99 research outputs found
Asymptotic enumeration of magic series
A magic series is a set of natural numbers that, by virtue of its size, sum,
and maximum value, could fill a row of a normal magic square. In this paper, we
derive an exact two-dimensional integral representation for the number of magic
series of order N. By applying the stationary phase approximation, we develop
an expansion in powers of 1/N for the number of magic series and calculate the
first few terms. We find excellent agreement between our approximation and the
known exact values. Related results are presented for magic cube and hypercube
series, bimagic series, and trimagic series
Polyhedral Cones of Magic Cubes and Squares
Using computational algebraic geometry techniques and Hilbert bases of
polyhedral cones we derive explicit formulas and generating functions for the
number of magic squares and magic cubes.Comment: 14 page
Mathematical Magic: A Study of Number Puzzles
Within this paper, we will briefly review the history of a collection of number puzzles which take the shape of squares, polygons, and polyhedra in both modular and nonmodular arithmetic. Among other results, we develop construction techniques for solutions of both Modulo and regular Magic Squares. For other polygons in nonmodular arithmetic, specifically of order 3, we present a proof of why there are only four Magic Triangles using linear algebra, disprove the existence of the Magic Tetrahedron in two ways, and utilizing the infamous 3-SUM combinatorics problem we disprove the existence of the Magic Octahedron
On orthogonal Latin -dimensional cubes
summary:We give a construction of orthogonal Latin -dimensional cubes (or Latin hypercubes) of order for every natural number and . Our result generalizes the well known result about orthogonal Latin squares published in 1960 by R. C. Bose, S. S. Shikhande and E. T. Parker
Magic graphs and the faces of the Birkhoff polytope
Magic labelings of graphs are studied in great detail by Stanley and Stewart.
In this article, we construct and enumerate magic labelings of graphs using
Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes.
We define polytopes of magic labelings of graphs and digraphs. We give a
description of the faces of the Birkhoff polytope as polytopes of magic
labelings of digraphs.Comment: 9 page
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