Magic Squares of odd order

Magic Squares

Magic Squares of odd order by Marios Mamzeris
 


The following is a method of creating any odd-order associative magic square. I had the idea of this technique many years ago, back in 1988, and I finally decided to show it in public (all rights reserved).

In brief, a Magic Square is a n x n square grid (where n is the number of cells on each side) filled with distinct positive integers in the range 1,2...,n2 such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal.

An associative magic square is a magic square with a further property that every number added to the number equidistant, in a straight line, from the center gives n2 + 1. They are also called symmetric magic squares. All associated magic square is self-complementary magic squares as well. (Wikipedia)


To create any odd-order associative magic square quickly and hassle-free, we just need to make two simple passes, that I describe as table transformations (shifts). The following three tables are depicting the creation of a 31x31 magic square.

1) We first get the full table filled with all numbers from 1 to n2 as we go (1st table below showing the cells shifting in green color). Then we make a left shift of all rows, skipping the middle one, starting with moving one number (cell) in the first row while moving an extra more number (cell) in each next row to n-1 numbers in the last (nth ) row.

2) The second action is similar to the 1st, but instead of shifting cells in rows, we now shift cells in columns. It starts with moving one number (cell) upwards from the left side while moving an extra more number (cell) in each next column to n-1 numbers in the last (nth ) column, skipping the middle column this time. That is the 2nd table below, showing the cells shifting in blue color)

That’s it. A magic square is ready. That is the 3rd table below, with light and dark grey cells and yellow diagonals.


1. Initial table with all the numbers from 1 to n2 in sequence (The green arrow shows the direction of the cell shift that will follow and the green cells are the ones that will shift.)


Left green arrow

    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
   
1   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
2   32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
3   63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93
4   94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124
5   125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155
6   156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186
7   187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217
8   218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248
9   249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279
10   280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310
11   311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341
12   342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372
13   373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403
14   404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434
15   435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465
16   466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496
17   497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527
18   528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558
19   559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589
20   590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620
21   621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651
22   652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682
23   683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713
24   714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744
25   745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775
26   776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806
27   807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837
28   838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868
29   869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899
30   900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930
31   931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961

 


2. First transformation (primary cell shift). This table is the result of the shift depicted in green color above. (The blue arrow shows the direction of the 2nd and final cell shift and the blue cells are the ones that will shift.)

Up blue arrow

    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
   
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 1
2   34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 32 33
3   66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 63 64 65
4   98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 94 95 96 97
5   130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 125 126 127 128 129
6   162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 156 157 158 159 160 161
7   194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 187 188 189 190 191 192 193
8   226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 218 219 220 221 222 223 224 225
9   258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 249 250 251 252 253 254 255 256 257
10   290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 280 281 282 283 284 285 286 287 288 289
11   322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 311 312 313 314 315 316 317 318 319 320 321
12   354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 342 343 344 345 346 347 348 349 350 351 352 353
13   386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 373 374 375 376 377 378 379 380 381 382 383 384 385
14   418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 404 405 406 407 408 409 410 411 412 413 414 415 416 417
15   450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449
16   466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496
17   513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512
18   545 546 547 548 549 550 551 552 553 554 555 556 557 558 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544
19   577 578 579 580 581 582 583 584 585 586 587 588 589 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576
20   609 610 611 612 613 614 615 616 617 618 619 620 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608
21   641 642 643 644 645 646 647 648 649 650 651 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640
22   673 674 675 676 677 678 679 680 681 682 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672
23   705 706 707 708 709 710 711 712 713 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704
24   737 738 739 740 741 742 743 744 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
25   769 770 771 772 773 774 775 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768
26   801 802 803 804 805 806 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800
27   833 834 835 836 837 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832
28   865 866 867 868 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864
29   897 898 899 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896
30   929 930 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928
31   961 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960

 


3. Final transformation (secondary cell shift) that gives the Magic 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
   
