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Magic Squares
Charles Ng Cheuk Kin
6S (17)
14-10-2002
One day I happened to stop at Times Square in Causeway Bay to meet
my friend. While I was waiting, I suddenly thought about ‘squares’, the
shapes that we often encounter in Mathematics problems. ‘Ummm…
squares are quite interesting…’ I thought. Therefore I did some research and
wrote an article about ‘Magic Squares’.
A magic square is a square in which its cells add up to the same
amount, by each column, row, or diagonal, whichever way it is taken. One
legend says that a magic square is originated from China. It is brought from
the time of a great emperor, Yu, 2000 B.C from China. Then from China, the
magic square has passed to Japan, Southeast Asia, India, Arabs and the
knowledge of magic square kept traveling westward in a certain period of
time.
The magic square is divided into two types: odd and even. It is odd if
its side consists of an odd number of cells, and even for an even number of
cells. The even squares are further divided into two classes: evenly even and
unevenly even. Note that a 2x2 square is not counted as a magic square.
The following diagram shows their relationship:
The methods of producing these squares are all different, so each one
must be treated separately.
The ‘Odd Squares’ are the easiest to be filled up, and the same
technique may be applied to all odd squares. There are many ways to fill up
odd squares, and the following is the easiest method. For example, let’s
consider a 5x5 square. The sequence of numbers are 1, 2, 3 … 24, 25. First
of all, place the first number, 1, immediately in the cell just below the center
cell. Then place the consecutive numbers, one by one, in a diagonal line
inclining downwards to the right. When beyond the side of the square, no
matter vertical or horizontal, carry the following number to the relative
position on the opposite side of the square. When the diagonal march meets
a cell already occupied, take a diagonal direction from the cell towards the
left, and then proceed as before to the right. As a result to the rule, the mean
number always occupies the center cell, and the highest number in the cell
just above the center.
3
4
5
1
1
1
2
11
4
17
10
23
24 7
12 25
5 13
18 1
6 19
2
7
20 3
8 16
21 9
14 22
2 15
4
3
8
5
9
1
6
2
For ‘Evenly even Squares’, let’s start with a 4x4 square. First, place
the numbers in their natural order. Then here comes the trick: change the top
and bottom central numbers alternately. Finally, change the left and right
central numbers alternately. Look! A magic square is now created.
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
1 15 14 4
5 6 7 8
9 10 11 12
13 3 2 16
1 15 14 4
12 6 7 9
8 10 11 5
13 3 2 16
For any larger evenly even squares, the rule is similar. First we should
divide the square into 4x4 squares. Then, place the numbers in their natural
order just like one whole square. Next, passing over the other cells, change
the top, bottom, left, right central numbers alternately. Eventually, a magic
square comes out! The following 8x8 square illustrates the process.
1
9
17
25
33
41
49
57
2
10
18
26
34
42
50
58
3
11
19
27
35
43
51
59
4
12
20
28
36
44
52
60
5
13
21
29
37
45
53
61
6
14
22
30
38
46
54
62
7
15
23
31
39
47
55
63
8
16
24
32
40
48
56
64
1
9
17
25
33
41
49
57
63
10
18
39
31
42
50
7
62
11
19
38
30
43
51
6
4
12
20
28
36
44
52
60
5
13
21
29
37
45
53
61
59
14
22
35
27
46
54
3
58
15
23
34
26
47
55
2
8
16
24
32
40
48
56
64
1
56
48
25
33
24
16
57
63
10
18
39
31
42
50
7
62
11
19
38
30
43
51
6
4
53
45
28
36
21
13
60
5
52
44
29
37
20
12
61
59
14
22
35
27
46
54
3
58
15
23
34
26
47
55
2
8
49
41
32
40
17
9
64
1
56
48
25
33
24
16
57
63
10
18
39
31
42
50
7
62
11
19
38
30
43
51
6
4
53
45
28
36
21
13
60
5
52
44
29
37
20
12
61
59
14
22
35
27
46
54
3
58
15
23
34
26
47
55
2
8
49
41
32
40
17
9
64
Notice that the four corner cells, and the center four inside the 4x4
squares are unmoved all the time.
As for the ‘Unevenly even Squares’, it has hitherto been found much
more difficult to produce than the previous squares. Let us now explore one
of the ways to construct a 6x6 square. First set up a square with side 6.
Divide it into 4 equal square regions labeled A to D.
A B
C D
Then start in square A and use the technique for producing ‘Odd
Squares’, starting from the smallest number, say, from 1 to 9. Then by using
the next number after the last number of square A, work in square D using
the same technique. Next, work in square B then to square C using the same
pattern. We should obtain a 6x6 square like this:
4 9 2 22
3 5 7 21
8 1 6 26
31 36 29 13
30 32 34 12
35 28 33 17
27
23
19
18
14
10
20
25
24
11
16
15
After that, we can highlight the center cell and two corner cells on the
left of square region A and C. Exchange the numbers, moving them to their
exact relative positions. So, there we have, a 6x6 magic square!!!
31 9 2 22 27
3 32 7 21 23
35 1 6 26 19
4 36 29 13 18
30 5 34 12 14
8 28 33 17 10
20
25
24
11
16
15
Can readers find out how to construct a 10x10 or 14x14 magic square?
Although the construction of ‘Unevenly even Squares’ seems much more
difficult, it actually follows some kind of rules. The rules are to be
discovered by readers.
Bear in mind: these are just one of the methods to produce the three
basic kinds of magic squares ( Odd, Evenly even, Unevenly even ). There
are many other ways to fill up the squares. This is similar to learning, there
are infinite number of new things and knowledge to be discovered.
Therefore, we should learn the old, basic knowledge and develop the new.