A Special Order 13 Magic Square:
This is an order 13 magic square with the normal “magic
square properties”, but it also has some special additions.
157

13

23

147

109

31

111

138

36

66

102

100

72

145

25

17

153

61

139

59

32

134

104

68

98

70

16

154

144

26

57

56

30

112

136

99

105

60

110

22

148

156

14

113

114

140

58

34

65

71

133

37

97

73

94

76

151

18

21

89

146

135

35

29

141

79

91

78

92

27

82

150

155

11

63

107

33

137

74

96

75

95

143

159

15

20

88

115

55

101

69

90

80

93

77

19

24

81

149

152

54

116

103

67

164

6

3

167

85

142

158

12

28

64

106

108

62

7

163

168

86

1

132

44

39

125

50

48

118

124

162

8

84

2

169

38

126

131

45

120

122

52

46

5

83

161

10

166

129

43

40

128

123

117

49

51

87

165

9

160

4

41

127

130

42

47

53

121

119

The square has 169 spaces for number, and contains all
of the numbers from 1 to 169 (13 squared) one time each. All of the rows have the same sum (1105), all
of the columns have the same sum (1105), and both of the main diagonals have
the same sum.
What makes it special?
It also contains 12 smaller magic squares (well, one of them is only
semimagic because only one of its diagonals adds up to the magic constant for
that square) ranging in size from 3 x 3 to 11 x 11. So this magic square is really a total of 13
magic squares.
A 3 x3 semimagic square with 3 in the
upper left corner. All rows and columns,
but only 1 main diagonal have a magic constant of 255.
Four 4 x 4 magic squares with 157, 97, 132
or 50 in the upper left corner of each square.
All rows, columns and the two main diagonals sum to 340.
Two 5 x 5 magic squares with 164 or 151 in
the upper left corner. All rows, columns
and the two main diagonals sum to 425.
Two 7 x 7 magic squares with 94 or 57 in
the upper left corner. All rows,
columns, and the two main diagonals sum to 595.
Two 9 x 9 magic squares with 109 and 97 in
the upper left corner. Their magic
constant is 765.
An 11 x 11 magic square with 23 in the
upper left corner. Its magic constant is
935.
And finally, a 13 x 13 magic square with
157 in the upper left corner (the entire array). Its magic constant is 1105.
My
daughter was born on the 13th of the month, turned 13 (became a
teenager) on a Friday the 13th (with 12 friends attending a sleepover,
all 13 dressed in black). The last time I warned her about a Friday the
13th coming up (so that she could avoid hospital duty that nigh) she
said "Why? That's when all the fun stuff happens." What can I say 
she has a thing for the number 13.
David
Hi,
ReplyDeleteQuestion is how many 13x13 tables exist? Any information is greatly appreciated.
Thank you,
RFR