Use the famous Khajuraho 4x4 panmagic square to construct larger magic squares which are a multiple of 4 (= 8x8, 12x12, 16x16, 20x20, … magic square).
Rewrite the Khajuraho magic square as follows:
Khajuraho magic square Basic magic square
7 |
12 |
1 |
14 |
7 |
h-4 |
1 |
h-2 |
||
2 |
13 |
8 |
11 |
2 |
h-3 |
8 |
h-5 |
||
16 |
3 |
10 |
5 |
h |
3 |
h-6 |
5 |
||
9 |
6 |
15 |
4 |
h-7 |
6 |
h-1 |
4 |
To construct an 24x24 panmagic square, you need the basic square and 35 extending magic squares:
7
|
h-4 | 1 | h-2 | 8 | -8 | 8 | -8 | 16 | -16 | 16 | -16 | 24 | -24 | 24 | -24 | 32 | -32 | 32 | -32 | 40 | -40 | 40 | -40 |
2 | h-3 | 8 | h-5 | 8 | -8 | 8 | -8 | 16 | -16 | 16 | -16 | 24 | -24 | 24 | -24 | 32 | -32 | 32 | -32 | 40 | -40 | 40 | -40 |
h | 3 | h-6 | 5 | -8 | 8 | -8 | 8 | -16 | 16 | -16 | 16 | -24 | 24 | -24 | 24 | -32 | 32 | -32 | 32 | -40 | 40 | -40 | 40 |
h-7 | 6 | h-1 | 4 | -8 | 8 | -8 | 8 | -16 | 16 | -16 | 16 | -24 | 24 | -24 | 24 | -32 | 32 | -32 | 32 | -40 | 40 | -40 | 40 |
48 | -48 | 48 | -48 | 56 | -56 | 56 | -56 | 64 | -64 | 64 | -64 | 72 | -72 | 72 | -72 | 80 | -80 | 80 | -80 | 88 | -88 | 88 | -88 |
48 | -48 | 48 | -48 | 56 | -56 | 56 | -56 | 64 | -64 | 64 | -64 | 72 | -72 | 72 | -72 | 80 | -80 | 80 | -80 | 88 | -88 | 88 | -88 |
-48 | 48 | -48 | 48 | -56 | 56 | -56 | 56 | -64 | 64 | -64 | 64 | -72 | 72 | -72 | 72 | -80 | 80 | -80 | 80 | -88 | 88 | -88 | 88 |
-48 | 48 | -48 | 48 | -56 | 56 | -56 | 56 | -64 | 64 | -64 | 64 | -72 | 72 | -72 | 72 | -80 | 80 | -80 | 80 | -88 | 88 | -88 | 88 |
96 | -96 | 96 | -96 | 104 | -104 | 104 | -104 | 112 | -112 | 112 | -112 | 120 | -120 | 120 | -120 | 128 | -128 | 128 | -128 | 136 | -136 | 136 | -136 |
96 | -96 | 96 | -96 | 104 | -104 | 104 | -104 | 112 | -112 | 112 | -112 | 120 | -120 | 120 | -120 | 128 | -128 | 128 | -128 | 136 | -136 | 136 | -136 |
-96 | 96 | -96 | 96 | -104 | 104 | -104 | 104 | -112 | 112 | -112 | 112 | -120 | 120 | -120 | 120 | -128 | 128 | -128 | 128 | -136 | 136 | -136 | 136 |
-96 | 96 | -96 | 96 | -104 | 104 | -104 | 104 | -112 | 112 | -112 | 112 | -120 | 120 | -120 | 120 | -128 | 128 | -128 | 128 | -136 | 136 | -136 | 136 |
144 | -144 | 144 | -144 | 152 | -152 | 152 | -152 | 160 | -160 | 160 | -160 | 168 | -168 | 168 | -168 | 176 | -176 | 176 | -176 | 184 | -184 | 184 | -184 |
144 | -144 | 144 | -144 | 152 | -152 | 152 | -152 | 160 | -160 | 160 | -160 | 168 | -168 | 168 | -168 | 176 | -176 | 176 | -176 | 184 | -184 | 184 | -184 |
-144 | 144 | -144 | 144 | -152 | 152 | -152 | 152 | -160 | 160 | -160 | 160 | -168 | 168 | -168 | 168 | -176 | 176 | -176 | 176 | -184 | 184 | -184 | 184 |
