Take a 15x15 magic square and construct the second, third and fourth 15x15 magic square by adding (15 x 15 =) 225, (2 x 225 = ) 450 respectively (3 x 225 = ) 675 to all numbers of the first 15x15 magic square. Put the first square in the top left corner, put the second square in the bottom right corner, put the third square in the top right corner and put the fourth square in the bottom left corner.
67 | 190 | 79 | 65 | 194 | 82 | 70 | 184 | 80 | 74 | 187 | 85 | 64 | 185 | 89 | 517 | 640 | 529 | 515 | 644 | 532 | 520 | 634 | 530 | 524 | 637 | 535 | 514 | 635 | 539 |
210 | 3 | 128 | 208 | 1 | 135 | 198 | 8 | 133 | 196 | 15 | 123 | 203 | 13 | 121 | 660 | 453 | 578 | 658 | 451 | 585 | 648 | 458 | 583 | 646 | 465 | 573 | 653 | 463 | 571 |
47 | 116 | 177 | 51 | 114 | 167 | 56 | 117 | 171 | 54 | 107 | 176 | 57 | 111 | 174 | 497 | 566 | 627 | 501 | 564 | 617 | 506 | 567 | 621 | 504 | 557 | 626 | 507 | 561 | 624 |
97 | 220 | 19 | 95 | 224 | 22 | 100 | 214 | 20 | 104 | 217 | 25 | 94 | 215 | 29 | 547 | 670 | 469 | 545 | 674 | 472 | 550 | 664 | 470 | 554 | 667 | 475 | 544 | 665 | 479 |
150 | 33 | 158 | 148 | 31 | 165 | 138 | 38 | 163 | 136 | 45 | 153 | 143 | 43 | 151 | 600 | 483 | 608 | 598 | 481 | 615 | 588 | 488 | 613 | 586 | 495 | 603 | 593 | 493 | 601 |
62 | 191 | 87 | 66 | 189 | 77 | 71 | 192 | 81 | 69 | 182 | 86 | 72 | 186 | 84 | 512 | 641 | 537 | 516 | 639 | 527 | 521 | 642 | 531 | 519 | 632 | 536 | 522 | 636 | 534 |
202 | 10 | 124 | 200 | 14 | 127 | 205 | 4 | 125 | 209 | 7 | 130 | 199 | 5 | 134 | 652 | 460 | 574 | 650 | 464 | 577 | 655 | 454 | 575 | 659 | 457 | 580 | 649 | 455 | 584 |
60 | 108 | 173 | 58 | 106 | 180 | 48 | 113 | 178 | 46 | 120 | 168 | 53 | 118 | 166 | 510 | 558 | 623 | 508 | 556 | 630 | 498 | 563 | 628 | 496 | 570 | 618 | 503 | 568 | 616 |
92 | 221 | 27 | 96 | 219 | 17 | 101 | 222 | 21 | 99 | 212 | 26 | 102 | 216 | 24 | 542 | 671 | 477 | 546 | 669 | 467 | 551 | 672 | 471 | 549 | 662 | 476 | 552 | 666 | 474 |
142 | 40 | 154 | 140 | 44 | 157 | 145 | 34 | 155 | 149 | 37 | 160 | 139 | 35 | 164 | 592 | 490 | 604 | 590 | 494 | 607 | 595 | 484 | 605 | 599 | 487 | 610 | 589 | 485 | 614 |
75 | 183 | 83 | 73 | 181 | 90 | 63 | 188 | 88 | 61 | 195 | 78 | 68 | 193 | 76 | 525 | 633 | 533 | 523 | 631 | 540 | 513 | 638 | 538 | 511 | 645 | 528 | 518 | 643 | 526 |
197 | 11 | 132 | 201 | 9 | 122 | 206 | 12 | 126 | 204 | 2 | 131 | 207 | 6 | 129 | 647 | 461 | 582 | 651 | 459 | 572 | 656 | 462 | 576 | 654 | 452 | 581 | 657 | 456 | 579 |
