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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 Important1. There is evidence to date magic squares as early as the 6th century due to Chinese

mathematicians

History is Very Important1. There is evidence to date magic squares as early as the 6th century due to Chinese

mathematicians2. It was later discovered by the Arabs in the 7th century

History is Very Important1. There is evidence to date magic squares as early as the 6th century due to Chinese

mathematicians2. It was later discovered by the Arabs in the 7th century3. The “Lo Shu” square is the first recorded magic square

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History is Very Important1. There is evidence to date magic squares as early as the 6th century due to Chinese

mathematicians2. It was later discovered by the Arabs in the 7th century3. 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

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History is Very Important1. There is evidence to date magic squares as early as the 6th century due to Chinese

mathematicians2. It was later discovered by the Arabs in the 7th century3. 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)

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So what exactly is a Magic Square?• A magic square is an x table containing integers such that the

numbers in each row, column, or diagonal sums to the same number

So what exactly is a Magic Square?• A magic square is an x table containing integers such that the

numbers in each row, column, or diagonal sums to the same number

• The order of a magic square is the size of the square

So what exactly is a Magic Square?• A magic square is an x table containing integers such that the

numbers in each row, column, or diagonal sums to the same number

• The order of a magic square is the size of the square• The above definition is rather broad and we usually will be using

what is called a normal magic square

So what exactly is a Magic Square?• A magic square is an x table containing integers such that the

numbers in each row, column, or diagonal sums to the same number

• The order of a magic square is the size of the square• The above definition is rather broad and we usually will be using

what is called a normal magic square• A normal magic square is a magic square containing the numbers

1 through

So what exactly is a Magic Square?• A magic square is an x table containing integers such that the

numbers in each row, column, or diagonal sums to the same number

• The order of a magic square is the size of the square• The above definition is rather broad and we usually will be using

what is called a normal magic square• A normal magic square is a magic square containing the numbers

1 through • Normal magic squares exist for all except for

So what exactly is a Magic Square?• A magic square is an x table containing integers such that the

numbers in each row, column, or diagonal sums to the same number

• The order of a magic square is the size of the square• The above definition is rather broad and we usually will be using

what is called a normal magic square• A normal magic square is a magic square containing the numbers

1 through • Normal magic squares exist for all except for • For we simply get the trivial square containing 1

So what exactly is a Magic Square?• A magic square is an x table containing integers such that the

numbers in each row, column, or diagonal sums to the same number

• The order of a magic square is the size of the square• The above definition is rather broad and we usually will be using

what is called a normal magic square• A normal magic square is a magic square containing the numbers

1 through • Normal magic squares exist for all except for • For we simply get the trivial square containing 1• For we would have the following square

Which would imply that

But then this is not a normal magic square.

So what exactly is a Magic Square?• A magic square is an x table containing integers such that the

numbers in each row, column, or diagonal sums to the same number

• The order of a magic square is the size of the square• The above definition is rather broad and we usually will be using

what is called a normal magic square• A normal magic square is a magic square containing the numbers

1 through • Normal magic squares exist for all except for • For we simply get the trivial square containing 1• For we would have the following square

Which would imply that

But then this is not a normal magic square.• For we will prove that a normal magic square exists

Before we Start1. The sum of numbers in each row, column, and diagonal is called the

magic constant and is equal to

Before we Start1. The sum of numbers in each row, column, and diagonal is called the

magic constant and is equal to

This is true because the sum of all the numbers in the magic square is equal to

and because there are rows we can divide by to obtain the above result

Before we Start1. The sum of numbers in each row, column, and diagonal is called the

magic constant and is equal to

This is true because the sum of all the numbers in the magic square is equal to

and because there are rows we can divide by to obtain the above result2. For ,… the magic constants are

Before we Start1. The sum of numbers in each row, column, and diagonal is called the

magic constant and is equal to

This is true because the sum of all the numbers in the magic square is equal to

and because there are rows we can divide by to obtain the above result2. For ,… the magic constants are 3. For odd the middle number is equal to

Types of Magic Squares• Singly even - • Doubly even - • Odd - • Antimagic - the rows, columns, diagonals are consecutive integers

(mostly open problems)• Bimagic - if the numbers are squared we still have a magic square• Word - 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

• Cube - the equivalent of a two dimensional magic square but in three dimensions

• Panmagic - the broken diagonals also add up to the magic constant• Trimagic - if the numbers are either squares or cubed we still end up

with a magic square• Prime - all the numbers are prime• Product - the product instead of the sum is the same across all rows,

columns, diagonals• And many more

Construction MethodsOdd orders (De la Loubère)

Construction MethodsOdd orders (De la Loubère)

Construction MethodsOdd orders

Construction MethodsDoubly Even• 1st step is to write the numbers in consecutive order from the top left to the bottom

right and delete all the numbers that are not on the diagonals• 2nd step is to start writing the numbers the numbers that are not on the diagonals in

consecutive order starting from the bottom right to the top left in the available spots. For example for

Construction MethodsDoubly Even• 1st step is to write the numbers in consecutive order from the top left to the bottom

right and delete all the numbers that are not on the diagonals• 2nd step is to start writing the numbers the numbers that are not on the diagonals in

consecutive order starting from the bottom right to the top left in the available spots. For example for

Construction MethodsSingly Even• The Ralph Strachey Method

Construction MethodsSingly Even• The Ralph Strachey Method for orders of the form • 1st Step – Divide the square into four smaller subsquares ABCD

A C

D B

Construction MethodsSingly Even• The Ralph Strachey Method• 2nd Step – Exchange the leftmost columns in subsquare A with the corresponding

columns of subsquare D and exchange the rightmost columns in subsquare C with the corresponding columns of subsquare B

Construction MethodsSingly Even• The Ralph Strachey Method• 3rd 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• 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.• The smallest non-trivial panmagic squares are squares such as

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.• The smallest non-trivial panmagic squares are squares such as

• Any 2 by 2 square including the ones warping around edges, the corners of 3 by 3 squares, displacement by a (2,2) vector, all add up to the magic constant!!!

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.• The smallest non-trivial panmagic squares are squares such as

• Any 2 by 2 square including the ones warping around edges, the corners of 3 by 3 squares, displacement by a (2,2) vector, all add up to the magic constant!!!

• The above three panmagic squares are the only 3 that exist for the numbers 1 through 16.

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=6521+7+13+19+5=654+10+13+16+22=6520+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• A magic cube is a magic square but in 3-D. All of the properties are translated to 3-D.• The magic constant is . Why?

Magic Cube• A magic cube is a magic square but in 3-D. All of the properties are translated to 3-D.• The magic constant is . Why?• Because there are rows and the total sum is .

Magic Cube• A magic cube is a magic square but in 3-D. All of the properties are translated to 3-D.• The magic constant is . Why?• Because there are rows and the total sum is .

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• 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

• Proof: Consider the following magic square and note that because

. In addition, by the same reasoning we have that Thus

Hence In the same way we get that all other numbers are equal as well.

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)

B I T C A R D A C H A L A S I AI C E A R E A C R E N I D E N ST E N R E A R H E X A N D R I C

D A R T A N A B O L I T E L I N O L E N I N A D D L E H E A D S E R I N E T T E I N I T I A T O R A S C E N D E R S

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 WordsMasonic Cipher

Final WordsMasonic Cipher

Durer Magic Square

Final WordsThe message is

Final WordsThe message is

I Love Mathematics