-144 | 144 | -144 | 144 | -152 | 152 | -152 | 152 | -160 | 160 | -160 | 160 | -168 | 168 | -168 | 168 | -176 | 176 | -176 | 176 | -184 | 184 | -184 | 184 |
192 | -192 | 192 | -192 | 200 | -200 | 200 | -200 | 208 | -208 | 208 | -208 | 216 | -216 | 216 | -216 | 224 | -224 | 224 | -224 | 232 | -232 | 232 | -232 |
192 | -192 | 192 | -192 | 200 | -200 | 200 | -200 | 208 | -208 | 208 | -208 | 216 | -216 | 216 | -216 | 224 | -224 | 224 | -224 | 232 | -232 | 232 | -232 |
-192 | 192 | -192 | 192 | -200 | 200 | -200 | 200 | -208 | 208 | -208 | 208 | -216 | 216 | -216 | 216 | -224 | 224 | -224 | 224 | -232 | 232 | -232 | 232 |
-192 | 192 | -192 | 192 | -200 | 200 | -200 | 200 | -208 | 208 | -208 | 208 | -216 | 216 | -216 | 216 | -224 | 224 | -224 | 224 | -232 | 232 | -232 | 232 |
240 | -240 | 240 | -240 | 248 | -248 | 248 | -248 | 256 | -256 | 256 | -256 | 264 | -264 | 264 | -264 | 272 | -272 | 272 | -272 | 280 | -280 | 280 | -280 |
240 | -240 | 240 | -240 | 248 | -248 | 248 | -248 | 256 | -256 | 256 | -256 | 264 | -264 | 264 | -264 | 272 | -272 | 272 | -272 | 280 | -280 | 280 | -280 |
-240 | 240 | -240 | 240 | -248 | 248 | -248 | 248 | -256 | 256 | -256 | 256 | -264 | 264 | -264 | 264 | -272 | 272 | -272 | 272 | -280 | 280 | -280 | 280 |
-240 | 240 | -240 | 240 | -248 | 248 | -248 | 248 | -256 | 256 | -256 | 256 | -264 | 264 | -264 | 264 | -272 | 272 | -272 | 272 | -280 | 280 | -280 | 280 |
The highest number in the 24x24 square is 576. Fill in 576 for h and calculate all the numbers. You get the following 24x24 panmagic square.
Panmagic 24x24 square
7 | 572 | 1 | 574 | 15 | 564 | 9 | 566 | 23 | 556 | 17 | 558 | 31 | 548 | 25 | 550 | 39 | 540 | 33 | 542 | 47 | 532 | 41 | 534 |
2 | 573 | 8 | 571 | 10 | 565 | 16 | 563 | 18 | 557 | 24 | 555 | 26 | 549 | 32 | 547 | 34 | 541 | 40 | 539 | 42 | 533 | 48 | 531 |
576 | 3 | 570 | 5 | 568 | 11 | 562 | 13 | 560 | 19 | 554 | 21 | 552 | 27 | 546 | 29 | 544 | 35 | 538 | 37 | 536 | 43 | 530 | 45 |
569 | 6 | 575 | 4 | 561 | 14 | 567 | 12 | 553 | 22 | 559 | 20 | 545 | 30 | 551 | 28 | 537 | 38 | 543 | 36 | 529 | 46 | 535 | 44 |
55 | 524 | 49 | 526 | 63 | 516 | 57 | 518 | 71 | 508 | 65 | 510 | 79 | 500 | 73 | 502 | 87 | 492 | 81 | 494 | 95 | 484 | 89 | 486 |
50 | 525 | 56 | 523 | 58 | 517 | 64 | 515 | 66 | 509 | 72 | 507 | 74 | 501 | 80 | 499 | 82 | 493 | 88 | 491 | 90 | 485 | 96 | 483 |
528 | 51 | 522 | 53 | 520 | 59 | 514 | 61 | 512 | 67 | 506 | 69 | 504 | 75 | 498 | 77 | 496 | 83 | 490 | 85 | 488 | 91 | 482 | 93 |