1   34 67 100 133 166 199 232 265 298 331 364 397 430 463 480 17 498 531 564 597 630 663 696 729 762 795 828 861 894 927 960
2   66 99 132 165 198 231 264 297 330 363 396 429 462 479 527 49 530 563 596 629 662 695 728 761 794 827 860 893 926 959 1
3   98 131 164 197 230 263 296 329 362 395 428 461 478 526 528 81 562 595 628 661 694 727 760 793 826 859 892 925 958 31 33
4   130 163 196 229 262 295 328 361 394 427 460 477 525 558 560 113 594 627 660 693 726 759 792 825 858 891 924 957 30 32 65
5   162 195 228 261 294 327 360 393 426 459 476 524 557 559 592 145 626 659 692 725 758 791 824 857 890 923 956 29 62 64 97
6   194 227 260 293 326 359 392 425 458 475 523 556 589 591 624 177 658 691 724 757 790 823 856 889 922 955 28 61 63 96 129
7   226 259 292 325 358 391 424 457 474 522 555 588 590 623 656 209 690 723 756 789 822 855 888 921 954 27 60 93 95 128 161
8   258 291 324 357 390 423 456 473 521 554 587 620 622 655 688 241 722 755 788 821 854 887 920 953 26 59 92 94 127 160 193
9   290 323 356 389 422 455 472 520 553 586 619 621 654 687 720 273 754 787 820 853 886 919 952 25 58 91 124 126 159 192 225
10   322 355 388 421 454 471 519 552 585 618 651 653 686 719 752 305 786 819 852 885 918 951 24 57 90 123 125 158 191 224 257
11   354 387 420 453 470 518 551 584 617 650 652 685 718 751 784 337 818 851 884 917 950 23 56 89 122 155 157 190 223 256 289
12   386 419 452 469 517 550 583 616 649 682 684 717 750 783 816 369 850 883 916 949 22 55 88 121 154 156 189 222 255 288 321
13   418 451 468 516 549 582 615 648 681 683 716 749 782 815 848 401 882 915 948 21 54 87 120 153 186 188 221 254 287 320 353
14   450 467 515 548 581 614 647 680 713 715 748 781 814 847 880 433 914 947 20 53 86 119 152 185 187 220 253 286 319 352 385
15   466 514 547 580 613 646 679 712 714 747 780 813 846 879 912 465 946 19 52 85 118 151 184 217 219 252 285 318 351 384 417
16   513 546 579 612 645 678 711 744 746 779 812 845 878 911 944 481 18 51 84 117 150 183 216 218 251 284 317 350 383 416 449
17   545 578 611 644 677 710 743 745 778 811 844 877 910 943 16 497 50 83 116 149 182 215 248 250 283 316 349 382 415 448 496
18   577 610 643 676 709 742 775 777 810 843 876 909 942 15 48 529 82 115 148 181 214 247 249 282 315 348 381 414 447 495 512
19   609 642 675 708 741 774 776 809 842 875 908 941 14 47 80 561 114 147 180 213 246 279 281 314 347 380 413 446 494 511 544
20   641 674 707 740 773 806 808 841 874 907 940 13 46 79 112 593 146 179 212 245 278 280 313 346 379 412 445 493 510 543 576
21   673 706 739 772 805 807 840 873 906 939 12 45 78 111 144 625 178 211 244 277 310 312 345 378 411 444 492 509 542 575 608
22   705 738 771 804 837 839 872 905 938 11 44 77 110 143 176 657 210 243 276 309 311 344 377 410 443 491 508 541 574 607 640
23   737 770 803 836 838 871 904 937 10 43 76 109 142 175 208 689 242 275 308 341 343 376 409 442 490 507 540 573 606 639 672
24   769 802 835 868 870 903 936 9 42 75 108 141 174 207 240 721 274 307 340 342 375 408 441 489 506 539 572 605 638 671 704
25   801 834 867 869 902 935 8 41 74 107 140 173 206 239 272 753 306 339 372 374 407 440 488 505 538 571 604 637 670 703 736
26   833 866 899 901 934 7 40 73 106 139 172 205 238 271 304 785 338 371 373 406 439 487 504 537 570 603 636 669 702 735 768
27   865 898 900 933 6 39 72 105 138 171 204 237 270 303 336 817 370 403 405 438 486 503 536 569 602 635 668 701 734 767 800
28   897 930 932 5 38 71 104 137 170 203 236 269 302 335 368 849 402 404 437 485 502 535 568 601 634 667 700 733 766 799 832
29   929 931 4 37 70 103 136 169 202 235 268 301 334 367 400 881 434 436 484 501 534 567 600 633 666 699 732 765 798 831 864
30   961 3 36 69 102 135 168 201 234 267 300 333 366 399 432 913 435 483 500 533 566 599 632 665 698 731 764 797 830 863 896
31   2 35 68 101 134 167 200 233 266 299 332 365 398 431 464 945 482 499 532 565 598 631 664 697 730 763 796 829 862 895 928

 



An alternative way (based on the above shift pattern) is to create a magic square by performing the cell shifting in opposite directions for each half of the table, always skipping the middle row or column.


