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? A magic square is an arrangement of numbers from 1 to n 2 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(n 2 +1) ⃪ called the magic constant 2

Another way to find the sum Number the square from 1 to n 2 then add the numbers going down either of the main diagonals. This is equal to the sum.

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.

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.

Numbering the Magic Square

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.

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 = = Works for any 3 x 3 magic square.

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.

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

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

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