### 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 18 175 66 254 19 176 65 253 20 116 157 33 208 115 158 34 207 114 159 35 206 113 160 36 205 5 236 88 185 6 235 87 186 7 234 86 187 8 233 85 188 220 53 137 104 219 54 138 103 218 55 139 102 217 56 140 101 169 72 252 21 170 71 251 22 171 70 250 23 172 69 249 24 120 153 37 204 119 154 38 203 118 155 39 202 117 156 40 201 9 232 92 181 10 231 91 182 11 230 90 183 12 229 89 184 216 57 133 108 215 58 134 107 214 59 135 106 213 60 136 105 165 76 248 25 166 75 247 26 167 74 246 27 168 73 245 28 124 149 41 200 123 150 42 199 122 151 43 198 121 152 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 each 1/4 row/column/diagonal gives 1/4 of the magic sum. The 4x4 sub-squares are perfectly 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). The conclusion is that the 16x16 magic square is (Franklin panmagic) most perfect.

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

 241 273 241 273 241 273 241 273 273 241 273 241 273 241 273 241 241 273 241 273 241 273 241 273 273 241 273 241 273 241 273 241 241 273 241 273 241 273 241 273 273 241 273 241 273 241 273 241 241 273 241 273 241 273 241 273 273 241 273 241 273 241 273 241 241 273 241 273 241 273 241 273 273 241 273 241 273 241 273 241 241 273 241 273 241 273 241 273 273 241 273 241 273 241 273 241 241 273 241 273 241 273 241 273 273 241 273 241 273 241 273 241 241 273 241 273 241 273 241 273 273 241 273 241 273 241 273 241

 225 289 225 289 225 289 225 289 225 289 225 289 225 289 225 289 289 225 289 225 289 225 289 225 289 225 289 225 289 225 289 225 225 289 225 289 225 289 225 289 225 289 225 289 225 289 225 289 289 225 289 225 289 225 289 225 289 225 289 225 289 225 289 225 225 289 225 289 225 289 225 289 225 289 225 289 225 289 225 289 289 225 289 225 289 225 289 225 289 225 289 225 289 225 289 225 225 289 225 289 225 289 225 289 225 289 225 289 225 289 225 289 289 225 289 225 289 225 289 225 289 225 289 225 289 225 289 225

(3) In each 4x4 sub-square you can find a number 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 respectively. The numbers of each sequence are in order from low to high, starting from one of the four corners. And as a 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 do bottom left corner and the fourth four sequences start from the bottom right corner.

A more perfect square does not exist.

16x16, the perfect magic square.xls