521 | 54 | 527 | 52 | 513 | 62 | 519 | 60 | 505 | 70 | 511 | 68 | 497 | 78 | 503 | 76 | 489 | 86 | 495 | 84 | 481 | 94 | 487 | 92 |
103 | 476 | 97 | 478 | 111 | 468 | 105 | 470 | 119 | 460 | 113 | 462 | 127 | 452 | 121 | 454 | 135 | 444 | 129 | 446 | 143 | 436 | 137 | 438 |
98 | 477 | 104 | 475 | 106 | 469 | 112 | 467 | 114 | 461 | 120 | 459 | 122 | 453 | 128 | 451 | 130 | 445 | 136 | 443 | 138 | 437 | 144 | 435 |
480 | 99 | 474 | 101 | 472 | 107 | 466 | 109 | 464 | 115 | 458 | 117 | 456 | 123 | 450 | 125 | 448 | 131 | 442 | 133 | 440 | 139 | 434 | 141 |
473 | 102 | 479 | 100 | 465 | 110 | 471 | 108 | 457 | 118 | 463 | 116 | 449 | 126 | 455 | 124 | 441 | 134 | 447 | 132 | 433 | 142 | 439 | 140 |
151 | 428 | 145 | 430 | 159 | 420 | 153 | 422 | 167 | 412 | 161 | 414 | 175 | 404 | 169 | 406 | 183 | 396 | 177 | 398 | 191 | 388 | 185 | 390 |
146 | 429 | 152 | 427 | 154 | 421 | 160 | 419 | 162 | 413 | 168 | 411 | 170 | 405 | 176 | 403 | 178 | 397 | 184 | 395 | 186 | 389 | 192 | 387 |
432 | 147 | 426 | 149 | 424 | 155 | 418 | 157 | 416 | 163 | 410 | 165 | 408 | 171 | 402 | 173 | 400 | 179 | 394 | 181 | 392 | 187 | 386 | 189 |
425 | 150 | 431 | 148 | 417 | 158 | 423 | 156 | 409 | 166 | 415 | 164 | 401 | 174 | 407 | 172 | 393 | 182 | 399 | 180 | 385 | 190 | 391 | 188 |
199 | 380 | 193 | 382 | 207 | 372 | 201 | 374 | 215 | 364 | 209 | 366 | 223 | 356 | 217 | 358 | 231 | 348 | 225 | 350 | 239 | 340 | 233 | 342 |
194 | 381 | 200 | 379 | 202 | 373 | 208 | 371 | 210 | 365 | 216 | 363 | 218 | 357 | 224 | 355 | 226 | 349 | 232 | 347 | 234 | 341 | 240 | 339 |
384 | 195 | 378 | 197 | 376 | 203 | 370 | 205 | 368 | 211 | 362 | 213 | 360 | 219 | 354 | 221 | 352 | 227 | 346 | 229 | 344 | 235 | 338 | 237 |
377 | 198 | 383 | 196 | 369 | 206 | 375 | 204 | 361 | 214 | 367 | 212 | 353 | 222 | 359 | 220 | 345 | 230 | 351 | 228 | 337 | 238 | 343 | 236 |
247 | 332 | 241 | 334 | 255 | 324 | 249 | 326 | 263 | 316 | 257 | 318 | 271 | 308 | 265 | 310 | 279 | 300 | 273 | 302 | 287 | 292 | 281 | 294 |
242 | 333 | 248 | 331 | 250 | 325 | 256 | 323 | 258 | 317 | 264 | 315 | 266 | 309 | 272 | 307 | 274 | 301 | 280 | 299 | 282 | 293 | 288 | 291 |
336 | 243 | 330 | 245 | 328 | 251 | 322 | 253 | 320 | 259 | 314 | 261 | 312 | 267 | 306 | 269 | 304 | 275 | 298 | 277 | 296 | 283 | 290 | 285 |
329 | 246 | 335 | 244 | 321 | 254 | 327 | 252 | 313 | 262 | 319 | 260 | 305 | 270 | 311 | 268 | 297 | 278 | 303 | 276 | 289 | 286 | 295 | 284 |
This magic square is almost Franklin panmagic. Only not all 2x2 sub-squares give 1/6 of the magic sum (1/6 x 6924 = 1154). If you swap the colours you get the following most perfect (Franklin pan)magic 24x24 square:
