Office Applications and Entertainment, Magic Squares (Natural Numbers)

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Contents

1.0 Introduction

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2.0 Magic Squares (4 x 4)
2.1 Analytic Solution     Pan Magic Squares
2.2 Analytic Solution     Simple Magic Squares
2.3 Analytic Solution     Associated Magic Squares
2.4 Analytic Solution     Most Perfect Magic Squares
2.5 Other Magic Constants
2.6 Spreadsheet Solutions
2.7 Summary, Enumeration

16 9 3 6
2 7 13 12
5 4 10 15
11 14 8 1

3.0 Magic Squares (5 x 5)
3.1 Analytic Solution     Pan Magic Squares
3.2 Analytic Solution     Simple Magic Squares
3.3 Other Magic Constants
3.4 Rotation, Reflection, Row and Column Shifts
3.5 Transformations
3.6 Quinary Composition
5.4 Further Analysis      Symmetric Squares
5.5 Further Analysis      Concentric/Eccentric
5.6 Further Analysis      Miscellaneous Inlays
5.7 Spreadsheet Solutions
5.8 Summary, Enumeration

25 11 4 8 17
3 7 20 21 14
16 24 13 2 10
12 5 6 19 23
9 18 22 15 1

6.0  Magic Squares (6 x 6)
6.1  Analytic Solution     Pan Magic Squares (Don't Exist)
6.2  Analytic Solution     Simple Magic Squares
6.3  Transformations
6.4  Further Analysis      Symmetrical Main Diagonals
6.6  Further Analysis      Medjig Solutions
6.7  Further Analysis      Concentric/Eccentric/Inlaid
6.8  Spreadsheet Solutions
6.9  Non Cons. Integers    Pan Magic, Part 1
6.10 Further Analysis      Rectangular Compact
6.11 Further Analysis      Symmetries
6.12 Non Cons. Integers    Pan Magic, Part 2
6.13 Non Cons. Integers    Bimagic

6 32 3 34 35 1
7 11 27 28 8 30
19 14 16 15 23 24
18 20 22 21 17 13
25 29 10 9 26 12
36 5 33 4 2 31

7.0 Magic Squares (7 x 7)
7.1 Analytic Solution     Pan Magic Squares
7.2 Sudoku Comparable Method
7.3 Further Analysis      Ultra Magic Squares
7.5 Further Analysis      Concentric/Eccentric
7.6 Further Analysis      Miscellaneous Inlays
7.7 Further Analysis      Overlapping Sub Squares
7.8 Further Analysis      Non Overlapping Sub Squares
7.9 Further Analysis      Bent Diagonals

16 41 18 2 33 35 30
45 1 38 42 3 7 39
29 23 14 31 46 26 6
10 28 37 25 13 22 40
44 24 4 19 36 27 21
11 43 47 8 12 49 5
20 15 17 48 32 9 34

8.0 Magic Squares (8 x 8)
8.1 Pan Magic Squares     General
8.2 Magic Squares         Composed of Sub Squares
8.3 Medjig Solutions      Main Diagonals
8.4 Franklin Squares      Bent Diagonals
8.5 Pan Magic Squares     Based on Franklin Properties
8.6 Further Analysis      Miscellaneous Properties
8.7 More Solutions        Sudoku Comparable
8.8 Magic Squares         Concentric/Eccentric/Inlaid
8.9 Intermezzo

46 41 44 47 23 20 17 22
3 32 50 5 40 10 61 59
27 2 29 56 58 37 16 35
51 53 8 26 13 64 34 11
54 25 55 4 33 15 60 14
30 7 28 49 63 36 9 38
6 52 1 31 12 57 39 62
43 48 45 42 18 21 24 19

9.0 Magic Squares (9 x 9)
9.1 Analytic Solution     Pan Magic Squares
9.2 Further Analysis      Matrix Operation
9.4 Further Analysis      Compact/Symmetric
9.5 More Solutions        Sudoku Comparable
9.6 Magic Squares         Concentric/Eccentric
9.7 Magic Squares         Overlapping Sub Squares
9.7 Ass Mgc Squares       Miscellaneous Inlays
9.8 Intermezzo

