# Franklin Magic Squares of order 4k

It turns out that Franklin Magic Squares exist except for n=4 and n=12. An extensive report about general construction methods for Franklin Squares is found in an internal TUe report, in pdf format. It also describes the brute force search for a 12 by 12 square which failed, thereby proving that no such square exists. It takes merely 160 hours of computing time. A program in C is available at gen12x.c. I have given a presentation about Franklin magic squares at the Dutch Cube Day (Eindhoven, October 14, 2007). Here you find the slides in PDF-format. The report has many examples of nice Franklin or almost Franklin Squares, some of which can be found below.

## Magic square built by 4x4 perfect blocks

 1 140 109 40 9 132 101 48 49 92 61 88 143 6 35 106 135 14 43 98 95 54 83 58 36 105 144 5 44 97 136 13 84 57 96 53 110 39 2 139 102 47 10 131 62 87 50 91 3 138 111 38 11 130 103 46 51 90 63 86 141 8 33 108 133 16 41 100 93 56 81 60 34 107 142 7 42 99 134 15 82 59 94 55 112 37 4 137 104 45 12 129 64 85 52 89 17 124 125 24 25 116 117 32 65 76 77 72 127 22 19 122 119 30 27 114 79 70 67 74 20 121 128 21 28 113 120 29 68 73 80 69 126 23 18 123 118 31 26 115 78 71 66 75
4x4 blocks are pan-diagonal, contain complementary entries;
lines of 4 entries, starting at odd row/column are magic;
each 8x8 subsquare is Franklin.

## Franklin 12x12, except for magic half columns

 4 143 10 139 78 61 5 142 11 133 76 68 105 38 99 42 31 120 104 39 98 48 33 113 52 95 58 91 126 13 53 94 59 85 124 20 129 14 123 18 55 96 128 15 122 24 57 89 28 119 34 115 102 37 29 118 35 109 100 44 81 62 75 66 7 144 80 63 74 72 9 137 64 83 70 79 138 1 65 82 71 73 136 8 117 26 111 30 43 108 116 27 110 36 45 101 16 131 22 127 90 49 17 130 23 121 88 56 93 50 87 54 19 132 92 51 86 60 21 125 40 107 46 103 114 25 41 106 47 97 112 32 141 2 135 6 67 84 140 3 134 12 69 77
Instead of magic half columns, we have magic third columns;
complementary entries reflect along horizontal midline.

## Franklin Square of order 20

 1 395 21 320 141 240 326 180 341 40 86 360 101 295 181 280 201 160 261 80 386 20 366 95 246 175 61 235 46 375 301 55 286 120 206 135 186 255 126 335 2 394 22 319 142 239 327 179 342 39 87 359 102 294 182 279 202 159 262 79 388 18 368 93 248 173 63 233 48 373 303 53 288 118 208 133 188 253 128 333 11 385 31 310 151 230 336 170 351 30 96 350 111 285 191 270 211 150 271 70 396 10 376 85 256 165 71 225 56 365 311 45 296 110 216 125 196 245 136 325 12 384 32 309 152 229 337 169 352 29 97 349 112 284 192 269 212 149 272 69 397 9 377 84 257 164 72 224 57 364 312 44 297 109 217 124 197 244 137 324 14 382 34 307 154 227 339 167 354 27 99 347 114 282 194 267 214 147 274 67 398 8 378 83 258 163 73 223 58 363 313 43 298 108 218 123 198 243 138 323 3 393 23 318 143 238 328 178 343 38 88 358 103 293 183 278 203 158 263 78 387 19 367 94 247 174 62 234 47 374 302 54 287 119 207 134 187 254 127 334 4 392 24 317 144 237 329 177 344 37 89 357 104 292 184 277 204 157 264 77 389 17 369 92 249 172 64 232 49 372 304 52 289 117 209 132 189 252 129 332 5 391 25 316 145 236 330 176 345 36 90 356 105 291 185 276 205 156 265 76 390 16 370 91 250 171 65 231 50 371 305 51 290 116 210 131 190 251 130 331 13 383 33 308 153 228 338 168 353 28 98 348 113 283 193 268 213 148 273 68 399 7 379 82 259 162 74 222 59 362 314 42 299 107 219 122 199 242 139 322 15 381 35 306 155 226 340 166 355 26 100 346 115 281 195 266 215 146 275 66 400 6 380 81 260 161 75 221 60 361 315 41 300 106 220 121 200 241 140 321
True Franklin Magic Square of order 20;
magic half rows, magic half columns, magic bent-diagonals;
complementarity along horizontal midline