Initial square with numbers 1 - n2 in sequence

Blue arrow left
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 37 38 39 40 41 42
43 44 45 46 47 48 49
                                   blue arrow right 


After the first pass. Notice how each column is equal to Σ

    Green arrow up
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2 3 4 5 6 7 1   















 
 
 



 Green arrow down
10 11 12 13 14 8 9
18 19 20 21 15 16 17
22 23 24 25 26 27 28
33 34 35 29 30 31 32
41 42 36 37 38 39 40
49 43 44 45 46 47 48

 

Second pass completed and now each row is also equal to Σ. The Magic Square is ready

 10  19  24  5  30  39  48
 18  23  35  13  38  47  1
 22  34  36  21  46  7  9
 33  42  44  25  6  8  17
 41  43  4  29  14  16  28
 49  3  12  37  15  27  32
 2  11  20  45  26  31  40




Some Notes

Below are some simple notes to help the reader understand what is happening to the sums of rows and columns of the initial (1) table
n x n, and how each pass (series of shifts) affects them, and leads to the construction of a symmetrical magic square.

n = square side size (magic square is n x n in size)
M = (n2+1)/2 (middle number, in the center)
Σ = Mn (Sum of each row, column, diagonal of the magic square) 
r = the number of rows or columns away from the middle.

As we create the initial square (1) with all the numbers from 1 to
n2 in sequence, we notice the ‘weight’ difference in each row and column, compared to the middle row and column accordingly. For columns, going left from the middle one, the column’s sum is Σ - rn. For columns going to the right from the middle one, the sum is Σ + rn. For rows, going up from the middle row, the sum is  Σ - (rn2). For rows, going down from the middle row, the sum is  Σ + (rn2).

The first pass, shifting cells in rows to the left, starting with one cell (1st top left corner) and incrementing one more cell in every next row, skipping the middle one, gives us a square (2) where all the row sums are equal to Σ. Notice how the left row has gained + 1 to the first cell on the top, adding one extra in every next cell going to the last cell on the bottom, skipping the middle one. Each next row is following a decremental pattern of gains, and they all affect the row sums in a way that compensates the initial (1) sums to equalize with the middle row.

In a similar fashion, we observe the same behavior in column sums. While they remain as detailed in the first paragraph above, in squares (1) and (2), when we perform the equivalent cell shifting in columns as we previously did in rows, we notice how the column sums are compensated and become equal to Σ.

Below we can see:

On the left side, the squares with each cell value in relation to n.

On the right side, the square progress with actual values. Square (3) is the final and magic one.

In the green-colored cells, the sums of each row and column. The diagonals are always equal to Σ.

In the blue-colored cells, are the initial middle row and column cells, and their location throughout the passes.

