René Chrétien had noticed the 15x15 composite (4) magic square and showed me it is possible to use the method to construct magic squares of even orders as well.
Construct the 24x24 magic square by using 36 proportional 4x4 panmagic squares. The squares are proportional because all 36 panmagic 4x4 squares have the same magic sum of (1/6 x 6924 = ) 1154. We use the basic key method (4x4) to produce the panmagic 4x4 squares. As row coordinates don't use 0 up to 3 but use 0 up to (36x4 -/- 1 = ) 143 instead. Take care that the sum of the row coordinates in each 4x4 square is the same (0+71+72+143 = 1+70+73+142 = ... = 35+36+107+108 = 286) to get proportional squares.
1x row coordinate +144x column coordinate + 1 = panmagic 4x4 square
0 | 71 | 72 | 143 | 0 | 3 | 1 | 2 | 1 | 504 | 217 | 432 | ||
72 | 143 | 0 | 71 | 3 | 0 | 2 | 1 | 505 | 144 | 289 | 216 | ||
71 | 0 | 143 | 72 | 2 | 1 | 3 | 0 | 360 | 145 | 576 | 73 | ||
143 | 72 | 71 | 0 | 1 | 2 | 0 | 3 | 288 | 361 | 72 | 433 | ||
1 | 70 | 73 | 142 | 0 | 3 | 1 | 2 | 2 | 503 | 218 | 431 | ||
73 | 142 | 1 | 70 | 3 | 0 | 2 | 1 | 506 | 143 | 290 | 215 | ||
70 | 1 | 142 | 73 | 2 | 1 | 3 | 0 | 359 | 146 | 575 | 74 | ||
142 | 73 | 70 | 1 | 1 | 2 | 0 | 3 | 287 | 362 | 71 | 434 | ||
2 | 69 | 74 | 141 | 0 | 3 | 1 | 2 | 3 | 502 | 219 | 430 | ||
74 | 141 | 2 | 69 | 3 | 0 | 2 | 1 | 507 | 142 | 291 | 214 | ||
69 | 2 | 141 | 74 | 2 | 1 | 3 | 0 | 358 | 147 | 574 | 75 | ||
141 | 74 | 69 | 2 | 1 | 2 | 0 | 3 | 286 | 363 | 70 | 435 | ||
3 | 68 | 75 | 140 | 0 | 3 | 1 | 2 | 4 | 501 | 220 | 429 | ||
75 | 140 | 3 | 68 | 3 | 0 | 2 | 1 | 508 | 141 | 292 | 213 | ||
68 | 3 | 140 | 75 | 2 | 1 | 3 | 0 | 357 | 148 | 573 | 76 | ||
140 | 75 | 68 | 3 | 1 | 2 | 0 | 3 | 285 | 364 | 69 | 436 | ||
4 | 67 | 76 | 139 | 0 | 3 | 1 | 2 | 5 | 500 | 221 | 428 | ||
76 | 139 | 4 | 67 | 3 | 0 | 2 | 1 | 509 | 140 | 293 | 212 | ||
67 | 4 | 139 | 76 | 2 | 1 | 3 | 0 | 356 | 149 | 572 | 77 | ||
139 | 76 | 67 | 4 | 1 | 2 | 0 | 3 | 284 | 365 | 68 | 437 | ||
