Magic Squares A 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. 1 2 Magic Number = ? 3 6 5 4 7 8 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 Generalising 1 + 2 + 3 +……………+ + + n n-2 n-1

Magic Squares A 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 = ? 1 2 3 6 5 4 7 8 9 9 2 8 6 4 3 7 1 5 15

3 x 3 Magic Square Which one of these did you get? Why are they all the same as the first? 4 3 8 9 5 1 2 7 6 2 7 6 9 5 1 4 3 8 8 3 4 1 5 9 6 7 2 6 1 8 7 5 3 2 9 4 4 9 2 3 5 7 8 1 6 8 1 6 3 5 7 4 9 2 4 3 8 9 5 1 2 7 6 2 9 4 7 5 3 6 1 8 4 Reflections 3 Rotations

The History of Magic Squares Historically, 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. 6 2 1 8 3 4 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 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 6 7 8 9 10 11 12 13 14 15 16 880 Solutions! 16 1 13 4 3 2 5 10 11 6 7 8 9 12 15 14 34

Melancholia Engraving by Albrecht Durer (1514) 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 Durer 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

The Melancholia 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 16 3 2 13 16 3 2 13 5 10 11 8 5 10 11 8 5 10 11 8 9 6 7 12 9 6 7 12 9 6 7 12 4 15 14 1 4 15 14 1 4 15 14 1 16 3 2 13 16 3 2 13 16 3 2 13 5 10 11 8 5 10 11 8 5 10 11 8 9 6 7 12 9 6 7 12 9 6 7 12 4 15 14 1 4 15 14 1 4 15 14 1

Constructing a 4 x 4 Magic Square 1. Enter the numbers in serial order. 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 2. Reverse the entries in the diagonals 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!) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A 4 x4 straight off 3 2 5 8 11 6 10 7 9 12 13 4 15 14 16 1 Durers Melancholia 16 1 13 4 3 2 5 8 10 7 11 6 9 12 15 14

16 2 3 13 11 10 14 15 12 5 8 9 7 6 4 1 By interchanging rows, columns, or corner groups can you find some other distinct magic squares? 16 5 9 4 13 12 8 1 2 11 14 7 3 10 15 6 16 2 3 13 11 10 14 15 12 5 8 9 7 6 4 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. Build the pyramid A 3 x 3 Construction 3. Fill the holes 1 2 3 2. Fill the diagonals 2 4 5 6 8 4 5 6 7 9 1 3 7 8 9

Check the magic constant Constructing n x n Magic Squares (n odd) Pyramid Method 1. Build the pyramid A 5 x 5 Construction 1 2 3 4 5 3. Fill the holes 6 7 8 9 10 2. Fill the diagonals 11 12 13 14 15 3 7 8 9 11 12 13 14 15 17 18 19 23 21 22 16 16 17 18 19 20 20 25 24 1 2 6 21 22 23 24 25 5 4 10 Check the magic constant

Construct a 7 x 7 magic Square! 5 4 10 21 22 16 1 2 6 20 25 24 3 7 8 9 11 12 13 14 15 17 18 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! Construct a 7 x 7 magic Square!

Check the magic constant Constructing n x n Magic Squares (n odd) Pyramid Method 1. Build the pyramid A 7 x 7 Construction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 3. Fill the holes 2. Fill the diagonals 15 16 17 18 19 20 21 22 23 24 25 26 27 28 4 10 11 12 16 17 18 19 20 22 23 24 25 26 27 28 30 31 32 33 34 38 39 40 46 29 36 37 43 44 45 29 30 31 32 33 34 35 35 41 42 47 48 49 1 2 3 8 9 15 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5 6 7 13 14 21 Check the magic constant

The diagonals do not add to 260 A Knights Tour of an 8 x 8 Chessboard Euler’s Magic Square Solution 2 1 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 23 25 26 27 28 29 30 31 32 33 35 37 38 40 41 42 43 44 45 46 47 48 50 51 52 53 54 60 64 63 56 59 57 61 62 58 55 49 24 39 34 36 22 11 260 What’s the magic number? 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 - 1790 a 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)

260 Franklins 8 x 8 Magic Square 52 61 4 13 20 29 36 45 14 3 62 51 46 35 30 19 53 60 5 12 21 28 37 44 11 6 59 54 43 38 27 22 55 58 7 10 23 26 39 42 9 8 57 56 41 40 25 24 50 63 2 15 18 31 34 47 16 1 64 49 48 33 32 17 Magic Number? 260 Check 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 45 14 3 62 51 46 35 30 19 53 60 5 12 21 28 37 44 11 6 59 54 43 38 27 22 55 58 7 10 23 26 39 42 9 8 57 56 41 40 25 24 50 63 2 15 18 31 34 47 16 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.

By interchanging rows, columns, and corner groups, can you find some other distinct magic squares? 16 2 3 13 11 10 14 15 12 5 8 9 7 6 4 1