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Let S be partitioned into r×s disjoint sets E_i and F_j where the general subset is denoted E_i intersection F_j. Then the marginal probability of E_i is ...

257 is a Fermat prime, and the 257-gon is therefore a constructible polygon using compass and straightedge, as proved by Gauss. An illustration of the 257-gon is not included ...

If replacing each number by its square in a magic square produces another magic square, the square is said to be a bimagic square. Bimagic squares are also called doubly ...

It is always possible to "fairly" divide a cake among n people using only vertical cuts. Furthermore, it is possible to cut and divide a cake such that each person believes ...

An official chessboard is an 8×8 board containing squares alternating in color between olive green and buff (where "buff" is a color variously defined as a moderate orange ...

Let P(G) denote the chromatic polynomial of a finite simple graph G. Then G is said to be chromatically unique if P(G)=P(H) implies that G and H are isomorphic graphs, in ...

Construct a square equal in area to a circle using only a straightedge and compass. This was one of the three geometric problems of antiquity, and was perhaps first attempted ...

A curve with polar coordinates, r=b+asectheta (1) studied by the Greek mathematician Nicomedes in about 200 BC, also known as the cochloid. It is the locus of points a fixed ...

A convex polyhedron can be defined algebraically as the set of solutions to a system of linear inequalities mx<=b, where m is a real s×3 matrix and b is a real s-vector. ...

Let s_1, s_2, ... be an infinite series of real numbers lying between 0 and 1. Then corresponding to any arbitrarily large K, there exists a positive integer n and two ...