5 | 66 | 77 | 138 | 0 | 3 | 1 | 2 | 6 | 499 | 222 | 427 | ||
77 | 138 | 5 | 66 | 3 | 0 | 2 | 1 | 510 | 139 | 294 | 211 | ||
66 | 5 | 138 | 77 | 2 | 1 | 3 | 0 | 355 | 150 | 571 | 78 | ||
138 | 77 | 66 | 5 | 1 | 2 | 0 | 3 | 283 | 366 | 67 | 438 | ||
6 | 65 | 78 | 137 | 0 | 3 | 1 | 2 | 7 | 498 | 223 | 426 | ||
78 | 137 | 6 | 65 | 3 | 0 | 2 | 1 | 511 | 138 | 295 | 210 | ||
65 | 6 | 137 | 78 | 2 | 1 | 3 | 0 | 354 | 151 | 570 | 79 | ||
137 | 78 | 65 | 6 | 1 | 2 | 0 | 3 | 282 | 367 | 66 | 439 | ||
7 | 64 | 79 | 136 | 0 | 3 | 1 | 2 | 8 | 497 | 224 | 425 | ||
79 | 136 | 7 | 64 | 3 | 0 | 2 | 1 | 512 | 137 | 296 | 209 | ||
64 | 7 | 136 | 79 | 2 | 1 | 3 | 0 | 353 | 152 | 569 | 80 | ||
136 | 79 | 64 | 7 | 1 | 2 | 0 | 3 | 281 | 368 | 65 | 440 | ||
8 | 63 | 80 | 135 | 0 | 3 | 1 | 2 | 9 | 496 | 225 | 424 | ||
80 | 135 | 8 | 63 | 3 | 0 | 2 | 1 | 513 | 136 | 297 | 208 | ||
63 | 8 | 135 | 80 | 2 | 1 | 3 | 0 | 352 | 153 | 568 | 81 | ||
135 | 80 | 63 | 8 | 1 | 2 | 0 | 3 | 280 | 369 | 64 | 441 | ||
9 | 62 | 81 | 134 | 0 | 3 | 1 | 2 | 10 | 495 | 226 | 423 | ||
81 | 134 | 9 | 62 | 3 | 0 | 2 | 1 | 514 | 135 | 298 | 207 | ||
62 | 9 | 134 | 81 | 2 | 1 | 3 | 0 | 351 | 154 | 567 | 82 | ||
134 | 81 | 62 | 9 | 1 | 2 | 0 | 3 | 279 | 370 | 63 | 442 | ||
10 | 61 | 82 | 133 | 0 | 3 | 1 | 2 | 11 | 494 | 227 | 422 | ||
82 | 133 | 10 | 61 | 3 | 0 | 2 | 1 | 515 | 134 | 299 | 206 | ||
61 | 10 | 133 | 82 | 2 | 1 | 3 | 0 | 350 | 155 | 566 | 83 | ||
133 | 82 | 61 | 10 | 1 | 2 | 0 | 3 | 278 | 371 | 62 | 443 | ||
11 | 60 | 83 | 132 | 0 | 3 | 1 | 2 | 12 | 493 | 228 | 421 | ||
83 | 132 | 11 | 60 | 3 | 0 | 2 | 1 | 516 | 133 | 300 | 205 | ||
60 | 11 | 132 | 83 | 2 | 1 | 3 | 0 | 349 | 156 | 565 | 84 | ||
132 | 83 | 60 | 11 | 1 | 2 | 0 | 3 | 277 | 372 | 61 | 444 | ||
12 | 59 | 84 | 131 | 0 | 3 | 1 | 2 | 13 | 492 | 229 | 420 | ||
84 | 131 | 12 | 59 | 3 | 0 | 2 | 1 | 517 | 132 | 301 | 204 | ||
59 | 12 | 131 | 84 | 2 | 1 | 3 | 0 | 348 | 157 | 564 | 85 | ||
131 | 84 | 59 | 12 | 1 | 2 | 0 | 3 | 276 | 373 | 60 | 445 | ||
