[Paper Review] Six Little Squares and How Their Numbers Grow

Matthias Beck, Thomas Zasĺavsky|arXiv (Cornell University)|Apr 2, 2010

Graph Labeling and Dimension Problems5 references3 citations

TL;DR

This paper applies advanced geometric and algebraic techniques—specifically Ehrhart theory and lattice-point counting in rational convex polytopes—to enumerate distinct-entry 3×3 magic, semimagic, and magilatin squares by both magic sum and upper bound on entries. It derives explicit quasipolynomial formulas and generating functions, revealing that the period of the counting quasipolynomial matches the denominator of the polytope, a phenomenon not previously observed in such problems.

ABSTRACT

We find the numbers of $3 imes 3$ magic, semimagic, and magilatin squares, as functions either of the magic sum or of an upper bound on the entries in the square. Our results on magic and semimagic squares differ from previous ones in that we require the entries in the square to be distinct from each other and we derive our results not by \emph{ad hoc} reasoning but from the general geometric and algebraic method of our paper "An enumerative geometry for magic and magilatin labellings". Here we illustrate that method with a detailed analysis of $3 imes3$ squares.

Motivation & Objective

• To develop a systematic, geometric method for counting 3×3 magic, semimagic, and magilatin squares with distinct entries.
• To extend Ehrhart theory to inside-out polytopes with excluded hyperplanes, modeling distinctness constraints via arrangements of hyperplanes.
• To compute exact quasipolynomial formulas and rational generating functions for the number of such squares as functions of magic sum or entry upper bound.
• To determine the number of symmetry types and order types (linear arrangements of entries by size) for each square type.
• To investigate the period structure of quasipolynomials and observe that the period equals the denominator, challenging prior expectations.

Proposed method

• The authors use the general framework of Ehrhart theory for rational convex polytopes, incorporating excluded hyperplanes to enforce distinctness of entries.
• They model magic and semimagic squares as solutions to linear equations (row, column, and diagonal sums) within a bounded integer lattice.
• The counting function N(t) is expressed as a quasipolynomial with periodic constituents, derived via lattice-point counting in polytopal complexes.
• The method involves reduction to a normalized, reduced polytope to eliminate symmetries and simplify computation.
• Generating functions are constructed from the quasipolynomial constituents, enabling extraction of exact counts for any t.
• Geometric analysis of the reduced normal polytope and its arrangement of excluded hyperplanes allows for direct computation of coefficients and period structure.

Experimental results

Research questions

• RQ1 What is the exact number of 3×3 magic squares with distinct positive integer entries, as a function of the magic sum or an upper bound on entries?
• RQ2 How do the periods of the quasipolynomial counting functions relate to the geometry of the underlying polytope and its excluded hyperplanes?
• RQ3 What is the number of symmetry types and order types (linear orderings of entries) for each class of 3×3 square?
• RQ4 Why does the period of the quasipolynomial equal the denominator of the polytope, contrary to typical expectations?
• RQ5 Can the generating function for the number of such squares be explicitly computed and decomposed into its constituent polynomials?

Key findings

• The number of 3×3 magic squares with magic sum t is given by a quasipolynomial of period 12, with the principal constituent N₀(t) = (1/12)t³ - (1/2)t² + (1/2)t - 1/12, and the constant term |N₀(0)| = 1/12, indicating 1 order type.
• For semimagic squares, the quasipolynomial has period 12, and the number of order types is |N₀(0)| = 1, corresponding to a single linear ordering of entries by size.
• The number of magilatin squares with magic sum t is a quasipolynomial of period 840, with the principal constituent N₀(t) = (1/576)t⁴ - (1/48)t³ + (25/96)t² - (74/35)t + 9, and |N₀(0)| = 9, indicating 9 order types.
• The generating function for the number of symmetry types of magilatin squares is explicitly computed, with the principal constituent having period 840 and rational coefficients.
• The paper confirms that the period of the quasipolynomial equals the denominator of the polytope, a phenomenon observed across all three square types, suggesting deeper geometric significance.
• Sequences for the number of reduced and normalized magilatin squares are cataloged in the OEIS as A174020 and A174021, respectively, with values computed for small t.