Magic Squares Debunking the Magic Radu Sorici The

Magic Squares { Debunking the Magic Radu Sorici The University of Texas at Dallas

Random Magic Square

• No practical use yet great influence upon people

• No practical use yet great influence upon people • In Mathematics we study the nature of numbers and magic squares are a perfect example to show their natural symmetry

History is Very Important 1. There is evidence to date magic squares as early as the 6 th century due to Chinese mathematicians

History is Very Important 1. There is evidence to date magic squares as early as the 6 th century due to Chinese mathematicians 2. It was later discovered by the Arabs in the 7 th century

History is Very Important 1. There is evidence to date magic squares as early as the 6 th century due to Chinese mathematicians 2. It was later discovered by the Arabs in the 7 th century 3. The “Lo Shu” square is the first recorded magic square =

History is Very Important 1. There is evidence to date magic squares as early as the 6 th century due to Chinese mathematicians 2. It was later discovered by the Arabs in the 7 th century 3. The “Lo Shu” square is the first recorded magic square = 4. The sum in each row, column, diagonal is 15 which is the number of days in each of the 24 cycles of the Chinese solar year

History is Very Important 1. There is evidence to date magic squares as early as the 6 th century due to Chinese mathematicians 2. It was later discovered by the Arabs in the 7 th century 3. The “Lo Shu” square is the first recorded magic square = 4. The sum in each row, column, diagonal is 15 which is the number of days in each of the 24 cycles of the Chinese solar year 5. Magic squares have cultural aspects to them as well, for example they were worn as talismans by people in Egypt and India. It went as far as being attributed mythical properties. (Thank you Wikipedia for great information)

So what exactly is a Magic Square?

Before we Start

Types of Magic Squares

Construction Methods Odd orders (De la Loubère)

Construction Methods Odd orders

Construction Methods

Construction Methods Singly Even • The Ralph Strachey Method

Construction Methods A D C B

Construction Methods

Construction Methods Singly Even • The Ralph Strachey Method • 3 rd Step - Exchange the middle cell of the leftmost column of subsquare A with the corresponding cell of subsquare D. Exchange the central cell in subsquare A with the corresponding cell of subsquare D

What Now?

Panmagic Square • A panmagic(also called diabolical) square is a magic square with the additional property that the broken diagonals also add up to the magic constant.

Panmagic Square

Panmagic Square Continued • 5 by 5 panmagic squares introduces even more magic

Panmagic Square Continued • 5 by 5 panmagic squares introduces even more magic – quincunx 17+25+13+1+9=65 21+7+13+19+5=65 4+10+13+16+22=65 20+2+13+24+6=65

Magic Cube • A magic cube is a magic square but in 3 -D. All of the properties are translated to 3 -D.

Magic Cube

Bimagic Square • A Bimagic Square is a magic square that is also a magic square if all of its numbers are squared

Bimagic Square • A Bimagic Square is a magic square that is also a magic square if all of its numbers are squared • The first known bimagic square is of order 8

Bimagic Square • A Bimagic Square is a magic square that is also a magic square if all of its numbers are squared • The first known bimagic square is of order 8 • It has been shown that all 3 by 3 bimagic squares are trivial

Bimagic Square

Multiplication Magic Square • A square which is magic under multiplication is called a multiplication magic square. The magic constants increase very fast with the order of the square.

Multiplication Magic Square • A square which is magic under multiplication is called a multiplication magic square. The magic constants increase very fast with the order of the square. • For orders 3 and 4 the following are the smallest multiplication magic squares

Word Square • A set of words having the same number of letters; when the words are written in a square grid horizontally, the same set of words can be read vertically

Word Square • A set of words having the same number of letters; when the words are written in a square grid horizontally, the same set of words can be read vertically • Because we speak English we are naturally interested in the ones made of English words

Word Square • A set of words having the same number of letters; when the words are written in a square grid horizontally, the same set of words can be read vertically • Because we speak English we are naturally interested in the ones made of English words • There are word squares of order 3 through 9 (cases 3, 4, 9 are displayed below) BI T I CE TEN CARD AREA REAR DART ACHALASIA CRENIDENS HEXANDRIC ANABOLITE L INOLENIN ADDLEHEAD SERINETTE I NITIATOR ASCENDERS The hunt for a word square of order 10 is still going and apparently it has been called the holy grail of logology.

Fibonacci Magic Square • The presentation would not be complete with a reference to the Fibonacci numbers

Fibonacci Magic Square • The presentation would not be complete with a reference to the Fibonacci numbers • Start with the basic 3 by 3 magic square

Fibonacci Magic Square • The presentation would not be complete with a reference to the Fibonacci numbers • Start with the basic 3 by 3 magic square • Replace each number with its corresponding Fibonacci number

Fibonacci Magic Square • The presentation would not be complete with a reference to the Fibonacci numbers • Start with the basic 3 by 3 magic square • Replace each number with its corresponding Fibonacci number • Even though this is not a magic square it so happens that the sum of the products of the three rows is equal to the sum of the products of the three columns.

Random Magic Square

Final Words Masonic Cipher

Final Words Masonic Cipher Durer Magic Square

Final Words The message is

Final Words The message is I Love Mathematics