Magic SquaresA 3 x 3 magic Square

Put the numbers 1 to 9 into the square so that all rows, columns and diagonals add to the magic

number.

12

Magic Number = ?

3

6 5

4

78

9

4

7

5 3

9

Adding Successive Numbers

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

Sum (1 10) = 5 x 11 = 55

1 + 2 + 3 +………………………+ 18 + 19 + 20

Sum (1 20) = 10 x 21 = 210

1 + 2 + 3 +………………………+ 98 + 99 + 100

Sum (1 100) = 50 x 101 = 5050

1 + 2 + 3 +……………+ + + n

n-1n-2( 1)(1 ) 2

n nSum n

Generalising

15

( 1)(1 ) 2n nSum n

Magic SquaresA 3 x 3 magic Square

Put the numbers 1 to 9 into the square so that all rows, columns and diagonals add to the magic

number.

Magic Number = ?

12 3

6 5

4

78

9

51 9

2

8

6

43

7

438951276 2 7 6

9 5 14 3 8

8 3 41 5 96 7 2

6 1 87 5 32 9 4

4 9 23 5 78 1 6

8 1 63 5 74 9 2

4 3 89 5 12 7 6

2 9 47 5 36 1 8

3 x 3 Magic Square

Which one of these did you get? Why are they all the same as the first?

3 Rotations

4 Reflections

The History of Magic SquaresHistorically, the first magic square was supposed to have been marked on the back of a divine tortoise before Emperor Yu (about 2200 B.C) when he was standing on the bank of the Yellow River.

Even (feminine) numbers or yin. Odd (masculine) numbers or yang.

lo-shu

Water

Fire

Metal Wood

The 4 elements evenly balanced

With the Earth at the centre.

62

1

834

9 5

7

In the Middle Ages magic squares were believed to give protection against the plague!In the 16th Century, the Italian mathematician, Cardan, made an extensive study of the properties of magic squares and in the following century they were extensively studied by several leading Japanese mathematicians. During this century they have been used as amulets in India, as well as been found in oriental fortune bowls and medicine cups.Even today they are widespread in Tibet, (appearing in the “Wheel of Life) and in other countries such as Malaysia, that have close connections with China and India.

A 4 x 4 Magic Square

( 1)(1 ) 2n nSum n

Put the numbers 1 to 16 into the square so that all rows, columns and diagonals add to

the magic number.1

Magic Number = ?

2

3 4

5 67 8

9 1011 12

13 14

15 16880 Solutions!

34

10 11

6 7

15 14

8

129

16

1

13

4

5

3 2

Melancholia

Engraving by Albrecht Durer (1514)

16

3 2 135 1

011

89 6 7 1

24 15

14

1Durer never explained the rich symbolism of his masterpiece but most authorities agree that it depicts the sullen mood of the thinker, unable to engage in action. In the Renaissance the melancholy temperament was thought characteristic of the creative genius. In Durers’ picture unused tools of science and carpentry lie in disorder about the dishevelled, brooding figure of Melancholy. There is nothing in the balance scale, no one mounts the ladder, the sleeping hound is half starved, the winged cherub is waiting for dictation, whist time is running out in the hour glass above. (thanks to Martin Gardner)

Order 4 magic squares were linked to Jupiter by Renaissance astrologers and were thought to combat melancholy.

A Famous Magic Square

16

3 2 13

5 10

11

8

9 6 7 12

4 15

14

1

The Melancholia Magic Square

The melancholia magic square is highly symmetrically with regard to its magic constant of 34. Can you find other groups of cells that give the same value?

34

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

1. Enter the numbers in serial order.

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Constructing a 4 x 4 Magic Square

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

Swapping columns 2 and 3 gives a different magic square. (Durers Melancholia!)

16 2 3 13

5 11 10 8

9 7 6 12

4 14 15 1

2. Reverse the entries in the diagonals

32

5 811

6

10

79 12

13

4 1514

16

1

A 4 x4 straight off

3 2

5 811

6

10

79 12

13

4 15 14

16

1

Durers Melancholia

16

2 3 131

110

14

15

12

5 89 7 64 1

By interchanging rows, columns, or corner groups can you find some other distinct magic

squares?

16594

1312

8

1

21114

7

31015

6 11

10

5 812

9 7 616

2 3 13

14

15

4 15 81

110

9 12

7616

13

23

4 114

15

16

2

3 13

11

10

14

15

12

5

89 7

6

4

116 23 1

3 1110

14

1512

58

9 7641

16 2

3 13

11

10

14

1512

5

8

9 7

64

1

An Amazing Magic Square!

7 12 1 14

2 13 8 11

16 3 10 5

9 6 15 4

This magic square originated in India in the 11th or 12th century

How many 34’s can you find?

