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The average number of regions N(n) into which n lines divide a square is N^_(n)=1/(16)n(n-1)pi+n+1 (Santaló 1976; Finch 2003, p. 481). The maximum number of sequences is ...

The problem of finding in how many ways E_n a plane convex polygon of n sides can be divided into triangles by diagonals. Euler first proposed it to Christian Goldbach in ...

Consider the plane figure obtained by drawing each diagonal in a regular polygon. If each point of intersection is associated with a node and diagonals are split ar each ...

In general, a remainder is a quantity "left over" after performing a particular algorithm. The term is most commonly used to refer to the number left over when two integers ...

The recurrence relation E_n=E_2E_(n-1)+E_3E_(n-2)+...+E_(n-1)E_2 which gives the solution to Euler's polygon division problem.

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 ...

The maximum number of regions that can be created by n cuts using space division by planes, cube division by planes, cylinder cutting, etc., is given by N_(max)=1/6(n^3+5n+6) ...

The maximum number of pieces into which a cylinder can be divided by n oblique cuts is given by f(n) = (n+1; 3)+n+1 (1) = 1/6(n+1)(n^2-n+6) (2) = 1/6(n^3+5n+6), (3) where (a; ...

Ruffini's rule a shortcut method for dividing a polynomial by a linear factor of the form x-a which can be used in place of the standard long division algorithm. This method ...

With n cuts of a torus of genus 1, the maximum number of pieces which can be obtained is N(n)=1/6(n^3+3n^2+8n). The first few terms are 2, 6, 13, 24, 40, 62, 91, 128, 174, ...