52 | 115 | 169 | 50 | 119 | 172 | 55 | 109 | 170 | 59 | 112 | 175 | 49 | 110 | 179 | 502 | 565 | 619 | 500 | 569 | 622 | 505 | 559 | 620 | 509 | 562 | 625 | 499 | 560 | 629 |
105 | 213 | 23 | 103 | 211 | 30 | 93 | 218 | 28 | 91 | 225 | 18 | 98 | 223 | 16 | 555 | 663 | 473 | 553 | 661 | 480 | 543 | 668 | 478 | 541 | 675 | 468 | 548 | 673 | 466 |
137 | 41 | 162 | 141 | 39 | 152 | 146 | 42 | 156 | 144 | 32 | 161 | 147 | 36 | 159 | 587 | 491 | 612 | 591 | 489 | 602 | 596 | 492 | 606 | 594 | 482 | 611 | 597 | 486 | 609 |
742 | 865 | 754 | 740 | 869 | 757 | 745 | 859 | 755 | 749 | 862 | 760 | 739 | 860 | 764 | 292 | 415 | 304 | 290 | 419 | 307 | 295 | 409 | 305 | 299 | 412 | 310 | 289 | 410 | 314 |
885 | 678 | 803 | 883 | 676 | 810 | 873 | 683 | 808 | 871 | 690 | 798 | 878 | 688 | 796 | 435 | 228 | 353 | 433 | 226 | 360 | 423 | 233 | 358 | 421 | 240 | 348 | 428 | 238 | 346 |
722 | 791 | 852 | 726 | 789 | 842 | 731 | 792 | 846 | 729 | 782 | 851 | 732 | 786 | 849 | 272 | 341 | 402 | 276 | 339 | 392 | 281 | 342 | 396 | 279 | 332 | 401 | 282 | 336 | 399 |
772 | 895 | 694 | 770 | 899 | 697 | 775 | 889 | 695 | 779 | 892 | 700 | 769 | 890 | 704 | 322 | 445 | 244 | 320 | 449 | 247 | 325 | 439 | 245 | 329 | 442 | 250 | 319 | 440 | 254 |
825 | 708 | 833 | 823 | 706 | 840 | 813 | 713 | 838 | 811 | 720 | 828 | 818 | 718 | 826 | 375 | 258 | 383 | 373 | 256 | 390 | 363 | 263 | 388 | 361 | 270 | 378 | 368 | 268 | 376 |
737 | 866 | 762 | 741 | 864 | 752 | 746 | 867 | 756 | 744 | 857 | 761 | 747 | 861 | 759 | 287 | 416 | 312 | 291 | 414 | 302 | 296 | 417 | 306 | 294 | 407 | 311 | 297 | 411 | 309 |
877 | 685 | 799 | 875 | 689 | 802 | 880 | 679 | 800 | 884 | 682 | 805 | 874 | 680 | 809 | 427 | 235 | 349 | 425 | 239 | 352 | 430 | 229 | 350 | 434 | 232 | 355 | 424 | 230 | 359 |
735 | 783 | 848 | 733 | 781 | 855 | 723 | 788 | 853 | 721 | 795 | 843 | 728 | 793 | 841 | 285 | 333 | 398 | 283 | 331 | 405 | 273 | 338 | 403 | 271 | 345 | 393 | 278 | 343 | 391 |
767 | 896 | 702 | 771 | 894 | 692 | 776 | 897 | 696 | 774 | 887 | 701 | 777 | 891 | 699 | 317 | 446 | 252 | 321 | 444 | 242 | 326 | 447 | 246 | 324 | 437 | 251 | 327 | 441 | 249 |
817 | 715 | 829 | 815 | 719 | 832 | 820 | 709 | 830 | 824 | 712 | 835 | 814 | 710 | 839 | 367 | 265 | 379 | 365 | 269 | 382 | 370 | 259 | 380 | 374 | 262 | 385 | 364 | 260 | 389 |
