Magic Squares

1. Magic Squares Debunking the Magic RaduSorici The University of Texas at Dallas

2. Random Magic Square

3. No practical use yet great influence upon people

4. 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

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

6. 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 History is Very Important

7. 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 = History is Very Important

8. 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 = History is Very Important

9. 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) = History is Very Important

10. 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?

11. 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?

12. 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?

13. 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?

14. 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?

15. 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?

16. 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?

17. 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 So what exactly is a Magic Square?

18. The sum of numbers in each row, column, and diagonal is called the magic constant and is equal to Before we Start

19. 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 Start

20. 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 Before we Start

21. 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 Before we Start

22. 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 Types of Magic Squares

23. Odd orders (De la Loubère) Construction Methods

24. Odd orders (De la Loubère) Construction Methods

25. Odd orders Construction Methods

26. 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 Construction Methods

27. 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 Construction Methods

28. Singly Even • The Ralph Strachey Method Construction Methods

29. 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 Construction Methods

30. 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 Construction Methods

31. 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 Construction Methods

32. What Now?

33. 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

34. 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

35. 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

36. 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

37. 5 by 5 panmagic squares introduces even more magic Panmagic Square Continued

38. 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 Panmagic Square Continued

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

40. 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

41. 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

42. 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

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

44. 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

45. 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

46. 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. Bimagic Square

47. 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

48. 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 Multiplication Magic Square

49. 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

50. 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