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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 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 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.

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 Dürers 4 x 4 square. - Dürers 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 churchs 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!

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).