Transcript Magic Squares - Suffolk Maths

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
7
9
2
5
6
Magic Number = ?
4
7
9
3
3
4
8
5
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
Generalising
Sum (1  100) = 50 x 101 = 5050
1 + 2 + 3 +……………+ n-2
Sum (1  n ) 
+ n-1
n (n  1)
2
+ n
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.
2
1
7
6
Magic Number = ?
6
7
2
1
5
9
8
3
4
Sum (1  n ) 
3
4
8
5
n (n  1)
2
9
15
3 x 3 Magic Square
6
7
2
2
7
6
8
3
4
6
1
8
1
5
9
9
5
1
1
5
9
7
5
3
8
3
4
4
3
8
6
7
2
2
9
4
Which one of these did you get? Why are they all the same as the first?
4
9
2
8
1
6
4
3
8
2
9
4
3
5
7
3
5
7
9
5
1
7
5
3
8
1
6
4
9
2
2
7
6
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.
lo-shu
The 4 elements evenly balanced
2
Fire
6
7
Water
9
5
Metal
4
Even (feminine) numbers or yin.
Odd (masculine) numbers or yang.
1
Wood
3
8
With the Earth at the centre.
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
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
880 Solutions!
Magic Number = ?
Sum (1  n ) 
n (n  1)
2
34
Making a 4 x 4 magic
square with any total.
Begin with this basic 4 x 4 square
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.
Suppose we wanted a 4 x 4
Magic square with a total of 51.
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.
A Famous Magic Square
Melancholia
16 3 2 13
5 10 11 8
9 6
7 12
4 15 14 1
Engraving by Albrecht Durer (1514)
Order 4 magic
squares were linked
to Jupiter by
Renaissance
astrologers and were
thought to combat
melancholy.
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)
The Melancholia Magic Square
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
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
5
10 11
9
6
4
15 14
16
3
5
10 11
9
6
4
15 14
2
7
2
7
13
16
3
8
5
10 11
12
9
6
1
4
15 14
13
16
3
8
5
10 11
12
9
6
1
4
15 14
2
7
2
7
13
16
3
8
5
10 11
12
9
6
1
4
15 14
13
16
3
8
5
10 11
12
9
6
1
4
15 14
2
7
2
7
13
8
12
1
13
8
12
1
Constructing a 4 x 4 Magic Square
1. Enter the numbers in serial order.
2. Reverse the entries in the diagonals
13
16
3
8
5
10 11
12
9
6
4
15 14 1
1
2
3
4
16
2
5
6
7
8
5
11 10
9
10
11 12
9
7
16
4
14 15 1
13 14 15
3
6
2
7
13
8
12
Swapping columns 2 and 3 gives a different magic
square. (Durers Melancholia!)
A 4 x4 straight off
16
2
3
13
16
3
2
13
5
11
10
8
5
10
11
8
9
7
6
12
9
6
7
12
4
14
15
1
4
15
14
1
Durers Melancholia
By interchanging rows, columns, or corner groups can
you find some other distinct magic squares?
16 2 3 13
5 11 10 8
9
7
6 12
4 14 15 1
16 3 2 13
9 6 7 12
16 3 2 13
5 10 11 8
16 2 3 13
9 7 6 12
5 10 11 8
4 15 14 1
9 6 7 12
5 11 10 8
4 15 14 1
4 14 15 1
6 12 3 13
6 12 9
15 1 10 8
15 1 4 14
3 13 16 2
5 11 4 14
3 13 6 12
10 8
10 8 15 1
9
7 16 2
4 14 5 11
7
5 11
16 2 9
7
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
2. Fill the diagonals
3
2
6
9
5
1
4
8
7
3. Fill the holes
A 3 x 3 Construction
2
7
6
9
5
1
4
3
8
Constructing n x n Magic Squares (n odd)
Pyramid Method
1. Build the pyramid
A 5 x 5 Construction
5
4
3
2
1
9
7
6
2. Fill the diagonals
10
8
15
14
13
11
19
16 9 22 15
20 8
25
21 14
7 25 13
24
24 12 5
23
17
16
3
20
18
12
3. Fill the holes
11
4
2
1 19
18
6
17 10 23
22
21
Check the magic constant
Sum (1  n ) 
n (n  1)
2
Sum (1  25) 
25x 26
 325
2
325
 65
5
Check these statements
3
16 9 22 15
20 8
21 14
2
7 25 13
1
19
24 12 5
18
6
11
4
17 10 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.
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!
Constructing n x n Magic Squares (n odd)
Pyramid Method
1. Build the pyramid
A 7 x 7 Construction
7
14
6
5
4
3
2
9
1
8
16
24
32
31
23
30
22
29
34
33
25
35 11 36 19 44 27 3
35
27
10 42 18 43 26 2 34
42
41 17 49 25 1 33 9
49
41
16 48 24 7 32 8 40
47 23 6 31 14 39 15
22 5 30 13 38 21 46
48
40
47
39
46
38
3. Fill the holes
4 29 12 37 20 45 28
28
26
18
17
15
20
19
11
10
21
13
12
2. Fill the diagonals
45
37
44
36
43
Check the magic constant
Sum (1  n ) 
n (n  1)
2
Sum (1  49) 
49x 50
 1225
2
1225
 175
7
A Knights Tour of an 8 x 8 Chessboard
Euler’s
Magic
Square
Solution
16
260
3 62 19 14 35
260
49 32 15 34 17 64
260
30 51 46
47
2
52 29
4
44 25
45 20 61 36 13
260
28 53
8 41 24 57 12 37
260
43 6
55 26 39 10 59 22
260
54 27 42
56 9
7
58 23 38 11
260 260 260 260 260 260 260 260
What’s
the
magic
number?
260
40 21 60
5
n (n  1)
1 48 31 50
33 16 63 18
260
The
diagonals
do not add
to 260
64 x 65
16
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
1706 - 1790
life he investigated the physics of kite flying, he invented
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)
Franklins 8 x 8 Magic Square
52 61
14
3
53 60
11
6
55 58
9
8
50 63
16
1
4
13 20 29 36 45
Magic Number?
62 51 46 35 30 19
5
12 21 28 37 44
59 54 43 38 27 22
7
10 23 26 39 42
57 56 41 40 25 24
2
15 18
31 34 47
64 49 48 33 32 17
Sum (1  n ) 
n (n  1)
2
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
14
3
53 60
11
6
55 58
9
8
50 63
16
1
4
13 20 29 36 45
62 51 46 35 30 19
5
12 21 28 37 44
59 54 43 38 27 22
7
10 23 26 39 42
57 56 41 40 25 24
2
15 18
31 34 47
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
5 11 10 8
9
7
6 12
4 14 15 1