Magic Squares

1. Magic Squares By Miles Sherman & Dan Kelley

2. 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

3. 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 square.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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

9. Thank You!

10. 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, 2006. • Zimmerman, George. “The Subirachs Magic Square.” (2004).