                  1st stage of magic square                      
                Σ                   175   Σ
n-6 n-5 n-4 n-3 n-2 n-1 n Σ-(3n2) 1 2 3 4 5 6 7 28 Σ-(3n2)
n+1 n+2 n+3 n+4 n+5 n+6 n+7 Σ-(2n2) 8 9 10 11 12 13 14 77 Σ-(2n2)
n+8 n+9 n+10 n+11 n+12 n+13 n+14 Σ-(1n2) 15 16 17 18 19 20 21 126 Σ-(1n2)
n+15 n+16 n+17 n+18 n+19 n+20 n+21 Σ 22 23 24 25 26 27 28 175 Σ
n+22 n+23 n+24 n+25 n+26 n+27 n+28 Σ+(1n2) 29 30 31 32 33 34 35 224 Σ+(1n2)
n+29 n+30 n+31 n+32 n+33 n+34 n+35 Σ+(2n2) 36 37 38 39 40 41 42 273 Σ+(2n2)
n+36 n+37 n+38 n+39 n+40 n+41 n+42 Σ+(3n2) 43 44 45 46 47 48 49 322 Σ+(3n2)
Σ 175 Σ
                154 161 168 175 182 189 196  
Σ-(3n) Σ-(2n) Σ-(1n) Σ Σ+(1n) Σ+(2n) Σ+(3n) Σ-(3n) Σ-(2n) Σ-(1n) Σ Σ+(1n) Σ+(2n) Σ+(3n)
                  2nd stage of magic square                      
Σ   175 Σ
n-5 n-4 n-3 n-2 n-1 n n-6 Σ-(3n2) 2 3 4 5 6 7 1 28 Σ-(3n2)
n+3 n+4 n+5 n+6 n+7 n+1 n+2 Σ-(2n2) 10 11 12 13 14 8 9 77 Σ-(2n2)
n+11 n+12 n+13 n+14 n+8 n+9 n+10 Σ-(1n2) 18 19 20 21 15 16 17 126 Σ-(1n2)
n+15 n+16 n+17 n+18 n+19 n+20 n+21 Σ 22 23 24 25 26 27 28 175 Σ
n+26 n+27 n+28 n+22 n+23 n+24 n+25 Σ+(1n2) 33 34 35 29 30 31 32 224 Σ+(1n2)
n+34 n+35 n+29 n+30 n+31 n+32 n+33 Σ+(2n2) 41 42 36 37 38 39 40 273 Σ+(2n2)
n+42 n+36 n+37 n+38 n+39 n+40 n+41 Σ+(3n2) 49 43 44 45 46 47 48 322 Σ+(3n2)
Σ 175 Σ
                175 175 175 175 175 175 175
Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ
                  3rd stage of magic square                      
Σ   175 Σ
n+3 n+12 n+17 n-2 n+23 n+32 n+41 Σ 10 19 24 5 30 39 48 175 Σ
n+11 n+16 n+28 n+6 n+31 n+40 n-6 Σ 18 23 35 13 38 47 1 175 Σ
n+15 n+27 n+29 n+14 n+39 n n+2 Σ 22 34 36 21 46 7 9 175 Σ
n+26 n+35 n+37 n+18 n-1 n+1 n+10 Σ 33 42 44 25 6 8 17 175 Σ
n+34 n+36 n-3 n+22 n+7 n+9 n+21 Σ 41 43 4 29 14 16 28 175 Σ
n+42 n-4 n+5 n+30 n+8 n+20 n+25 Σ 49 3 12 37 15 27 32 175 Σ
n-5 n+4 n+13 n+38 n+19 n+24 n+33 Σ 2 11 20 45 26 31 40 175 Σ
Σ 175 Σ
                175 175 175 175 175 175 175  
Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ



 

Magic Square Alternatives

Variations

To get different Magic Squares for the same odd order, you may swap the actions. For example, instead of making a Left-shift followed by an Up-shift, you may try starting with a Right-shift followed by a Down-shift or another combination. Similarly for the alternative way shown above, you may swap the left-right dual-action per pass with right-left and the down-up with up-down.

Below are the four (4) variations of a 19x19 associative Magic Square.
Each is constructed with the same technique but a different action sequence. Each cell is color-coded according to its 'weight'.

Magis Square 1   Magis Square 2
     
Magis Square 1   Magis Square 3



Ppermutations of Magic Squares

Permutations

Any Magic Square generated from the above technique, including the four (4) variations, have multiple permutations. The permutations derive from shifting rows or columns symmetrically. You may also reposition the four quarters of any magic square. Furthermore, a resulted permutation from symmetrically moving rows can also generate more permutations from moving columns or quarters and vice versa.

The number of permutations for each variation (we have four possible variations for each odd-order) is given by the equation below

  Permutation Equation

As this technique allows for four (4) different variations, we have a total of 4xP permutations for every odd-order.
For example, for order 7 we get 4x576=2.304 magic squares.
Some cases are shown below (not exhausted) for a 7x7 Magic Square.