750 | 858 | 758 | 748 | 856 | 765 | 738 | 863 | 763 | 736 | 870 | 753 | 743 | 868 | 751 | 300 | 408 | 308 | 298 | 406 | 315 | 288 | 413 | 313 | 286 | 420 | 303 | 293 | 418 | 301 |
872 | 686 | 807 | 876 | 684 | 797 | 881 | 687 | 801 | 879 | 677 | 806 | 882 | 681 | 804 | 422 | 236 | 357 | 426 | 234 | 347 | 431 | 237 | 351 | 429 | 227 | 356 | 432 | 231 | 354 |
727 | 790 | 844 | 725 | 794 | 847 | 730 | 784 | 845 | 734 | 787 | 850 | 724 | 785 | 854 | 277 | 340 | 394 | 275 | 344 | 397 | 280 | 334 | 395 | 284 | 337 | 400 | 274 | 335 | 404 |
780 | 888 | 698 | 778 | 886 | 705 | 768 | 893 | 703 | 766 | 900 | 693 | 773 | 898 | 691 | 330 | 438 | 248 | 328 | 436 | 255 | 318 | 443 | 253 | 316 | 450 | 243 | 323 | 448 | 241 |
812 | 716 | 837 | 816 | 714 | 827 | 821 | 717 | 831 | 819 | 707 | 836 | 822 | 711 | 834 | 362 | 266 | 387 | 366 | 264 | 377 | 371 | 267 | 381 | 369 | 257 | 386 | 372 | 261 | 384 |
The columns and the diagonals give already the magic sum. To get the right sum in the rows, you must swap numbers, as follows. We split the 15x15 square in the top left corner and the 15x15 square in the bottom left corner in 'quarters' (marked by the blue numbers). The ‘quarters’ top left and bottom left of the 15x15 square in the top left corner must be swapped with the ‘quarters’ top left and bottom left of the 15x15 square in the bottom left corner. Also the (blue) numbers on the border between the two 'quarters’ from the second cell up to the crossing point must be swapped. Finally the numbers of the top half of the last column(s) must be swapped with the numbers of the bottom half of the last column(s). Because the numbers of the first two columns must be swapped, the numbers of the last (7 – 1 = ) 6 columns must be swapped. See below the result.
30x30 magic square
742 | 865 | 754 | 740 | 869 | 757 | 745 | 184 | 80 | 74 | 187 | 85 | 64 | 185 | 89 | 517 | 640 | 529 | 515 | 644 | 532 | 520 | 634 | 530 | 299 | 412 | 310 | 289 | 410 | 314 |
885 | 678 | 803 | 883 | 676 | 810 | 873 | 8 | 133 | 196 | 15 | 123 | 203 | 13 | 121 | 660 | 453 | 578 | 658 | 451 | 585 | 648 | 458 | 583 | 421 | 240 | 348 | 428 | 238 | 346 |
722 | 791 | 852 | 726 | 789 | 842 | 731 | 117 | 171 | 54 | 107 | 176 | 57 | 111 | 174 | 497 | 566 | 627 | 501 | 564 | 617 | 506 | 567 | 621 | 279 | 332 | 401 | 282 | 336 | 399 |
772 | 895 | 694 | 770 | 899 | 697 | 775 | 214 | 20 | 104 | 217 | 25 | 94 | 215 | 29 | 547 | 670 | 469 | 545 | 674 | 472 | 550 | 664 | 470 | 329 | 442 | 250 | 319 | 440 | 254 |