Most perfect (Franklin pan)magic 24x24 square
47 | 572 | 1 | 534 | 39 | 564 | 9 | 542 | 31 | 556 | 17 | 550 | 23 | 548 | 25 | 558 | 15 | 540 | 33 | 566 | 7 | 532 | 41 | 574 |
2 | 533 | 48 | 571 | 10 | 541 | 40 | 563 | 18 | 549 | 32 | 555 | 26 | 557 | 24 | 547 | 34 | 565 | 16 | 539 | 42 | 573 | 8 | 531 |
576 | 43 | 530 | 5 | 568 | 35 | 538 | 13 | 560 | 27 | 546 | 21 | 552 | 19 | 554 | 29 | 544 | 11 | 562 | 37 | 536 | 3 | 570 | 45 |
529 | 6 | 575 | 44 | 537 | 14 | 567 | 36 | 545 | 22 | 559 | 28 | 553 | 30 | 551 | 20 | 561 | 38 | 543 | 12 | 569 | 46 | 535 | 4 |
95 | 524 | 49 | 486 | 87 | 516 | 57 | 494 | 79 | 508 | 65 | 502 | 71 | 500 | 73 | 510 | 63 | 492 | 81 | 518 | 55 | 484 | 89 | 526 |
50 | 485 | 96 | 523 | 58 | 493 | 88 | 515 | 66 | 501 | 80 | 507 | 74 | 509 | 72 | 499 | 82 | 517 | 64 | 491 | 90 | 525 | 56 | 483 |
528 | 91 | 482 | 53 | 520 | 83 | 490 | 61 | 512 | 75 | 498 | 69 | 504 | 67 | 506 | 77 | 496 | 59 | 514 | 85 | 488 | 51 | 522 | 93 |
481 | 54 | 527 | 92 | 489 | 62 | 519 | 84 | 497 | 70 | 511 | 76 | 505 | 78 | 503 | 68 | 513 | 86 | 495 | 60 | 521 | 94 | 487 | 52 |
143 | 476 | 97 | 438 | 135 | 468 | 105 | 446 | 127 | 460 | 113 | 454 | 119 | 452 | 121 | 462 | 111 | 444 | 129 | 470 | 103 | 436 | 137 | 478 |
98 | 437 | 144 | 475 | 106 | 445 | 136 | 467 | 114 | 453 | 128 | 459 | 122 | 461 | 120 | 451 | 130 | 469 | 112 | 443 | 138 | 477 | 104 | 435 |
480 | 139 | 434 | 101 | 472 | 131 | 442 | 109 | 464 | 123 | 450 | 117 | 456 | 115 | 458 | 125 | 448 | 107 | 466 | 133 | 440 | 99 | 474 | 141 |
433 | 102 | 479 | 140 | 441 | 110 | 471 | 132 | 449 | 118 | 463 | 124 | 457 | 126 | 455 | 116 | 465 | 134 | 447 | 108 | 473 | 142 | 439 | 100 |
191 | 428 | 145 | 390 | 183 | 420 | 153 | 398 | 175 | 412 | 161 | 406 | 167 | 404 | 169 | 414 | 159 | 396 | 177 | 422 | 151 | 388 | 185 | 430 |
146 | 389 | 192 | 427 | 154 | 397 | 184 | 419 | 162 | 405 | 176 | 411 | 170 | 413 | 168 | 403 | 178 | 421 | 160 | 395 | 186 | 429 | 152 | 387 |
432 | 187 | 386 | 149 | 424 | 179 | 394 | 157 | 416 | 171 | 402 | 165 | 408 | 163 | 410 | 173 | 400 | 155 | 418 | 181 | 392 | 147 | 426 | 189 |
385 | 150 | 431 | 188 | 393 | 158 | 423 | 180 | 401 | 166 | 415 | 172 | 409 | 174 | 407 | 164 | 417 | 182 | 399 | 156 | 425 | 190 | 391 | 148 |
239 | 380 | 193 | 342 | 231 | 372 | 201 | 350 | 223 | 364 | 209 | 358 | 215 | 356 | 217 | 366 | 207 | 348 | 225 | 374 | 199 | 340 | 233 | 382 |
