# Magic Squares - PowerPoint PPT Presentation

PPT – Magic Squares PowerPoint presentation | free to download - id: 5b8d9c-YTk3M

The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Magic Squares

Description:

### 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

Number of Views:476
Avg rating:3.0/5.0
Slides: 11
Provided by: MilesS6
Category:
Tags:
Transcript and Presenter's Notes

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).