Student Seminar
Speaker: Aditya Dwivedi, IIT Bombay
Host: Suman Kumar Sahoo
Title: The Algebra of Magic Squares
Time, day and date: 3:30:00 PM – 4:30:00 PM, Friday, August 21
Venue: Ramanujan Hall
Abstract: Magic squares have been studied by thinkers for centuries, often appearing in classical texts across cultures, such as the Lo Shu square, Buduh square, or the bhadra chaturashra. The problem of counting magic squares was also pursued by many thinkers over centuries, often leading to calculations in specific cases (see Al-Ashabb et al., Mandadi et al., Mattingly et al., Dvivedi et al.).
But what happens when we move beyond these specific historical constructions and ask a generalised question: exactly how many n x n magic squares exist for a given line sum r?
This talk explores the surprising bridge between this ancient combinatorial problem and modern commutative algebra. After reviewing the elementary 1 x 1 and 2 x 2 matrices, we will introduce the celebrated Anand-Dumir-Gupta conjectures. Using the Birkhoff-von Neumann theorem, we will construct a specific subring generated by permutation matrices. By analysing the Hilbert series of this ring, we arrive at the elegant final result: the number of n x n magic squares is simply a polynomial in the line sum r.