Magic Squares

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
 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 squares
 There 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 square
 The scalar multiple of a magic square is a magic
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 n2 − 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
Thank You!
 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, 2006.
 Zimmerman, George. “The Subirachs Magic Square.”
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