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Magic Squares

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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. – PowerPoint PPT presentation

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Title: 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 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.

7
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.

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).
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