13 | 58 | 85 | 130 | 0 | 3 | 1 | 2 | 14 | 491 | 230 | 419 | ||
85 | 130 | 13 | 58 | 3 | 0 | 2 | 1 | 518 | 131 | 302 | 203 | ||
58 | 13 | 130 | 85 | 2 | 1 | 3 | 0 | 347 | 158 | 563 | 86 | ||
130 | 85 | 58 | 13 | 1 | 2 | 0 | 3 | 275 | 374 | 59 | 446 | ||
14 | 57 | 86 | 129 | 0 | 3 | 1 | 2 | 15 | 490 | 231 | 418 | ||
86 | 129 | 14 | 57 | 3 | 0 | 2 | 1 | 519 | 130 | 303 | 202 | ||
57 | 14 | 129 | 86 | 2 | 1 | 3 | 0 | 346 | 159 | 562 | 87 | ||
129 | 86 | 57 | 14 | 1 | 2 | 0 | 3 | 274 | 375 | 58 | 447 | ||
15 | 56 | 87 | 128 | 0 | 3 | 1 | 2 | 16 | 489 | 232 | 417 | ||
87 | 128 | 15 | 56 | 3 | 0 | 2 | 1 | 520 | 129 | 304 | 201 | ||
56 | 15 | 128 | 87 | 2 | 1 | 3 | 0 | 345 | 160 | 561 | 88 | ||
128 | 87 | 56 | 15 | 1 | 2 | 0 | 3 | 273 | 376 | 57 | 448 | ||
16 | 55 | 88 | 127 | 0 | 3 | 1 | 2 | 17 | 488 | 233 | 416 | ||
88 | 127 | 16 | 55 | 3 | 0 | 2 | 1 | 521 | 128 | 305 | 200 | ||
55 | 16 | 127 | 88 | 2 | 1 | 3 | 0 | 344 | 161 | 560 | 89 | ||
127 | 88 | 55 | 16 | 1 | 2 | 0 | 3 | 272 | 377 | 56 | 449 | ||
17 | 54 | 89 | 126 | 0 | 3 | 1 | 2 | 18 | 487 | 234 | 415 | ||
89 | 126 | 17 | 54 | 3 | 0 | 2 | 1 | 522 | 127 | 306 | 199 | ||
54 | 17 | 126 | 89 | 2 | 1 | 3 | 0 | 343 | 162 | 559 | 90 | ||
126 | 89 | 54 | 17 | 1 | 2 | 0 | 3 | 271 | 378 | 55 | 450 | ||
18 | 53 | 90 | 125 | 0 | 3 | 1 | 2 | 19 | 486 | 235 | 414 | ||
90 | 125 | 18 | 53 | 3 | 0 | 2 | 1 | 523 | 126 | 307 | 198 | ||
53 | 18 | 125 | 90 | 2 | 1 | 3 | 0 | 342 | 163 | 558 | 91 | ||
125 | 90 | 53 | 18 | 1 | 2 | 0 | 3 | 270 | 379 | 54 | 451 | ||
19 | 52 | 91 | 124 | 0 | 3 | 1 | 2 | 20 | 485 | 236 | 413 | ||
91 | 124 | 19 | 52 | 3 | 0 | 2 | 1 | 524 | 125 | 308 | 197 | ||
52 | 19 | 124 | 91 | 2 | 1 | 3 | 0 | 341 | 164 | 557 | 92 | ||
124 | 91 | 52 | 19 | 1 | 2 | 0 | 3 | 269 | 380 | 53 | 452 | ||
20 | 51 | 92 | 123 | 0 | 3 | 1 | 2 | 21 | 484 | 237 | 412 | ||
92 | 123 | 20 | 51 | 3 | 0 | 2 | 1 | 525 | 124 | 309 | 196 | ||
51 | 20 | 123 | 92 | 2 | 1 | 3 | 0 | 340 | 165 | 556 | 93 | ||
