Reflecting grids (1)
Look how René Chrétien used reflecting grids to produce a 18x18 magic square. Notify that the second (column) grid is a reflection of the first (row) grid.
1x number + 1
| 0 | 1 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 16 | 17 |
| 0 | 1 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 16 | 17 |
| 0 | 1 | 2 | 14 | 13 | 12 | 11 | 7 | 9 | 8 | 10 | 6 | 5 | 4 | 3 | 15 | 16 | 17 |
| 17 | 1 | 15 | 14 | 13 | 12 | 11 | 7 | 9 | 8 | 10 | 6 | 5 | 4 | 3 | 2 | 16 | 0 |
| 17 | 16 | 2 | 3 | 13 | 12 | 11 | 7 | 8 | 9 | 10 | 6 | 5 | 4 | 14 | 15 | 1 | 0 |
| 17 | 16 | 2 | 3 | 4 | 5 | 11 | 7 | 8 | 9 | 10 | 6 | 12 | 13 | 14 | 15 | 1 | 0 |
| 17 | 16 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 1 | 0 |
| 0 | 16 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 1 | 17 |
| 0 | 16 | 15 | 14 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 3 | 2 | 1 | 17 |
| 0 | 16 | 15 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 2 | 1 | 17 |
| 17 | 16 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 1 | 0 |
| 17 | 16 | 2 | 3 | 4 | 5 | 6 | 10 | 8 | 9 | 7 | 11 | 12 | 13 | 14 | 15 | 1 | 0 |
| 17 | 16 | 2 | 3 | 4 | 5 | 6 | 10 | 8 | 9 | 7 | 11 | 12 | 13 | 14 | 15 | 1 | 0 |
| 17 | 1 | 15 | 3 | 13 | 5 | 6 | 10 | 9 | 8 | 7 | 11 | 12 | 4 | 14 | 2 | 16 | 0 |
| 0 | 1 | 15 | 14 | 4 | 12 | 6 | 10 | 9 | 8 | 7 | 11 | 5 | 13 | 3 | 2 | 16 | 17 |
| 0 | 1 | 2 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 15 | 16 | 17 |
| 17 | 1 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 16 | 0 |
| 0 | 1 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 16 | 17 |
+18x number
| 0 | 0 | 0 | 17 | 17 | 17 | 17 | 0 | 0 | 0 | 17 | 17 | 17 | 17 | 0 | 0 | 17 | 0 |
| 1 | 1 | 1 | 1 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 1 | 1 | 1 | 1 | 1 |
| 15 | 15 | 2 | 15 | 2 | 2 | 2 | 2 | 15 | 15 | 2 | 2 | 2 | 15 | 15 | 2 | 15 | 15 |
| 14 | 14 | 14 | 14 | 3 | 3 | 3 | 3 | 14 | 3 | 3 | 3 | 3 | 3 | 14 | 14 | 14 | 14 |
| 13 | 13 | 13 | 13 | 13 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | 4 | 13 | 13 | 13 |
| 12 | 12 | 12 | 12 | 12 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 12 | 12 | 12 | 12 |
| 11 | 11 | 11 | 11 | 11 | 11 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 11 | 11 | 11 |
| 10 | 10 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 10 | 10 | 10 |
| 9 | 9 | 9 | 9 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 |
| 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 8 | 8 | 8 | 8 | 8 |
| 7 | 7 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
| 6 | 6 | 6 | 6 | 6 | 6 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 6 | 6 | 6 |
| 5 | 5 | 5 | 5 | 5 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 5 | 5 | 5 | 5 |
| 4 | 4 | 4 | 4 | 4 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 4 | 13 | 4 | 4 | 4 |
| 3 | 3 | 3 | 3 | 14 | 14 | 14 | 14 | 3 | 14 | 14 | 14 | 14 | 14 | 3 | 3 | 3 | 3 |
| 2 | 2 | 15 | 2 | 15 | 15 | 15 | 15 | 2 | 2 | 15 | 15 | 15 | 2 | 2 | 15 | 2 | 2 |
| 16 | 16 | 16 | 16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 16 | 16 | 16 | 16 | 16 |
| 17 | 17 | 17 | 0 | 0 | 0 | 0 | 17 | 17 | 17 | 0 | 0 | 0 | 0 | 17 | 17 | 0 | 17 |
= (simple) 18x18 magic square
| 1 | 2 | 16 | 321 | 320 | 319 | 318 | 11 | 10 | 9 | 314 | 313 | 312 | 311 | 4 | 3 | 323 | 18 |
| 19 | 20 | 34 | 33 | 302 | 301 | 300 | 299 | 298 | 297 | 296 | 295 | 294 | 23 | 22 | 21 | 35 | 36 |
| 271 | 272 | 39 | 285 | 50 | 49 | 48 | 44 | 280 | 279 | 47 | 43 | 42 | 275 | 274 | 52 | 287 | 288 |
| 270 | 254 | 268 | 267 | 68 | 67 | 66 | 62 | 262 | 63 | 65 | 61 | 60 | 59 | 256 | 255 | 269 | 253 |
| 252 | 251 | 237 | 238 | 248 | 85 | 84 | 80 | 81 | 82 | 83 | 79 | 78 | 239 | 87 | 250 | 236 | 235 |
| 234 | 233 | 219 | 220 | 221 | 96 | 102 | 98 | 99 | 100 | 101 | 97 | 103 | 104 | 231 | 232 | 218 | 217 |
| 216 | 215 | 201 | 202 | 203 | 204 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 214 | 200 | 199 |
| 181 | 197 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 192 | 193 | 194 | 195 | 196 | 182 | 198 |
| 163 | 179 | 178 | 177 | 149 | 150 | 151 | 152 | 153 | 154 | 155 | 156 | 157 | 176 | 166 | 165 | 164 | 180 |
| 145 | 161 | 160 | 148 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 | 158 | 159 | 147 | 146 | 162 |
| 144 | 143 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 138 | 139 | 140 | 141 | 142 | 128 | 127 |
| 126 | 125 | 111 | 112 | 113 | 114 | 205 | 209 | 207 | 208 | 206 | 210 | 211 | 212 | 213 | 124 | 110 | 109 |
| 108 | 107 | 93 | 94 | 95 | 222 | 223 | 227 | 225 | 226 | 224 | 228 | 229 | 230 | 105 | 106 | 92 | 91 |
| 90 | 74 | 88 | 76 | 86 | 240 | 241 | 245 | 244 | 243 | 242 | 246 | 247 | 77 | 249 | 75 | 89 | 73 |
| 55 | 56 | 70 | 69 | 257 | 265 | 259 | 263 | 64 | 261 | 260 | 264 | 258 | 266 | 58 | 57 | 71 | 72 |
| 37 | 38 | 273 | 51 | 284 | 283 | 282 | 281 | 46 | 45 | 278 | 277 | 276 | 41 | 40 | 286 | 53 | 54 |
| 306 | 290 | 304 | 303 | 32 | 31 | 30 | 29 | 28 | 27 | 26 | 25 | 24 | 293 | 292 | 291 | 305 | 289 |
| 307 | 308 | 322 | 15 | 14 | 13 | 12 | 317 | 316 | 315 | 8 | 7 | 6 | 5 | 310 | 309 | 17 | 324 |
Use the method of reflecting grids (1) to construct magic squares of order is double odd. See 6x6, 10x10, 14x14, 18x18, 22x22, 26x26 en 30x30
