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The Frobenius number is the largest value b for which the Frobenius equation a_1x_1+a_2x_2+...+a_nx_n=b, (1) has no solution, where the a_i are positive integers, b is an ...

The grid shading problem is the problem of proving the unimodality of the sequence {a_1,a_2,...,a_(mn)}, where for fixed m and n, a_i is the number of partitions of i with at ...

The mean triangle area of a triangle picked inside a regular hexagon with unit area is A^_=289/3888 (Woolhouse 1867, Pfiefer 1989). This is a special case of a general ...

In finding the average area A^__R of a triangle chosen from a closed, bounded, convex region R of the plane, then A^__(T(R))=A^__R, for T any nonsingular affine ...

The signature of a non-degenerate quadratic form Q=y_1^2+y_2^2+...+y_p^2-y_(p+1)^2-y_(p+2)^2-...-y_r^2 of rank r is most often defined to be the ordered pair (p,q)=(p,r-p) of ...

Pick three points P=(x_1,y_1), Q=(x_2,y_2), and R=(x_3,y_3) distributed independently and uniformly in a unit disk K (i.e., in the interior of the unit circle). Then the ...

Given an arrangement of points, a line containing just two of them is called an ordinary line. Dirac (1951) conjectured that every sufficiently set of n noncollinear points ...

The problem of finding the mean triangle area of a triangle with vertices picked inside a triangle with unit area was proposed by Watson (1865) and solved by Sylvester. It ...

Erdős proved that there exist at least one prime of the form 4k+1 and at least one prime of the form 4k+3 between n and 2n for all n>6.

A transformation of the form g=D^(T)etaD, where det(D)!=0 and det(D) is the determinant. Isometries are also called congruence transformations.