10 19 24 5 30 39 48
18 23 35 13 38 47 1
22 34 36 21 46 7 9
33 42 44 25 6 8 17
41 43 4 29 14 16 28
49 3 12 37 15 27 32
2 11 20 45 26 31 40
 
41 43 4 29 14 16 28
49 3 12 37 15 27 32
2 11 20 45 26 31 40
33 42 44 25 6 8 17
10 19 24 5 30 39 48
18 23 35 13 38 47 1
22 34 36 21 46 7 9
 
14 16 28 29 41 43 4
15 27 32 37 49 3 12
26 31 40 45 2 11 20
6 8 17 25 33 42 44
30 39 48 5 10 19 24
38 47 1 13 18 23 35
46 7 9 21 22 34 36
 
30 39 48 5 10 19 24
38 47 1 13 18 23 35
46 7 9 21 22 34 36
6 8 17 25 33 42 44
14 16 28 29 41 43 4
15 27 32 37 49 3 12
26 31 40 45 2 11 20

24 19 10 5 48 39 30
35 23 18 13 1 47 38
36 34 22 21 9 7 46
44 42 33 25 17 8 6
4 43 41 29 28 16 14
12 3 49 37 32 27 15
20 11 2 45 40 31 26
 
24 10 19 5 39 48 30
35 18 23 13 47 1 38
36 22 34 21 7 9 46
44 33 42 25 8 17 6
4 41 43 29 16 28 14
12 49 3 37 27 32 15
20 2 11 45 31 40 26
 
19 24 10 5 48 30 39
23 35 18 13 1 38 47
34 36 22 21 9 46 7
42 44 33 25 17 6 8
43 4 41 29 28 14 16
3 12 49 37 32 15 27
11 20 2 45 40 26 31
 
10 24 19 5 39 30 48
18 35 23 13 47 38 1
22 36 34 21 7 46 9
33 44 42 25 8 6 17
41 4 43 29 16 14 28
49 12 3 37 27 15 32
2 20 11 45 31 26 40

18 23 35 13 38 47 1
10 19 24 5 30 39 48
22 34 36 21 46 7 9
33 42 44 25 6 8 17
41 43 4 29 14 16 28
2 11 20 45 26 31 40
49 3 12 37 15 27 32
 
10 19 24 5 30 39 48
22 34 36 21 46 7 9
18 23 35 13 38 47 1
33 42 44 25 6 8 17
49 3 12 37 15 27 32
41 43 4 29 14 16 28
2 11 20 45 26 31 40
 
2 11 20 45 26 31 40
49 3 12 37 15 27 32
41 43 4 29 14 16 28
33 42 44 25 6 8 17
22 34 36 21 46 7 9
18 23 35 13 38 47 1
10 19 24 5 30 39 48
 
41 43 4 29 14 16 28
2 11 20 45 26 31 40
49 3 12 37 15 27 32
33 42 44 25 6 8 17
18 23 35 13 38 47 1
10 19 24 5 30 39 48
22 34 36 21 46 7 9



Finally, each Magic Square can be turned anticlockwise (or clockwise if you prefer) to produce four (4) different perspectives as shown below.

 

 4  3  8    8  1  6
 9  5  1    3  5  7
 2  7  6    4  9  2
      Magic Square perspectives
 6  7  2    2  9  4
 1  5  9    7  5  3
 8  3  4    6  1  8

 

You may get another four (4) versions by mirroring the above magic squares. So, for example, the 3x3 magic square can have eight (8) different solutions.
The mirroring can work for any magic square.

 



Please visit my YouTube short videos, which show all the steps for 5x5 and 7x7 associative Magic Squares.

 



Magic Squares links


David Pleacher, https://www.pleacher.com/mp/puzzles/mathpuz/magmenu.html

Harry White, http://budshaw.ca/Associative.html, http://budshaw.ca/Download.html

William Walkington, https://carresmagiques.blogspot.com/,  https://carresmagiques.blogspot.com/p/magic-square-links.html,

Equations for Marios Mamzeris's odd order associative magic squares by William Walkington

Holger Danielsson, https://www.magic-squares.info/construction/odd.html,  https://www.magic-squares.info/info/book.html

Bogdan Golunski, http://www.number-galaxy.eu/

Inder Taneja, https://inderjtaneja.com/category/magic-squares/

Arie Breedijk, https://www.magischvierkant.com/

Wikipedia (Associative magic square), https://en.wikipedia.org/wiki/Associative_magic_square



For any comments or questions, please send me an email at "contact (@) oddmagicsquares.com".

Marios Mamzeris - Magic Squares