123 | 92 | 51 | 20 | 1 | 2 | 0 | 3 | 268 | 381 | 52 | 453 | ||
21 | 50 | 93 | 122 | 0 | 3 | 1 | 2 | 22 | 483 | 238 | 411 | ||
93 | 122 | 21 | 50 | 3 | 0 | 2 | 1 | 526 | 123 | 310 | 195 | ||
50 | 21 | 122 | 93 | 2 | 1 | 3 | 0 | 339 | 166 | 555 | 94 | ||
122 | 93 | 50 | 21 | 1 | 2 | 0 | 3 | 267 | 382 | 51 | 454 | ||
22 | 49 | 94 | 121 | 0 | 3 | 1 | 2 | 23 | 482 | 239 | 410 | ||
94 | 121 | 22 | 49 | 3 | 0 | 2 | 1 | 527 | 122 | 311 | 194 | ||
49 | 22 | 121 | 94 | 2 | 1 | 3 | 0 | 338 | 167 | 554 | 95 | ||
121 | 94 | 49 | 22 | 1 | 2 | 0 | 3 | 266 | 383 | 50 | 455 | ||
23 | 48 | 95 | 120 | 0 | 3 | 1 | 2 | 24 | 481 | 240 | 409 | ||
95 | 120 | 23 | 48 | 3 | 0 | 2 | 1 | 528 | 121 | 312 | 193 | ||
48 | 23 | 120 | 95 | 2 | 1 | 3 | 0 | 337 | 168 | 553 | 96 | ||
120 | 95 | 48 | 23 | 1 | 2 | 0 | 3 | 265 | 384 | 49 | 456 | ||
24 | 47 | 96 | 119 | 0 | 3 | 1 | 2 | 25 | 480 | 241 | 408 | ||
96 | 119 | 24 | 47 | 3 | 0 | 2 | 1 | 529 | 120 | 313 | 192 | ||
47 | 24 | 119 | 96 | 2 | 1 | 3 | 0 | 336 | 169 | 552 | 97 | ||
119 | 96 | 47 | 24 | 1 | 2 | 0 | 3 | 264 | 385 | 48 | 457 | ||
25 | 46 | 97 | 118 | 0 | 3 | 1 | 2 | 26 | 479 | 242 | 407 | ||
97 | 118 | 25 | 46 | 3 | 0 | 2 | 1 | 530 | 119 | 314 | 191 | ||
46 | 25 | 118 | 97 | 2 | 1 | 3 | 0 | 335 | 170 | 551 | 98 | ||
118 | 97 | 46 | 25 | 1 | 2 | 0 | 3 | 263 | 386 | 47 | 458 | ||
26 | 45 | 98 | 117 | 0 | 3 | 1 | 2 | 27 | 478 | 243 | 406 | ||
98 | 117 | 26 | 45 | 3 | 0 | 2 | 1 | 531 | 118 | 315 | 190 | ||
45 | 26 | 117 | 98 | 2 | 1 | 3 | 0 | 334 | 171 | 550 | 99 | ||
117 | 98 | 45 | 26 | 1 | 2 | 0 | 3 | 262 | 387 | 46 | 459 | ||
27 | 44 | 99 | 116 | 0 | 3 | 1 | 2 | 28 | 477 | 244 | 405 | ||
99 | 116 | 27 | 44 | 3 | 0 | 2 | 1 | 532 | 117 | 316 | 189 | ||
44 | 27 | 116 | 99 | 2 | 1 | 3 | 0 | 333 | 172 | 549 | 100 | ||
116 | 99 | 44 | 27 | 1 | 2 | 0 | 3 | 261 | 388 | 45 | 460 | ||
28 | 43 | 100 | 115 | 0 | 3 | 1 | 2 | 29 | 476 | 245 | 404 | ||
100 | 115 | 28 | 43 | 3 | 0 | 2 | 1 | 533 | 116 | 317 | 188 | ||
