Shift method (2)

 

It is possible to use the shiftmethod to get a more tight structure of the panmagic 15x15 square.

 

You have to puzzle to get the first row. Construct a 3x5 = 5x3 matrix:

 

 

  Matrix 3x5                               =         Matrix 5x3 

0

9

12

 

21

   

0

9

12

 

21

1

14

6

 

21

   

1

14

6

 

21

11

2

8

 

21

   

11

2

8

 

21

13

3

5

 

21

   

13

3

5

 

21

10

7

4

 

21

   

10

7

4

 

21

                       

35

35

35

       

35

35

35

   

 

 

The magic sum of 0 up to 14 is 105. In the matrix the sum of each column is (5/15 x 105 =) 35 and the sum of each row is (3/15 x 105 =) 21 is. Put the digits in the first row:

 

 First row according to 3x5 matrix

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

 

 

 First row according to 5x3 matrix

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

 

 

Construct row 2 up to 15 by shifting the first row each time 4 places to the left.

 

 

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

     

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

             

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

                     

 

 

We have constructed the first grid. The second grid is a reflection (rotated by a quarter and mirrored) of the first grid. Take 15x digit +1 from first grid and add 1x digit from second grid to get a 15x15 panmagic square.

 

 

 Take 15x digit +1 from first grid

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

5

9

11

4

14

13

12

2

10

6

3

0

8

7

1

14

13

12

2

10

6

3

0

8

7

1

5

9

11

4

10

6

3

0

8

7

1

5

9

11

4

14

13

12

2

8

7

1

5

9

11

4

14

13

12

2

10

6

3

0

9

11

4

14

13

12

2

10

6

3

0

8

7

1

5

13

12

2

10

6

3

0

8

7

1

5

9

11

4

14

6

3

0

8

7

1

5

9

11

4

14

13

12

2

10

7

1

5

9

11

4

14

13

12

2

10

6

3

0

8

11

4

14

13

12

2

10

6

3

0

8

7

1

5

9

12

2

10

6

3

0

8

7

1

5

9

11

4

14

13

3

0

8

7

1

5

9

11

4

14

13

12

2

10

6

1

5

9

11

4

14

13

12

2

10

6

3

0

8

7

4

14

13

12

2

10

6

3

0

8

7

1

5

9

11

2

10

6

3

0

8

7

1

5

9

11

4

14

13

12

 

 

 + 1x digit from second grid (= reflection of first grid)

3

1

4

2

0

5

14

10

8

9

13

6

7

11

12

6

7

11

12

3

1

4

2

0

5

14

10

8

9

13

10

8

9

13

6

7

11

12

3

1

4

2

0

5

14

2

0

5

14

10

8

9

13

6

7

11

12

3

1

4

12

3

1

4

2

0

5

14

10

8

9

13

6

7

11

13

6

7

11

12

3

1

4

2

0

5

14

10

8

9

14

10

8

9

13

6

7

11

12

3

1

4

2

0

5

4

2

0

5

14

10

8

9

13

6

7

11

12

3

1

11

12

3

1

4

2

0

5

14

10

8

9

13

6

7

9

13

6

7

11

12

3

1

4

2

0

5

14

10

8

5

14

10

8

9

13

6

7

11

12

3

1

4

2

0

1

4

2

0

5

14

10

8

9

13

6

7

11

12

3

7

11

12

3

1

4

2

0

5

14

10

8

9

13

6

8

9

13

6

7

11

12

3

1

4

2

0

5

14

10

0

5

14

10

8

9

13

6

7

11

12

3

1

4

2

 

 

 = 15x15 panmagic square 

4

122

110

18

76

141

180

71

219

205

194

37

158

102

58

82

143

177

73

214

197

185

33

151

96

60

11

129

115

29

221

204

190

44

157

98

57

13

124

107

20

78

136

171

75

153

91

51

15

131

114

25

89

142

173

72

223

199

182

35

133

109

17

80

138

166

66

225

206

189

40

164

97

53

12

149

172

68

222

208

184

32

155

93

46

6

135

116

24

85

210

191

39

160

104

52

8

132

118

19

77

140

168

61

216

95

48

1

126

120

26

84

145

179

67

218

207

193

34

152

117

28

79

137

170

63

211

201

195

41

159

100

59

7

128

175

74

217

203

192

43

154

92

50

3

121

111

30

86

144

186

45

161

99

55

14

127

113

27

88

139

167

65

213

196

47

5

123

106

21

90

146

174

70

224

202

188

42

163

94

23

87

148

169

62

215

198

181

36

165

101

54

10

134

112

69

220

209

187

38

162

103

49

2

125

108

16

81

150

176

31

156

105

56

9

130

119

22

83

147

178

64

212

200

183

 

 

Use this method to construct magic squares which are an odd multiple of 3 but no multiple of 9 (= 15x15, 21x21, 33x33, 39x39, ... magic square).

 

 

Download
15x15, shift method (2).xls
Microsoft Excel werkblad 116.0 KB