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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 4 7 5 3 9
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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
7
8
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 )
2n n
Sum n
Generalising
15
( 1)(1 )
2n n
Sum 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
7
8
9
51 9
2
8
6
43
7
438
951
276 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
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 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
MetalWood
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 n
Sum 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 6
78
9 10
11 12
13 14
1516
880 Solutions!
34
10 11
6 7
15 14
8
129
16
1
13
4
5
3 2
Subtract the 34 from the required total. Divide the result of Step 1 by 4.
Add the integer part (i.e. ignore any remainder) of the result of Step 2 to all cells in the "basic" Magic Square.
In addition, add the remainder (i.e. ignore the integers before the decimal point) of the result from Step 2 to the four cells
with the highest numbers.
Making a 4 x 4 magic square with any total.
Begin with this basic 4 x 4 square
The result of Step 1 would be 17 (i.e. 51 - 34). The result of Step 2 would be 4 remainder 1.
The result of Step 3, therefore, would be that 4 is added to all 16 cells in the "basic" Magic Square.
Step 4 would result in an additional 1 being added to the cells in the "basic" Magic Square that contain 13, 14, 15 and
16.
Suppose we wanted a 4 x 4 Magic square with a total of 51.
Melancholia
Engraving by Albrecht Durer (1514)
16
3 2 135 1
011
8
9 6 7 124 1
514
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
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 groups of cells that give the value of 34?
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 8
9 7 6
4 1
By interchanging rows, columns, or corner groups can you find some other distinct magic
squares?
165
9
4
13
12
8
1
2
11
14
7
3
10
15
6 11
10
5 8
12
9 7 6
16
2 3 13
14
15
4 1
5 811
10
9 12
76
16
13
23
4 114
15
16
2
3 13
11
10
14
15
12
5
8
9 7
6
4
1
16 23 13 111
0
14
15
12
58
9 76
41
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
2
3
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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5
4 10
21
2216
1
2
6
20
25
24
3
7
8
9
11
12
13
14
15
17
18
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) 325
2x
Sum
3255
65( 1)
(1 )2
n nSum n
Check the magic constant
Construct a 7 x 7 magic Square!
5
4 10
21
2216
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!
5
6
7
13
14
2122
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49 1
2
3
8
9
15
35
41
42
47
48
49
29
36
37
43
44
45
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
214
10
11
12
16
17
18
19
20
22
23
24
25
26
27
28
30
31
32
33
34
38
39
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) 1225
2x
Sum
12257
175( 1)
(1 )2
n 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
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
260260
260260
260
260
260
260
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 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?
( 1)(1 )
2n n
Sum n
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.
16
2 3 131
110
14
15
12
5 8
9 7 6
4 1
By interchanging rows, columns, and corner
groups, can you find some other distinct magic
squares?
