Magic Squares

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Magic Squares

• By Miles Sherman Dan Kelley

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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

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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.

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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.

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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.

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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.

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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.

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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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• Thank You!

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