10.0 Magic Squares (10 x 10)
10.1 Medjig Solutions
10.2 Magic Squares        Concentric/Eccentric
10.3 Magic Squares        Composed
10.4 Pan Magic Squares    Non Consecutive Integers
10.5 Further Analysis     Construction Methods

43 51 29 66 80 58 14 19 9
26 4 12 46 36 41 78 56 70
63 68 73 2 16 24 31 39 53
76 57 71 27 5 10 47 34 42
32 37 54 61 69 74 3 17 22
15 20 7 44 49 30 64 81 59
1 18 23 33 38 52 62 67 75
65 79 60 13 21 8 45 50 28
48 35 40 77 55 72 25 6 11

11.0 Magic Squares (11 x 11)
11.1 Magic Squares        Concentric/Eccentric
11.2 Magic Squares        Composed
11.3 Magic Squares        Associated

12.0 Magic Squares, Higher Order
12.1 Pan Magic Squares    12 x 12, Pan Magic Sub Squares
12.2 Pan Magic Squares    12 x 12, Franklin Properties
12.3 Franklin Squares     16 x 16, Bent Diagonals
12.4 Franklin Squares     16 x 16, Most Perfect Pan Magic Squares
12.5 Franklin Squares     16 x 16, Most Perfect/Barink Restrictions
12.6 Magic Squares        13 x 13, Overlapping Sub Squares

22.0 Magic Squares, Higher Order, Composed
22.1 4 x 4 Sub Squares    Magic Sums Identical
22.5 Misc. Sub Squares    Magic Sums Different
22.6 Inlaid Squares       Associated Border
23.0 Single Even Order    Center Cross, Bordered

13.0 Construction of Single Magic Squares
13.1 Introduction
13.2 Pan Magic Squares    Odd
13.3 Magic Squares        Odd
13.4 Pan Magic Squares    Even
13.5 Interactive Solutions

14.0 Special Squares, Prime Numbers

15.0 Special Squares, Bimagic Squares
15.1 Bimagic Squares (8 x 8) Part 1
15.3 Bimagic Squares (8 x 8) Part 2
15.2 Bimagic Squares (9 x 9) Part 1
15.4 Bimagic Squares (9 x 9) Part 2

4 5 139 142
137 144 2 7
6 3 141 140
143 138 8 1
12 13 131 134
129 136 10 15
14 11 133 132
135 130 16 9
20 21 123 126
121 128 18 23
22 19 125 124
127 122 24 17
28 29 115 118
113 120 26 31
30 27 117 116
119 114 32 25
36 37 107 110
105 112 34 39
38 35 109 108
111 106 40 33
44 45 99 102
97 104 42 47
46 43 101 100
103 98 48 41
52 53 91 94
89 96 50 55
54 51 93 92
95 90 56 49
60 61 83 86
81 88 58 63
62 59 85 84
87 82 64 57
68 69 75 78
73 80 66 71
70 67 77 76
79 74 72 65

16.0 Special Squares, Squares of Squares
16.1 Squares of Squares (3 x 3)
16.2 Squares of Squares (4 x 4)
16.3 Squares of Squares (5 x 5)
16.4 Squares of Squares (6 x 6)
16.5 Squares of Squares (7 x 7)

17.0 Special Squares, Two and Four Way V-ZigZag
17.2 V-Zigzag, Two Way (4 x 4)
17.3 V-Zigzag, Two Way (5 x 5)
17.4 V-Zigzag, Two and Four Way (6 x 6)
17.5 V-Zigzag, Two and Four Way (7 x 7)
17.6 V-Zigzag, Two and Four Way (8 x 8)

262 322 72 512
532 392 42 22
242 382 372 312
172 192 542 282

18.0 Special Squares, Lozenge
18.2 Lozenge Squares (5 x 5)
18.3 Lozenge Squares (7 x 7)
18.4 Lozenge Squares (9 x 9)
18.5 Lozenge Squares (11 x 11)
18.4 Lozenge Squares (13 x 13)
18.5 Lozenge Squares (15 x 15)

List of VBA Subroutines
List of Spreadsheet Solutions
List of Attachments

24 22 5 6 8
12 3 19 15 16
1 9 13 17 25
10 11 7 23 14
18 20 21 4 2

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