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The golden ratio phi can be written in terms of a nested radical in the beautiful form phi=sqrt(1+sqrt(1+sqrt(1+sqrt(1+...)))), (1) which can be written recursively as ...

In a normal n×n Latin square, the entries in each row and column are chosen from a "global" set of n objects. Like a Latin square, a partial Latin square has no two rows or ...

According to G. Pólya, the method of finding geometric objects by intersection. 1. For example, the centers of all circles tangent to a straight line s at a given point P lie ...

A perfect partition is a partition of a number n whose elements uniquely generate any number 1, 2, ..., n. {1,1,...,1_()_(n)} is always a perfect partition of n, and every ...

Consider n intersecting circles. The maximal number of regions into which these divide the plane are N(n)=n^2-n+2, giving values for n=1, 2, ... of 2, 4, 8, 14, 22, 32, 44, ...

Bubbles can meet only at angles of 120 degrees (for three bubbles) and cos^(-1)(-1/3) approx 109 degrees28^'16^('') (for four bubbles), where cos^(-1)(-1/3) is the ...

For the rational curve of an unperturbed system with rotation number r/s under a map T (for which every point is a fixed point of T^s), only an even number of fixed points ...

In its original form, the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, ...

Let u and v be any functions of a set of variables (q_1,...,q_n,p_1,...,p_n). Then the expression ...

The integral kernel in the Poisson integral, given by K(psi)=1/(2pi)(1-|z_0|^2)/(|z_0-e^(ipsi)|^2) (1) for the open unit disk D(0,1). Writing z_0=re^(itheta) and taking ...