43 | 28 | 115 | 100 | 2 | 1 | 3 | 0 | 332 | 173 | 548 | 101 | ||
115 | 100 | 43 | 28 | 1 | 2 | 0 | 3 | 260 | 389 | 44 | 461 | ||
29 | 42 | 101 | 114 | 0 | 3 | 1 | 2 | 30 | 475 | 246 | 403 | ||
101 | 114 | 29 | 42 | 3 | 0 | 2 | 1 | 534 | 115 | 318 | 187 | ||
42 | 29 | 114 | 101 | 2 | 1 | 3 | 0 | 331 | 174 | 547 | 102 | ||
114 | 101 | 42 | 29 | 1 | 2 | 0 | 3 | 259 | 390 | 43 | 462 | ||
30 | 41 | 102 | 113 | 0 | 3 | 1 | 2 | 31 | 474 | 247 | 402 | ||
102 | 113 | 30 | 41 | 3 | 0 | 2 | 1 | 535 | 114 | 319 | 186 | ||
41 | 30 | 113 | 102 | 2 | 1 | 3 | 0 | 330 | 175 | 546 | 103 | ||
113 | 102 | 41 | 30 | 1 | 2 | 0 | 3 | 258 | 391 | 42 | 463 | ||
31 | 40 | 103 | 112 | 0 | 3 | 1 | 2 | 32 | 473 | 248 | 401 | ||
103 | 112 | 31 | 40 | 3 | 0 | 2 | 1 | 536 | 113 | 320 | 185 | ||
40 | 31 | 112 | 103 | 2 | 1 | 3 | 0 | 329 | 176 | 545 | 104 | ||
112 | 103 | 40 | 31 | 1 | 2 | 0 | 3 | 257 | 392 | 41 | 464 | ||
32 | 39 | 104 | 111 | 0 | 3 | 1 | 2 | 33 | 472 | 249 | 400 | ||
104 | 111 | 32 | 39 | 3 | 0 | 2 | 1 | 537 | 112 | 321 | 184 | ||
39 | 32 | 111 | 104 | 2 | 1 | 3 | 0 | 328 | 177 | 544 | 105 | ||
111 | 104 | 39 | 32 | 1 | 2 | 0 | 3 | 256 | 393 | 40 | 465 | ||
33 | 38 | 105 | 110 | 0 | 3 | 1 | 2 | 34 | 471 | 250 | 399 | ||
105 | 110 | 33 | 38 | 3 | 0 | 2 | 1 | 538 | 111 | 322 | 183 | ||
38 | 33 | 110 | 105 | 2 | 1 | 3 | 0 | 327 | 178 | 543 | 106 | ||
110 | 105 | 38 | 33 | 1 | 2 | 0 | 3 | 255 | 394 | 39 | 466 | ||
34 | 37 | 106 | 109 | 0 | 3 | 1 | 2 | 35 | 470 | 251 | 398 | ||
106 | 109 | 34 | 37 | 3 | 0 | 2 | 1 | 539 | 110 | 323 | 182 | ||
37 | 34 | 109 | 106 | 2 | 1 | 3 | 0 | 326 | 179 | 542 | 107 | ||
109 | 106 | 37 | 34 | 1 | 2 | 0 | 3 | 254 | 395 | 38 | 467 | ||
35 | 36 | 107 | 108 | 0 | 3 | 1 | 2 | 36 | 469 | 252 | 397 | ||
107 | 108 | 35 | 36 | 3 | 0 | 2 | 1 | 540 | 109 | 324 | 181 | ||
36 | 35 | 108 | 107 | 2 | 1 | 3 | 0 | 325 | 180 | 541 | 108 | ||
108 | 107 | 36 | 35 | 1 | 2 | 0 | 3 | 253 | 396 | 37 | 468 |
Put the 36 panmagic 4x4 squares in sequence together.
