Presented To

Department of Mathematics

My final year project entitled ’Magic Squares with Additional Prop-
erties’ aims to research on one of the branches of mathematics which is
magic squares. Since magic squares have existed for a long time, there
are many interesting properties on magic squares that are found and yet to
be found. We shall learn about the history and origin of magic squares.
In addition, a magic square also has its uses that we can make use of.
By referring to some published materials, we can learn more about it and
therefore understand its properties better. For a magic square, the sum
of the row entries, column entries and entries of the main diagonals has
to be the same and therefore this constant sum is known as the magic
sum. With this property itself, it makes this magic square seems unique
and distinguished. And also in order to understand some other proper-
ties, the method of construction of magic squares must be known first.
With different methods to construct these magic squares of different order,
these properties can be identified throughout the way. These methods are
known and its construction are detailed and clear for someone with least
mathematical background to understand them. Not only that, there are
fairly many types of magic squares which are interesting enough to catch
anyone’s attention. Such special characteristics comprising of symmetric
properties to having broken diagonals in a square definitely made magic
squares a distinctive part of mathematics. Lastly, we will try to modify a
method of construction to produce magic squares as well.
CHAPTER 1 Introduction 1
1-1 Background of Magic Squares 1
1-2 Application of Magic Squares 2
CHAPTER 2 Objectives and Planning 4
2-1 Project Scopes 4
2-2 Planning 5
2-2-1 Action plan for Project I 5
2-2-2 Action plan for Project II 5
CHAPTER 3 Literature Review 6
CHAPTER 4 Methodology 9
4-1 Computation on Magic Sum 9
4-2 Construction of Magic Squares of Odd Order 10
4-3 Construction of Magic Squares of Singly-Even Order 12
4-4 Construction of Magic Squares of Doubly-Even Order 15
CHAPTER 5 Results and Discussion 18
5-1 Symmetrical Magic Squares 18
5-2 Self-Complementary Magic Squares 22
5-3 Other Types of Magic Squares 24
5-4 Modification in Generalized Doubly-Even Method 26
CHAPTER 6 Conclusion 30

11 Lo-Shu square 2
12 Magic square of order 3 2
21 Action plan for Project I 5
22 Action plan for Project II 5
31 Franklin Square of order 8 7
32 Magic square with initial entry ‘1’ 7
33 Magic square with initial entry ‘2’ 8
34 Magic square with increment of 2 8
41 Conditions for De la Loubere method 10
42 Magic square of order 7 11
43 Magic square of order 9 12
44 Sub-squares of order 2k + 1 12
45 Sub-squares using De la Loubere method 13
46 Magic square of order 10 14
47 Magic square of order 6 14
48 Sub-squares of order 4 16
49 Magic square of order 8 16
410 Magic square of order 12 17
51 Symmetrical magic square of order 5 19
52 All entries joined together for order 5 19
53 All even-valued entries joined together for order 5 19
54 All odd-valued entries joined together for order 5 20
55 All entries joined together for order 6 20
56 All even-valued entries joined together for order 6 20
57 All odd-valued entries joined together for order 6 21
58 All entries joined together for order 4 21
59 All even-valued entries joined together for order 4 21
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