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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 Important 1. There is evidence to date magic squares as early as the 6th century due to Chinese mathematicians History is Very Important 1. There is evidence to date magic squares as early as the 6th century due to Chinese mathematicians 2. It was later discovered by the Arabs in the 7th century History is Very Important 1. There is evidence to date magic squares as early as the 6th century due to Chinese mathematicians 2. It was later discovered by the Arabs in the 7th 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 6th century due to Chinese mathematicians 2. It was later discovered by the Arabs in the 7th 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 6th century due to Chinese mathematicians 2. It was later discovered by the Arabs in the 7th 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? • A magic square is an 𝑛 x 𝑛 table containing 𝑛2 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 𝑛 x 𝑛 table containing 𝑛2 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 𝑛 x 𝑛 table containing 𝑛2 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 𝑛 x 𝑛 table containing 𝑛2 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 𝑛2 So what exactly is a Magic Square? • A magic square is an 𝑛 x 𝑛 table containing 𝑛2 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 𝑛2 • Normal magic squares exist for all 𝑛 ≥ 1, except for 𝑛 = 2 So what exactly is a Magic Square? • A magic square is an 𝑛 x 𝑛 table containing 𝑛2 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 𝑛2 • Normal magic squares exist for all 𝑛 ≥ 1, except for 𝑛 = 2 • For 𝑛 = 1 we simply get the trivial square containing 1 So what exactly is a Magic Square? • A magic square is an 𝑛 x 𝑛 table containing 𝑛2 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 𝑛2 • Normal magic squares exist for all 𝑛 ≥ 1, except for 𝑛 = 2 • For 𝑛 = 1 we simply get the trivial square containing 1 • For 𝑛 = 2 we would have the following square Which would imply that 𝐴+𝐵 =𝐶+𝐷 𝐴 + 𝐶 = 𝐵 + 𝐷 ⇒ 𝐴 = 𝐵 = 𝐶 = 𝐷. 𝐴+𝐷 =𝐶+𝐵 But then this is not a normal magic square. So what exactly is a Magic Square? • A magic square is an 𝑛 x 𝑛 table containing 𝑛2 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 𝑛2 • Normal magic squares exist for all 𝑛 ≥ 1, except for 𝑛 = 2 • For 𝑛 = 1 we simply get the trivial square containing 1 • For 𝑛 = 2 we would have the following square Which would imply that 𝐴+𝐵 =𝐶+𝐷 𝐴 + 𝐶 = 𝐵 + 𝐷 ⇒ 𝐴 = 𝐵 = 𝐶 = 𝐷. 𝐴+𝐷 =𝐶+𝐵 But then this is not a normal magic square. • For 𝑛 ≥ 3 we will prove that a normal magic square exists Before we Start 1. The sum of numbers in each row, column, and diagonal is called the magic constant and is equal to 𝑛 𝑛2 + 1 𝑀= . 2 Before we Start 1. The sum of numbers in each row, column, and diagonal is called the magic constant and is equal to 𝑛 𝑛2 + 1 𝑀= . 2 This is true because the sum of all the numbers in the magic square is equal to 2 𝑛2 + 1 𝑛 1 + 2 + 3 + ⋯ + 𝑛2 = 2 and because there are 𝑛 rows we can divide by 𝑛 to obtain the above result Before we Start 1. The sum of numbers in each row, column, and diagonal is called the magic constant and is equal to 𝑛 𝑛2 + 1 𝑀= . 2 This is true because the sum of all the numbers in the magic square is equal to 2 𝑛2 + 1 𝑛 1 + 2 + 3 + ⋯ + 𝑛2 = 2 and because there are 𝑛 rows we can divide by 𝑛 to obtain the above result 2. For 𝑛 = 3, 4, 5, 6, 7, 8,… the magic constants are 15, 34, 65, 111, 175, 260, … Before we Start 1. The sum of numbers in each row, column, and diagonal is called the magic constant and is equal to 𝑛 𝑛2 + 1 𝑀= . 2 This is true because the sum of all the numbers in the magic square is equal to 2 𝑛2 + 1 𝑛 1 + 2 + 3 + ⋯ + 𝑛2 = 2 and because there are 𝑛 rows we can divide by 𝑛 to obtain the above result 2. For 𝑛 = 3, 4, 5, 6, 7, 8,… the magic constants are 15, 34, 65, 111, 175, 260, … 3. For odd 𝑛 the middle number is equal to 𝑛2 + 1 Types of Magic Squares • • • • • • • • • • • • Singly even - 𝑛 = 4𝑘 + 2 Doubly even - 𝑛 = 4𝑘 Odd - 𝑛 = 2𝑘 + 1 Antimagic - the 2𝑛 + 2 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 Methods Odd orders (De la Loubère) Construction Methods Odd orders (De la Loubère) Construction Methods Odd orders Construction Methods 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 𝑛 = 4 Construction Methods 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 𝑛 = 4 Construction Methods Singly Even • The Ralph Strachey Method Construction Methods Singly Even • The Ralph Strachey Method for orders of the form 4𝑛 + 2 • 1st Step – Divide the square into four smaller subsquares ABCD A D C B Construction Methods Singly Even • The Ralph Strachey Method • 2nd Step – Exchange the leftmost 𝑛 columns in subsquare A with the corresponding columns of subsquare D and exchange the rightmost 𝑛 − 1 columns in subsquare C with the corresponding columns of subsquare B Construction Methods 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 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 4𝑥4 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 4𝑥4 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 4𝑥4 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=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 • A magic cube is a magic square but in 3-D. All of the properties are translated to 3-D. • The magic constant is 𝑀 = 𝑛 𝑛3 +1 2 . 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 𝑀 = 2 𝑛 𝑛3 +1 2 . Why? • Because there are 𝑛 rows and the total sum is 𝑛3 𝑛3 +1 2 . 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 𝑀 = 2 𝑛 𝑛3 +1 2 . Why? • Because there are 𝑛 rows and the total sum is 𝑛3 𝑛3 +1 2 . 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 𝑎 + 𝑖 = 2𝑒 because 𝑎+𝑏+𝑐 + 𝑑+𝑒+𝑓 + 𝑔+ℎ+𝑖 + 𝑎+𝑒+𝑖 = (𝑎 + 𝑒 + 𝑖) + (𝑔 + 𝑒 + 𝑐) + (𝑑 + 𝑒 + 𝑓) + (𝑏 + 𝑒 + ℎ). In addition, by the same reasoning we have that 𝑎2 + 𝑖 2 + 2𝑒 2 . Thus 𝑎 − 𝑖 2 = 2 𝑎2 + 𝑖 2 − 𝑎 + 𝑖 2 = 4𝑒 2 − 4𝑒 2 = 0 Hence 𝑎 = 𝑒 = 𝑖. 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) 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