24x24 magic square
1 | 504 | 217 | 432 | 2 | 503 | 218 | 431 | 3 | 502 | 219 | 430 | 4 | 501 | 220 | 429 | 5 | 500 | 221 | 428 | 6 | 499 | 222 | 427 |
505 | 144 | 289 | 216 | 506 | 143 | 290 | 215 | 507 | 142 | 291 | 214 | 508 | 141 | 292 | 213 | 509 | 140 | 293 | 212 | 510 | 139 | 294 | 211 |
360 | 145 | 576 | 73 | 359 | 146 | 575 | 74 | 358 | 147 | 574 | 75 | 357 | 148 | 573 | 76 | 356 | 149 | 572 | 77 | 355 | 150 | 571 | 78 |
288 | 361 | 72 | 433 | 287 | 362 | 71 | 434 | 286 | 363 | 70 | 435 | 285 | 364 | 69 | 436 | 284 | 365 | 68 | 437 | 283 | 366 | 67 | 438 |
7 | 498 | 223 | 426 | 8 | 497 | 224 | 425 | 9 | 496 | 225 | 424 | 10 | 495 | 226 | 423 | 11 | 494 | 227 | 422 | 12 | 493 | 228 | 421 |
511 | 138 | 295 | 210 | 512 | 137 | 296 | 209 | 513 | 136 | 297 | 208 | 514 | 135 | 298 | 207 | 515 | 134 | 299 | 206 | 516 | 133 | 300 | 205 |
354 | 151 | 570 | 79 | 353 | 152 | 569 | 80 | 352 | 153 | 568 | 81 | 351 | 154 | 567 | 82 | 350 | 155 | 566 | 83 | 349 | 156 | 565 | 84 |
282 | 367 | 66 | 439 | 281 | 368 | 65 | 440 | 280 | 369 | 64 | 441 | 279 | 370 | 63 | 442 | 278 | 371 | 62 | 443 | 277 | 372 | 61 | 444 |
13 | 492 | 229 | 420 | 14 | 491 | 230 | 419 | 15 | 490 | 231 | 418 | 16 | 489 | 232 | 417 | 17 | 488 | 233 | 416 | 18 | 487 | 234 | 415 |
517 | 132 | 301 | 204 | 518 | 131 | 302 | 203 | 519 | 130 | 303 | 202 | 520 | 129 | 304 | 201 | 521 | 128 | 305 | 200 | 522 | 127 | 306 | 199 |
348 | 157 | 564 | 85 | 347 | 158 | 563 | 86 | 346 | 159 | 562 | 87 | 345 | 160 | 561 | 88 | 344 | 161 | 560 | 89 | 343 | 162 | 559 | 90 |
276 | 373 | 60 | 445 | 275 | 374 | 59 | 446 | 274 | 375 | 58 | 447 | 273 | 376 | 57 | 448 | 272 | 377 | 56 | 449 | 271 | 378 | 55 | 450 |
19 | 486 | 235 | 414 | 20 | 485 | 236 | 413 | 21 | 484 | 237 | 412 | 22 | 483 | 238 | 411 | 23 | 482 | 239 | 410 | 24 | 481 | 240 | 409 |
523 | 126 | 307 | 198 | 524 | 125 | 308 | 197 | 525 | 124 | 309 | 196 | 526 | 123 | 310 | 195 | 527 | 122 | 311 | 194 | 528 | 121 | 312 | 193 |
342 | 163 | 558 | 91 | 341 | 164 | 557 | 92 | 340 | 165 | 556 | 93 | 339 | 166 | 555 | 94 | 338 | 167 | 554 | 95 | 337 | 168 | 553 | 96 |
270 | 379 | 54 | 451 | 269 | 380 | 53 | 452 | 268 | 381 | 52 | 453 | 267 | 382 | 51 | 454 | 266 | 383 | 50 | 455 | 265 | 384 | 49 | 456 |
25 | 480 | 241 | 408 | 26 | 479 | 242 | 407 | 27 | 478 | 243 | 406 | 28 | 477 | 244 | 405 | 29 | 476 | 245 | 404 | 30 | 475 | 246 | 403 |
529 | 120 | 313 | 192 | 530 | 119 | 314 | 191 | 531 | 118 | 315 | 190 | 532 | 117 | 316 | 189 | 533 | 116 | 317 | 188 | 534 | 115 | 318 | 187 |
336 | 169 | 552 | 97 | 335 | 170 | 551 | 98 | 334 | 171 | 550 | 99 | 333 | 172 | 549 | 100 | 332 | 173 | 548 | 101 | 331 | 174 | 547 | 102 |
264 | 385 | 48 | 457 | 263 | 386 | 47 | 458 | 262 | 387 | 46 | 459 | 261 | 388 | 45 | 460 | 260 | 389 | 44 | 461 | 259 | 390 | 43 | 462 |
31 | 474 | 247 | 402 | 32 | 473 | 248 | 401 | 33 | 472 | 249 | 400 | 34 | 471 | 250 | 399 |