The perfect magic square

 

 The perfect magic square must be a 16x16 magic square. I put an almost perfect 16x16 magic square on my website and asked for a challenge.

 

Ot Ottenheim from the Netherlands won the challenge and produced the real perfect magic square below. 

 

 

The perfect magic square

1

240

84

189

2

239

83

190

3

238

82

191

4

237

81

192

224

49

141

100

223

50

142

99

222

51

143

98

221

52

144

97

173

68

256

17

174

67

255

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175

66

254

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176

65

253

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116

157

33

208

115

158

34

207

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113

160

36

205

5

236

88

185

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235

87

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234

86

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233

85

188

220

53

137

104

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138

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139

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140

101

169

72

252

21

170

71

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171

70

250

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172

69

249

24

120

153

37

204

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38

203

118

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39

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117

156

40

201

9

232

92

181

10

231

91

182

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230

90

183

12

229

89

184

216

57

133

108

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134

107

214

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135

106

213

60

136

105

165

76

248

25

166

75

247

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167

74

246

27

168

73

245

28

124

149

41

200

123

150

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199

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198

121

152

44

197

13

228

96

177

14

227

95

178

15

226

94

179

16

225

93

180

212

61

129

112

211

62

130

111

210

63

131

110

209

64

132

109

161

80

244

29

162

79

243

30

163

78

242

31

164

77

241

32

128

145

45

196

127

146

46

195

126

147

47

194

125

148

48

193

 

 

This magic 16x16 square is the real perfect magic square, because of the following magic features:

 

(1) The magic square consists of four by four proportional panmagic 4x4 sub-squares and because of that structure each1/4 row/column/diagonal gives 1/4 of the magic sum. The 4x4 sub-squares are perfect connected, so the 16x16 magic square is panmagic and fully 2x2 compact (which means that each random chosen 2x2 sub-square gives 1/4 of the magic sum = 514). Conclusion is that the 16x16 magic square is (Franklin panmagic) most perfect.

 

(2) The magic square has got the tight 'Willem Barink' structure. Horizontal the sum of 2 digits (of cell 1+2, 3+4, 5+6, 7+8. 9+10, 11+12, 13+14 and 15+16) gives 241 respectively 273. Vertical the sum of 2 digits (of cell 1+2, 3+4, 5+6, 7+8. 9+10, 11+12, 13+14 and 15+16) gives 225 respectively 289.

 

 

241

273

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241

 

 

225

289

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225

 
 

 

 

(3) In each 4x4 sub-square you can find a digit from each of the 16 sequences, 1 up to 16, 17 up to 32, 33 up to 48, 49 up to 64, 65 up to 80, 81 up to 96, 97 up to 112, 113 up to 128, 129 up to 144, 145 up to 160, 161 up to 176, 177 up to 192, 193 up to 208, 209 up to 224, 225 up to 240 and 241 up to 256. The digits of each sequence are in order from low to high, starting from one of the four corners. And as finishing touch, the first four sequences start from the top left corner, the second four sequences start from the top right corner, the third four sequences start from the down left corner and the fourth four sequences start from the down right corner.

 

A more perfect square does not exist.

 

For analysis of the perfect magic square, see the download below. 

 

Download
16x16, the perfect magic square of Ot Ot
Microsoft Excel werkblad 613.0 KB

 

To produce an Ot Ottenheim perfect magic square for each order is multiple of 4, from 8x8 (to infinite), see download (developed from 8x8 up to 32x32) below.

 

Download
Ot Ottenheim perfect magic squares, orde
Microsoft Excel werkblad 180.3 KB