825 | 708 | 833 | 823 | 706 | 840 | 813 | 38 | 163 | 136 | 45 | 153 | 143 | 43 | 151 | 600 | 483 | 608 | 598 | 481 | 615 | 588 | 488 | 613 | 361 | 270 | 378 | 368 | 268 | 376 |
737 | 866 | 762 | 741 | 864 | 752 | 746 | 192 | 81 | 69 | 182 | 86 | 72 | 186 | 84 | 512 | 641 | 537 | 516 | 639 | 527 | 521 | 642 | 531 | 294 | 407 | 311 | 297 | 411 | 309 |
877 | 685 | 799 | 875 | 689 | 802 | 880 | 4 | 125 | 209 | 7 | 130 | 199 | 5 | 134 | 652 | 460 | 574 | 650 | 464 | 577 | 655 | 454 | 575 | 434 | 232 | 355 | 424 | 230 | 359 |
60 | 783 | 848 | 733 | 781 | 855 | 723 | 788 | 178 | 46 | 120 | 168 | 53 | 118 | 166 | 510 | 558 | 623 | 508 | 556 | 630 | 498 | 563 | 628 | 271 | 345 | 393 | 278 | 343 | 391 |
767 | 896 | 702 | 771 | 894 | 692 | 776 | 222 | 21 | 99 | 212 | 26 | 102 | 216 | 24 | 542 | 671 | 477 | 546 | 669 | 467 | 551 | 672 | 471 | 324 | 437 | 251 | 327 | 441 | 249 |
817 | 715 | 829 | 815 | 719 | 832 | 820 | 34 | 155 | 149 | 37 | 160 | 139 | 35 | 164 | 592 | 490 | 604 | 590 | 494 | 607 | 595 | 484 | 605 | 374 | 262 | 385 | 364 | 260 | 389 |
750 | 858 | 758 | 748 | 856 | 765 | 738 | 188 | 88 | 61 | 195 | 78 | 68 | 193 | 76 | 525 | 633 | 533 | 523 | 631 | 540 | 513 | 638 | 538 | 286 | 420 | 303 | 293 | 418 | 301 |
872 | 686 | 807 | 876 | 684 | 797 | 881 | 12 | 126 | 204 | 2 | 131 | 207 | 6 | 129 | 647 | 461 | 582 | 651 | 459 | 572 | 656 | 462 | 576 | 429 | 227 | 356 | 432 | 231 | 354 |
727 | 790 | 844 | 725 | 794 | 847 | 730 | 109 | 170 | 59 | 112 | 175 | 49 | 110 | 179 | 502 | 565 | 619 | 500 | 569 | 622 | 505 | 559 | 620 | 284 | 337 | 400 | 274 | 335 | 404 |
780 | 888 | 698 | 778 | 886 | 705 | 768 | 218 | 28 | 91 | 225 | 18 | 98 | 223 | 16 | 555 | 663 | 473 | 553 | 661 | 480 | 543 | 668 | 478 | 316 | 450 | 243 | 323 | 448 | 241 |
812 | 716 | 837 | 816 | 714 | 827 | 821 | 42 | 156 | 144 | 32 | 161 | 147 | 36 | 159 | 587 | 491 | 612 | 591 | 489 | 602 | 596 | 492 | 606 | 369 | 257 | 386 | 372 | 261 | 384 |
67 | 190 | 79 | 65 | 194 | 82 | 70 | 859 | 755 | 749 | 862 | 760 | 739 | 860 | 764 | 292 | 415 | 304 | 290 | 419 | 307 | 295 | 409 | 305 | 524 | 637 | 535 | 514 | 635 | 539 |
210 | 3 | 128 | 208 | 1 | 135 | 198 | 683 | 808 | 871 | 690 | 798 | 878 | 688 | 796 | 435 | 228 | 353 | 433 | 226 | 360 | 423 | 233 | 358 | 646 | 465 | 573 | 653 | 463 | 571 |
