By Miles Sherman & Dan Kelley

What is a magic square?An n x n matrix, M, with the sum of the

entries the same in each column, row, and diagonal.

Weight: sum of columns, rows, and diagonals in magic square.

A classical magic square contains each of the entries 1, 2,…, n2 exactly once.

Sum (weight) of columns, rows, and diagonals in classical magic square:

wt(M) = [n(n2 + 1)]/2

Properties of magic squaresThere only exists one 3 x 3 classical magic

square.880 4 x 4 classical magic squares.275,305,224 5 x 5 classical magic squares.The sum of two magic squares is a magic

squareThe scalar multiple of a magic square is a

magic square.

Vector spaces of magic squaresThe dimension of the vector space of an n x n

magic square is: [(n−1)2/ 2] +1

If wt(M) = 0, M is a zero magic square.For each magic square, A with wt(A)=u, there

exists an associated zero magic square, M: M = A – (u/n)E, where E is n x n matrix with all entries

equal to 1The dimension of the vector space of an n x n

zero matrix is denoted by n2 − 2n − 1.

Pandiagonal magic squaresMagic squares where broken diagonals add

up to the weight of the magic square are called pandiagonal.

The set of n x n classical magic squares and the set of n x n pandiagonal magic squares are a subspace.

Proof.

Famous magic squaresThe first magic square seen in European art

was Albrecht Dürer’s 4 x 4 square.Dürer’s magic square is found in his

engraving entitled Melencolia I.It has a weight of 34.Gnomon magic square: sum of all entries for each 2 x 2 matrix within the square is 34.

Famous magic squares (cntd)The Sagrada family church’s magic square

was designed by Josep Subirachs.The weight of the square is 33, the age of

Jesus at the time of his crucifixion.This is not a classical magic square

as the numbers 10 and 14 are repeated and the numbers 12 and16 are absent.

Magic squares and sudokuThe now popular number game of sudoku has

its origins in magic squares.Given an n x n matrix with certain elements

filled inComposed of 9 3 x 3 matrices where each

matrix contains the integers 1 through 9 exactly once

The integers 1 through 9 can only appear once in each row and column

Thank You!

BibliographyLee,Michael, Elizabeth Love, and Elizabeth

Wascher. "Linear Algebra of Magic Squares." (2006).

Poole,David. Linear Algebra: A Modern Introduction. 2 ed. Thompson Brooks/Cole, 2006.

Zimmerman, George. “The Subirachs Magic Square.” (2004).