This document discusses the history and properties of magic squares. It begins by explaining that the earliest known magic square originated in China over 4000 years ago. It then provides details on the formation and uniqueness of 3x3 magic squares, before discussing the extensive efforts to systematically form and count higher order magic squares, especially 5x5 squares. It notes that the exact number of 5x5 magic squares, over 68 million, was only determined in the 1970s using computer programs.

1. Selected topic 2

2. Title
MAGIC SQUARES

3. Objectives:
After reading this module, you will be able to:
1. form the 3 x 3 magic square by recalling the
interesting characteristics of this order-3 square
using the natural numbers 1 to 9;
2. appreciate the prodigious efforts of
mathematicians to systematize the formation of
higher order magic squares;
3. derive 3 x 3 magic square algebraically; and
4. trace the unicursal path of the sequence of
numbers on a magic squares.

4. History:
The first record of a magic square appeared more
4000 years ago in China. The first magic square
was supposedly brought to man by a sacred turtle
from the River Lo in the days the legendary
Emperor Yii, reputed to be a hyraulic engineer
(Boyer, 1991, p. 197). The completed square is the
Lo Shu of ancient China.
According to legend, the pattern was first revealed
on the back of turtle in the twenty-third century BC,

5. History:
but the modern Chinese scholars trace reference to
it to the fourth century BC. From then on until the
10th century the pattern was a mystical Chinese
symbol enormous significance. (Gardner, 1988; p
215). The even numbers were identified with yin,
the female principle; and the odd numbers with
yang, the male principle. The central 5 represented
the earth around which, in evenly balanced yin and
yang, were the other four elements; 4 and 9
symbolizing metal, 2 and 7, fire; 1 and 6, water; and
3 and 8, wood.

6. History:
4 9 2
3 5 7
8 2 6
earth
metal
fire
wood
water
Fig. 1: The Magic Square of China

7. History:
In the magic square, the numbers from 1 to 9 are
arranged in a square so that the sum of the three
numbers in each row, column and diagonal is the
same. In the Fig. 1 the sum is always 15.
The number 5 is always in the central cell. The even
numbers are in the corner cell. If we black out the
odd numbers, we get a figure. If we link the odd
numbers, we get a Z in an oblique position. If we
link the even numbers, the path traced is a Z.

8. History:
The movement traced 1, 2, 3 7,8, 9 and 1,
1, 3, 5, 7, 9
Even numbers

9. History:
And the path formed by the four ( 7, 8, 9; 3, 6, 9; 1,
4, 7 and 1, 2, 3 sequences is:
4 9 2
3 5 7
8 1 6

10. History:
The magic square has other interesting
characteristics. Arithmetic sequences are formed by
the three numbers in the middle row, in the central
column, and in each diagonal.
3, 5, 7 : difference 2 middle row
1, 5, 9 : difference 4 central row
4, 5, 6 : difference 1 diagonal
2, 5, 8 : difference 3 diagonal

11. ? SAQ 8 -1
9 8
6
1. Supposed only three numbers are given in 3-by-
3 magic square. How are you going to place the
other six numbers using x and y?
2. Do you find arithetic sequence in the resulting
magic square?
9 8
x 6 y

12. History:
.
.
.
you can make your own 3-by-3 magic square by
using the patterns that appear in every such
square.
+
.
.
.
.
.
.
. . .
..
.
. . .
+ =
T T T 3T

13. Magic Square
In the diagram, the three spots in the first squares
indicate sets of numbers whose sum is the same
number T, which is the sum of the three numbers in
each row, column and diagonal of a 3-by-3 magic
square. When the nine spots are transferred to a
single square, their total is 3T. However, the spots in
the top and bottom rows each add up to T. In other
words, the number in the central cell is always one-
third of T.

14. Magic Square
If the number in the central cell is m, then those in
the diagonal must form a sequences m – a, m, m +
a ; m – b, m, m + b, that have a sum of 3m = T.
Hence, the algebraic pattern for every 3-by-3 magic
square is
m + a m - a - b m + b
m - a + b m
m + a -
b
m - b m + a + b m - a

15. Magic Square
where m, a, b are any numbers integral of fractional
m + a m - a - b m + b
m - a + b m
m + a -
b
m - b m + a + b m - a
Question: What will happen if a = 2b?
Which cells will be affected most?

16. Magic Square
To avoid repeated number in a magic a ≠ 2b and
b ≠ 2a are necessary. And to avoid zero or negative
numbers m must exceed (a + b). The smallest set of
positive integers fulfilling these conditions is m = 5,
a = 3, b= 1, and these produce the Chinese magic
square made with the numbers 1 to 9.

