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MAGIC SQUARES by Megan Duke

MAGIC SQUARES by Megan Duke. Mathematics Discrete Combinatorics Latin Squares. Review . Matrix – a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Rows – horizontal component Columns – vertical component. What is a Magic Square?.

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MAGIC SQUARES by Megan Duke

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  1. MAGIC SQUARESby Megan Duke Mathematics Discrete Combinatorics Latin Squares

  2. Review • Matrix – a rectangular array of numbers, symbols, or expressions arranged in rows and columns. • Rows – horizontal component • Columns – vertical component

  3. What is a Magic Square? • A magic square is an arrangement of numbers from 1 to n2 in an n x n matrix with each number occurring exactly once, and such that the sum of the entries of any row, column, or main diagonal is the same. • The sum adds up to n(n2+1) ⃪ called the magic constant2

  4. Another way to find the sum • Number the square from 1 to n2 then add the numbers going down either of the main diagonals. This is equal to the sum.

  5. Examples • The simplest magic square is the 1 x 1 whose only entry is 1. • The next is the 3 x 3 magic square. • According to the formula the rows, columns, and diagonals must add to 15. • There are 8 3x3 magic squares.

  6. How to complete the Magic Square • Method 1: • Begin with 1 in the top row, center column. Travel north east to place the next number wrapping around the square when you reach the end or top of a row or column respectively. • If you reach a square with a number already in it, place the number directly below the original number then continue in the north east pattern • Used most often when n is odd.

  7. Numbering the Magic Square

  8. How to complete the Magic Square • Method 2: • Use Linear Algebra • Set each column and row equal together • Will generate 27 equations • Row reduce the matrix to get your entries.

  9. Other “magical” properties • If you look at the rows as a 3 digit number from left to right, square each one, and add the three together you will get the same sum as if you look at the rows as a 3 digit number from right to left, square each one, and add the three together. • 8162 + 3572 + 4922 = 6182 + 7532 + 2942 • 1035369=1035369 • Works for any 3 x 3 magic square.

  10. History of Magic Squares • Dates back to 2200 B.C. in China • Arab astrologers used them to calculate horoscopes • German artist Albrecht Durer included a magic square in which he embedded the date (1415) in the form of two consecutive numbers in the bottom row.

  11. Other Magic Shapes! • There are many other shapes that are considered magic! • Dodecahedrons • Triangles • Stars • Hexagons • Cubes • Plus more!!

  12. Big Picture Problem • How many n x n magic squares exist when n>5? • It was shown in 1973 by R. Schroeppelthat when n=5, there are 275,305,224magic squares.

  13. References • http://en.wikipedia.org/wiki/Matrix_(mathematics) • http://mathforum.org/alejandre/magic.square/adler/adler.whatsquare.html • http://www.jcu.edu/math/vignettes/magicsquares.htm • http://askville.amazon.com/find-magic-squares-found-cells-side-numbers-1-25/AnswerViewer.do?requestId=2689137

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