194 | 341 | 240 | 379 | 202 | 349 | 232 | 371 | 210 | 357 | 224 | 363 | 218 | 365 | 216 | 355 | 226 | 373 | 208 | 347 | 234 | 381 | 200 | 339 |
384 | 235 | 338 | 197 | 376 | 227 | 346 | 205 | 368 | 219 | 354 | 213 | 360 | 211 | 362 | 221 | 352 | 203 | 370 | 229 | 344 | 195 | 378 | 237 |
337 | 198 | 383 | 236 | 345 | 206 | 375 | 228 | 353 | 214 | 367 | 220 | 361 | 222 | 359 | 212 | 369 | 230 | 351 | 204 | 377 | 238 | 343 | 196 |
287 | 332 | 241 | 294 | 279 | 324 | 249 | 302 | 271 | 316 | 257 | 310 | 263 | 308 | 265 | 318 | 255 | 300 | 273 | 326 | 247 | 292 | 281 | 334 |
242 | 293 | 288 | 331 | 250 | 301 | 280 | 323 | 258 | 309 | 272 | 315 | 266 | 317 | 264 | 307 | 274 | 325 | 256 | 299 | 282 | 333 | 248 | 291 |
336 | 283 | 290 | 245 | 328 | 275 | 298 | 253 | 320 | 267 | 306 | 261 | 312 | 259 | 314 | 269 | 304 | 251 | 322 | 277 | 296 | 243 | 330 | 285 |
289 | 246 | 335 | 284 | 297 | 254 | 327 | 276 | 305 | 262 | 319 | 268 | 313 | 270 | 311 | 260 | 321 | 278 | 303 | 252 | 329 | 286 | 295 | 244 |
Use the Khajuraho method to construct magic squares of order is multiple of 4 from 8x8 to infinity. See 8x8, 12x12, 16x16, 20x20, 24x24, 28x28 and 32x32
It is possible to use each 4x4 panmagic square to construct a 24x24 Franklin panmagic square.
See above how to construct the almost perfect 24x24 Franklin panmagic square (replace the numbers 9 up to 16 of the 4x4 panmagic square by 569 up to 576 to create the first 4x4 sub-square and add each time 8 to the eight low numbers and -/- 8 to the eight high numbers to create the 35 other 4x4 sub-squares).
You must swap half of the numbers to get a perfect 24x24 Franklin panmagic square. Which numbers you must swap and how to swap the numbers, depends on the place of the 1 and the 8 in the 4x4 panmagic square.
1
|
2 | 3 | 4 | 5 | 6 |
7 | 8 | 9 | 10 | 11 | 12 |
13 | 14 | 15 | 16 | 17 | 18 |
19 | 20 | 21 | 22 | 23 | 24 |
25 | 26 | 27 | 28 | 29 | 30 |
31 | 32 | 33 | 34 | 35 | 36 |
Holger Danielsson showed me how to swap numbers in the 16x16, 24x24, 32x32, ... square (see also on his website https://www.magic-squares.info/construction/pandiagonal
5.html).
If the 1 and the 8 are in the same column, than you must swap half of the numbers of sub-square 1/7/13/19/25/31 with 6/12/18/24/30/36, 2/8/14/20/26/32 with 5/11/17/23/29/35 and 3/9/15/22/27/33 with 4/10/16/22/28/34 (= horizontally).
If the 1 and the 8 are in the same row, than you must swap half of the numbers of sub-square 1/2/3/4/5/6/7/8 with 31/32/33/34/35/36, 7/8/9/10/11/12 with 25/26/27/28/29/30 and 13/14/15/16/17/18 with 19/20/21/22/23/24 (= vertically).
Correction sheet 1