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Magic Squares

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Magic Squares

Debunking the Magic

RaduSorici

The University of Texas at Dallas

- 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

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

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

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

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

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

The “Lo Shu” square is the first recorded magic square

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

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

The “Lo Shu” square is the first recorded magic square

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

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

The “Lo Shu” square is the first recorded magic square

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

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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- A magic square is an x table containing integers such that the numbers in each row, column, or diagonal sums to the same number

- 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

- 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

- 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

- 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

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

- The order of a magic square is the size of the 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.

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

The sum of numbers in each row, column, and diagonal is called the magic constant and is equal to

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

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

For ,… the magic constants are

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

For ,… the magic constants are

For odd the middle number is equal to

- 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

Odd orders (De la Loubère)

Odd orders (De la Loubère)

Odd orders

- Doubly 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

- Doubly 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

- Singly Even
- The Ralph Strachey Method

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

C

A

D

B

- Singly Even
- The Ralph Strachey Method
- 2nd Step – Exchange the leftmost columns in subsquare A with the corresponding columns of subsquareD and exchange the rightmost columns in subsquare C with the corresponding columns of subsquare B

- Singly 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

- 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.

- 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

- 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 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.

- 5 by 5 panmagic squares introduces even more magic

- 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

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

- 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?

- 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 .

- 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 .

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

- 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

- 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

- 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.

- 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.

- 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

- 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

- 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

- 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 A
- I 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.

- The presentation would not be complete with a reference to the Fibonacci numbers

- The presentation would not be complete with a reference to the Fibonacci numbers
- Start with the basic 3 by 3 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

- 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.

Masonic Cipher

Masonic Cipher

Durer Magic Square

The message is

The message is

I Love Mathematics