solid J_(15)), ...