Constructing n x n Magic Squares (n odd)

Pyramid Method

1

23

4

5

6

7

8

9

2

4

5

6

8

7

9 1

3

1. Build the pyramid

3. Fill the holes

2. Fill the diagonals

A 3 x 3 Construction

12

34

5

67

89

10

1112

1314

15

1617

1819

20

2122

2324

25

54 10

212216

1

2

6

2025

24

3

78

9

1112

1314

15

1718

19

23

Constructing n x n Magic Squares (n odd) Pyramid Method

A 5 x 5 Construction1. Build the pyramid

3. Fill the holes2. Fill the diagonals

25 26(1 25) 3252xSum

3255 65

( 1)(1 ) 2n nSum n

Check the magic constant

Construct a 7 x 7 magic

Square!

54 10

212216

12

6

2025

24

3

78

9

1112

1314

15

1718

19

23

1. Adding the same number to all entries maintains the magic.

2. Multiplying all entries by the same number maintains the magic.

3. Swapping a pair of rows or columns that are equidistant from the centre produces a different magic square.

Check these statements

Mathematicians have recently programmed a computer to calculate the number of 5 x 5 magic

squares.There are exactly 275 305 224 distinct solutions!

56

7

1314

212223

2425

2627

28

2930

3132

3334

35

3637

3839

4041

42

4344

4546

4748

49 12

3

89

15

35

4142

4748

49

2936

37

4344

45

12

34

56

7

89

1011

1213

14

1516

1718

1920

214

1011

12

16

1718

1920

2223

2425

2627

28

30

3132

3334

3839

40

46

Constructing n x n Magic Squares (n odd) Pyramid Method

A 7 x 7 Construction1. Build the pyramid

3. Fill the holes2. Fill the diagonals

49 50(1 49) 12252xSum

12257 175

( 1)(1 ) 2n nSum n

Check the magic constant

A Knights Tour of an 8 x 8 Chessboard

Euler’s Magic Square Solutio

n

2

1

3

4

5

6

7

89

1012

13

1415

16

17

18

19

2021

23

25

26

27

28

29

30

31

32

3335

37

38

40

41

42

43

4445

4647

48 50

51

52

53

54

60

64

63

56

59

57

61

62

5855

49

24

39

34

36

22

11

260260260260260260260260

What’s the

magic number

?

( 1)16

n n 64 6516x

260

260

260

260

260

260

260

260

The diagonals

do not add to

260

Benjamin Franklin’s Magic Square.The American statesman, scientist, philosopher, author and publisher created a magic square full of interesting features. Benjamin was born in Massachusetts and was the 15th child and youngest son of a family of seventeen. In a very full life he investigated the physics of kite flying, he invented

1706 - 1790a stove, bifocal glasses, he founded hospitals, libraries, and various postal systems and was a signer of the Declaration of Independence. He worked on street lighting systems, a description of lead poisoning, and experiments in electricity. In 1752 he flew a home-made kite in a thunderstorm and proved that lightning is electricity. A bolt of lightning struck the kite wire and travelled down to a key fastened at the end, where it caused a spark. He also charted the movement of the Gulf Stream in the Atlantic Ocean, recording its temperature, speed and depth. Franklin led all the men of his time in a lifelong concern for the happiness, well-being and dignity of mankind. His name appears on the list of the greatest Americans of all time. In recognition of his life’s work, his picture appears on some stamps and money of the United States. Lorraine Mottershead (Sources of Mathematical

Discovery)

Franklins 8 x 8 Magic Square

52 61 4 13 20 29 36 4514 3 62 51 46 35 30 1953 60 5 12 21 28 37 4411 6 59 54 43 38 27 2255 58 7 10 23 26 39 429 8 57 56 41 40 25 24

50 63 2 15 18 31 34 4716 1 64 49 48 33 32 17

Magic Number?

( 1)(1 ) 2n nSum n

260Check the sum of

the diagonals.

As in Euler’s chessboard solution, the square is not completely magic

Some Properties of Franklin’s Square

52 61 4 13 20 29 36 4514 3 62 51 46 35 30 1953 60 5 12 21 28 37 4411 6 59 54 43 38 27 2255 58 7 10 23 26 39 429 8 57 56 41 40 25 24

50 63 2 15 18 31 34 4716 1 64 49 48 33 32 17

(a) What is the sum of the numbers in each quarter?(b) What is the total of the diagonal cells 4 up and down 4 in each quarter?(c) Calculate the sum of the 4 corners plus the 4 middle cells.(d) Find the sum of any 4 cell sub square.

(e) Work out the sum of any 4 cells equidistant from the square’s centre.

16

2 3 131

110

14

15

12

5 89 7 64 1

By interchanging rows, columns, and corner

groups, can you find some other distinct magic

squares?