An Enumerative Geometry for Magic and Magilatin Labellings

Beck, Matthias; Zaslavsky, Thomas

doi:10.1007/s00026-006-0296-4

Cited by 26 publications

(12 citation statements)

References 11 publications

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“…number of n × n matrices whose entries are distinct positive integers less than t, whose sums along any row, column, or main diagonal are equal. The previous record consisted of the 3 × 3 counting functions [6, 20] (see also [5] for the computational implementation of [6])…”

Section: It's a Kind Of Magicmentioning

confidence: 99%

Enumeration of $4 \times 4$ magic squares

Beck

Herick

2010

Math. Comp.

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A magic square is an n × n array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for n = 4, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the n = 3 case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane arrangements and Ehrhart's theory of lattice-point enumeration.

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Section: It's a Kind Of Magicmentioning

confidence: 99%

Enumeration of $4 \times 4$ magic squares

Beck

Herick

2010

Math. Comp.

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“…The first nontrivial formulas addressing the counting problem, namely (6.4) and (6.13) for H 3 and M 3 , were established by Percy Macmahon (1854-1929) 5 [124] in 1915. Recently there has grown up a literature on exact formulas (see for example [80,168] for semimagic squares; for magic squares see [1,23]; for magic squares with distinct entries see [31,189]). …”

Section: The Enumeration Of Magic Squaresmentioning

confidence: 99%

Computing the Continuous Discretely

2007

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Section: The Enumeration Of Magic Squaresmentioning

confidence: 99%

“…Here each row and column has n different numbers, the same n numbers in every row/column (usually taken to be the first n positive integers). There are counting problems associated with latin squares, which can be attacked using Ehrhart theory [31] (see also [165, Sequence A002860]).…”

Section: The Enumeration Of Magic Squaresmentioning

confidence: 99%

Computing the continuous discretely: integer-point enumeration in polyhedra

Beck¹,

Robins²

2007

Choice Reviews Online

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We seek to bridge some critical gaps between various fields of mathematics by studying the interplay between the continuous volume and the discrete volume of polytopes. Examples of polytopes in three dimensions include crystals, boxes, tetrahedra, and any convex object whose faces are all flat. It is amusing to see how many problems in combinatorics, number theory, and many other mathematical areas can be recast in the language of polytopes that exist in some Euclidean space. Conversely, the versatile structure of polytopes gives us number-theoretic and combinatorial information that flows naturally from their geometry. The discrete volume of a body P can be described intuitively as the number of grid points that lie inside P, given a fixed grid in Euclidean space. The continuous volume of P has the usual intuitive meaning of volume that we attach to everyday objects we see in the real world. VIII PrefaceIndeed, the difference between the two realizations of volume can be thought of in physical terms as follows. On the one hand, the quantumlevel grid imposed by the molecular structure of reality gives us a discrete notion of space and hence discrete volume. On the other hand, the Newtonian notion of continuous space gives us the continuous volume. We see things continuously at the Newtonian level, but in practice we often compute things discretely at the quantum level. Mathematically, the grid we impose in space-corresponding to the grid formed by the atoms that make up an object-helps us compute the usual continuous volume in very surprising and charming ways, as we shall discover.In order to see the continuous/discrete interplay come to life among the three fields of combinatorics, number theory, and geometry, we begin our focus with the simple-to-state coin-exchange problem of Frobenius. The beauty of this concrete problem is that it is easy to grasp, it provides a useful computational tool, and yet it has most of the ingredients of the deeper theories that are developed here.In the first chapter, we give detailed formulas that arise naturally from the Frobenius coin-exchange problem in order to demonstrate the interconnections between the three fields mentioned above. The coin-exchange problem provides a scaffold for identifying the connections between these fields. In the ensuing chapters we shed this scaffolding and focus on the interconnections themselves:(1) Enumeration of integer points in polyhedra-combinatorics, (2) Dedekind sums and finite Fourier series-number theory, (3) Polygons and polytopes-geometry.We place a strong emphasis on computational techniques, and on computing volumes by counting integer points using various old and new ideas. Thus, the formulas we get should not only be pretty (which they are!) but should also allow us to efficiently compute volumes by using some nice functions. In the very rare instances of mathematical exposition when we have a formulation that is both "easy to write" and "quickly computable," we have found a mathematical nugget. We have endeavored to fill this boo...

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An Enumerative Geometry for Magic and Magilatin Labellings

Cited by 26 publications

References 11 publications

Enumeration of $4 \times 4$ magic squares

Enumeration of $4 \times 4$ magic squares

Computing the Continuous Discretely

Computing the continuous discretely: integer-point enumeration in polyhedra

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