Ultra magic 15x15 square

 

Use 2x the same of 2 different magic 5x3 (=3x5) rectangles to construct an ultramagic 15x15 square. The ultramagic 15x15 square is panmagic, symmetric, 3x3 and 5x5 compact.

 

 

= 1x number

                 

6

9

3

4

13

6

9

3

4

13

6

9

3

4

13

14

2

7

12

0

14

2

7

12

0

14

2

7

12

0

1

10

11

5

8

1

10

11

5

8

1

10

11

5

8

6

9

3

4

13

6

9

3

4

13

6

9

3

4

13

14

2

7

12

0

14

2

7

12

0

14

2

7

12

0

1

10

11

5

8

1

10

11

5

8

1

10

11

5

8

6

9

3

4

13

6

9

3

4

13

6

9

3

4

13

14

2

7

12

0

14

2

7

12

0

14

2

7

12

0

1

10

11

5

8

1

10

11

5

8

1

10

11

5

8

6

9

3

4

13

6

9

3

4

13

6

9

3

4

13

14

2

7

12

0

14

2

7

12

0

14

2

7

12

0

1

10

11

5

8

1

10

11

5

8

1

10

11

5

8

6

9

3

4

13

6

9

3

4

13

6

9

3

4

13

14

2

7

12

0

14

2

7

12

0

14

2

7

12

0

1

10

11

5

8

1

10

11

5

8

1

10

11

5

8

 


+ 15x number +1

             

13

0

8

13

0

8

13

0

8

13

0

8

13

0

8

4

12

5

4

12

5

4

12

5

4

12

5

4

12

5

3

7

11

3

7

11

3

7

11

3

7

11

3

7

11

9

2

10

9

2

10

9

2

10

9

2

10

9

2

10

6

14

1

6

14

1

6

14

1

6

14

1

6

14

1

13

0

8

13

0

8

13

0

8

13

0

8

13

0

8

4

12

5

4

12

5

4

12

5

4

12

5

4

12

5

3

7

11

3

7

11

3

7

11

3

7

11

3

7

11

9

2

10

9

2

10

9

2

10

9

2

10

9

2

10

6

14

1

6

14

1

6

14

1

6

14

1

6

14

1

13

0

8

13

0

8

13

0

8

13

0

8

13

0

8

4

12

5

4

12

5

4

12

5

4

12

5

4

12

5

3

7

11

3

7

11

3

7

11

3

7

11

3

7

11

9

2

10

9

2

10

9

2

10

9

2

10

9

2

10

6

14

1

6

14

1

6

14

1

6

14

1

6

14

1

 


= Ultra (pan)magic 15x15 square

           

202

10

124

200

14

127

205

4

125

209

7

130

199

5

134

75

183

83

73

181

90

63

188

88

61

195

78

68

193

76

47

116

177

51

114

167

56

117

171

54

107

176

57

111

174

142

40

154

140

44

157

145

34

155

149

37

160

139

35

164

105

213

23

103

211

30

93

218

28

91

225

18

98

223

16

197

11

132

201

9

122

206

12

126

204

2

131

207

6

129

67

190

79

65

194

82

70

184

80

74

187

85

64

185

89

60

108

173

58

106

180

48

113

178

46

120

168

53

118

166

137

41

162

141

39

152

146

42

156

144

32

161

147

36

159

97

220

19

95

224

22

100

214

20

104

217

25

94

215

29

210

3

128

208

1

135

198

8

133

196

15

123

203

13

121

62

191

87

66

189

77

71

192

81

69

182

86

72

186

84

52

115

169

50

119

172

55

109

170

59

112

175

49

110

179

150

33

158

148

31

165

138

38

163

136

45

153

143

43

151

92

221

27

96

219

17

101

222

21

99

212

26

102

216

24

 

 

It is also possible to use a different magic 5x3 (=3x5) rectangle to construct the second grid:

 

 

9

6

3

13

4

2

14

7

0

12

10

1

11

8

5

 


Use this method to construct magic squares which are an odd multiple of 3 but no multiple of 9 (21x21, 33x33, 39x39, ... magic square). You can find the necessary symmetric 3x7, 3x11 and 3x13 magic rectangles on the website of Aale de Winkel:

http://www.magichypercubes.com/Encyclopedia/DataBase/RectanglesSymmetric3byX.html

 

Download
15x15, Ultra magic.xls
Microsoft Excel werkblad 105.0 KB