47 | 116 | 177 | 51 | 114 | 167 | 56 | 792 | 846 | 729 | 782 | 851 | 732 | 786 | 849 | 272 | 341 | 402 | 276 | 339 | 392 | 281 | 342 | 396 | 504 | 557 | 626 | 507 | 561 | 624 |
97 | 220 | 19 | 95 | 224 | 22 | 100 | 889 | 695 | 779 | 892 | 700 | 769 | 890 | 704 | 322 | 445 | 244 | 320 | 449 | 247 | 325 | 439 | 245 | 554 | 667 | 475 | 544 | 665 | 479 |
150 | 33 | 158 | 148 | 31 | 165 | 138 | 713 | 838 | 811 | 720 | 828 | 818 | 718 | 826 | 375 | 258 | 383 | 373 | 256 | 390 | 363 | 263 | 388 | 586 | 495 | 603 | 593 | 493 | 601 |
62 | 191 | 87 | 66 | 189 | 77 | 71 | 867 | 756 | 744 | 857 | 761 | 747 | 861 | 759 | 287 | 416 | 312 | 291 | 414 | 302 | 296 | 417 | 306 | 519 | 632 | 536 | 522 | 636 | 534 |
202 | 10 | 124 | 200 | 14 | 127 | 205 | 679 | 800 | 884 | 682 | 805 | 874 | 680 | 809 | 427 | 235 | 349 | 425 | 239 | 352 | 430 | 229 | 350 | 659 | 457 | 580 | 649 | 455 | 584 |
735 | 108 | 173 | 58 | 106 | 180 | 48 | 113 | 853 | 721 | 795 | 843 | 728 | 793 | 841 | 285 | 333 | 398 | 283 | 331 | 405 | 273 | 338 | 403 | 496 | 570 | 618 | 503 | 568 | 616 |
92 | 221 | 27 | 96 | 219 | 17 | 101 | 897 | 696 | 774 | 887 | 701 | 777 | 891 | 699 | 317 | 446 | 252 | 321 | 444 | 242 | 326 | 447 | 246 | 549 | 662 | 476 | 552 | 666 | 474 |
142 | 40 | 154 | 140 | 44 | 157 | 145 | 709 | 830 | 824 | 712 | 835 | 814 | 710 | 839 | 367 | 265 | 379 | 365 | 269 | 382 | 370 | 259 | 380 | 599 | 487 | 610 | 589 | 485 | 614 |
75 | 183 | 83 | 73 | 181 | 90 | 63 | 863 | 763 | 736 | 870 | 753 | 743 | 868 | 751 | 300 | 408 | 308 | 298 | 406 | 315 | 288 | 413 | 313 | 511 | 645 | 528 | 518 | 643 | 526 |
197 | 11 | 132 | 201 | 9 | 122 | 206 | 687 | 801 | 879 | 677 | 806 | 882 | 681 | 804 | 422 | 236 | 357 | 426 | 234 | 347 | 431 | 237 | 351 | 654 | 452 | 581 | 657 | 456 | 579 |
52 | 115 | 169 | 50 | 119 | 172 | 55 | 784 | 845 | 734 | 787 | 850 | 724 | 785 | 854 | 277 | 340 | 394 | 275 | 344 | 397 | 280 | 334 | 395 | 509 | 562 | 625 | 499 | 560 | 629 |
105 | 213 | 23 | 103 | 211 | 30 | 93 | 893 | 703 | 766 | 900 | 693 | 773 | 898 | 691 | 330 | 438 | 248 | 328 | 436 | 255 | 318 | 443 | 253 | 541 | 675 | 468 | 548 | 673 | 466 |
137 | 41 | 162 | 141 | 39 | 152 | 146 | 717 | 831 | 819 | 707 | 836 | 822 | 711 | 834 | 362 | 266 | 387 | 366 | 264 | 377 | 371 | 267 | 381 | 594 | 482 | 611 | 597 | 486 | 609 |
Use the method of Strachey to construct magic squares of order is double odd. See 6x6, 10x10, 14x14, 18x18, 22x22, 26x26 en 30x30