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,…, n 2 exactly once. Sum (weight) of columns, rows, and diagonals in classical magic square: wt(M) = [n(n 2 + 1)]/2

Properties of magic squares There only exists one 3 x 3 classical magic square x 4 classical magic squares. 275,305,224 5 x 5 classical magic squares. The sum of two magic squares is a magic square The scalar multiple of a magic square is a magic square.

Vector spaces of magic squares The 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 1 The dimension of the vector space of an n x n zero matrix is denoted by n 2 − 2n − 1.

Pandiagonal magic squares Magic 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 squares The 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 and 16 are absent.

Magic squares and sudoku The now popular number game of sudoku has its origins in magic squares. Given an n x n matrix with certain elements filled in Composed 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

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Bibliography Lee,Michael, Elizabeth Love, and Elizabeth Wascher. "Linear Algebra of Magic Squares." (2006). Poole,David. Linear Algebra: A Modern Introduction. 2 ed. Thompson Brooks/Cole, Zimmerman, George. “The Subirachs Magic Square.” (2004).