17. Magic Square
A Standard magic square is a square array of
positive integers from 1 to 𝑁2, arranged so that the
sum of every row, every column, and each of the
two main diagonals is th same. N is the order of the
square. It is easy to see that the magic constant is
the sum of all numbers divided by N. the formula is
(1 + 2 + 3 + ⋯ + 𝑁2)
𝑁
=
(𝑁3 + 𝑁)
2

18. Magic Square
The trivial square of order is simply the number 1;
and of course, it is unique. It is equally trivial to
prove that no order-2 square is possible.
The order-3 magic square composed of the integers
1 to 9, when rotations and reflections are excluded,
is unique. The constant sum is
(1+2+3+4+5+6+7+8+9)
3
=
45
3
= 15 This is
𝑙(33+3)
2

19. Magic Square
There are eight ways in which the sum 15 can be
partitioned:
9 + 5 + 1
9 + 4 + 2
8 + 6 + 1
8 + 5 + 2
8 + 4 + 3
7 + 6 + 2
7 + 5 + 3
6 + 5 + 4
These correspond to the six
orthogonals (rows and columns)
and the two diagonals.

20. Magic Square
The eight lines exactly match the number of triples
we have available. Since the center number
belongs to a row, a column and both diagonals, it
clearly must be a digit that appears in four of the
eight triples. The only such digit is 5. Hence, we
know that 5 is the central number. The number 9
belongs to only two triples. We cannot place it in a
corner because each corner cell belongs to three
lines. Consequently, it must go in a side cell.

21. Magic Square
Because of the square’s symmetry, it does not
matter which cell we choose, so let us put it above
5. For the top corner on each side of 9, we have
choices, so except 2 and 4. Again, it does not
matter which digit goes where since one
arrangement is merely a mirror reflection of the
other. The rest of the square follows automatically.
This simple instruction proves the uniqueness of
this square.

22. FIVE-BY-FIVE-SQUARES
How many magic square are of order 5? According
to Gardner (1991), the best estimate was given by
Albert Candy in his Construction, Classification and
Census of Magic Square of the Order Five, privately
published in 1938. Candy arrived at a total of
13,288,952. The exact number was not known until
1973 when the counting was completed by a
computer programmer at Information International.
The program, using a standard backtracking
procedure,

23. FIVE-BY-FIVE-SQUARES
consists of about 3500 “words” and took about 100
hours of running time on a PDP-10. A final report
was issued in 1975.
Candy’s estimate was far below Schroeppel’s count.
Not counting rotations and reflections, there are
245,305,224 magic square order of 5. Schroeppel
prefers to divide that number 4 and give the total
68,826,306. The reason is that there are four other
variants generated by the following two
transformations which also preserve magic:

24. FIVE-BY-FIVE-SQUARES
1. Exchange the left and right border columns, then
exchange the top and bottom border rows.
2. Exchange rows 1 and 2 and rows 4 and 5. then
exchange columns 1 and 2 and columns 4 and 5.
When the two transformations are combined with
the two reflections and four rotations, the result is
2 x 4 x 2 x 2 =32 forms that can be called
isomorphic. With this definitions of isomorphic, the
count becomes 68,826,306.

25. FIVE-BY-FIVE-SQUARES
That number can be lowered even more by
considering another well known transformation. If
every number in a magic square is subtracted from
𝑁2 + 1 (in this case 26), the result called the
complement, is also magic. When the center of an
order-5 square is 13, the complement is isimorphic
with the original. If it is not 13, a difference square
results. If we broaden the term isimorphic to include
complements, the count of order-5 drops to about
35 million.

26. FIVE-BY-FIVE-SQUARES
Here is a 5-by-5 magic square:
1 15 24 8 17
23 7 16 5 14
20 4 13 22 6
12 21 10 19 3
9 18 2 11 25

27. FIVE-BY-FIVE-SQUARES
Consider these numbers of order-5 square whose
centers are number 1 through 13.
Number at Center
1
2
3
4
5
6
No. of different Order-5
squares possible
1,091,448
1,366,179
1,914,984
1,958,837
2,431,806
2,600,879

28. FIVE-BY-FIVE-SQUARES
Number at Center
7
8
9
10
11
12
13
No. of different Order-5
squares possible
3,016,881
3,112,161
3,472,542
3,344,034
3,933,818
3,784,618
4,769,936

29. FIVE-BY-FIVE-SQUARES
Note that the totals steadily increase from 1
through8 but the totals for 9 and 10 and 11 and 12
are reversed in rank. This is a surprising result.
These same trends occur in counts of squares with
centers 14 through 25, since these are
complements of the centers 1 to 12. There are as
many squares with 1 in the centers as there are with
25; and the same is true for all numbers except 13
(which is the same as its complement).

30. FIVE-BY-FIVE-SQUARES
The numbers of squares listed for center 1 through
13 add up to the earlier figure cited as Schroeppel’s
calculation with the help of the computer.
Do you think the study of the number of order-5
square could have been possible without the help of
computers? Explain your answer.
One of the major breakthrough in mathematics has
been made in the study of magic square. A further
development and breakthrough is on magic cubes.
However, that is not within the scope of this module.