Best Franklin Squares
 
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My Squares
Part 1

32x32

 

1008 15 1009 18 1006 13 1011 20 1004 11 1013 22 1002 9 1015 24 1000 7 1017 26 998 5 1019 28 996 3 1021 30 994 1 1023 32
753 274 752 271 755 276 750 269 757 278 748 267 759 280 746 265 761 282 744 263 763 284 742 261 765 286 740 259 767 288 738 257
48 975 49 978 46 973 51 980 44 971 53 982 42 969 55 984 40 967 57 986 38 965 59 988 36 963 61 990 34 961 63 992
305 722 304 719 307 724 302 717 309 726 300 715 311 728 298 713 313 730 296 711 315 732 294 709 317 734 292 707 319 736 290 705
80 943 81 946 78 941 83 948 76 939 85 950 74 937 87 952 72 935 89 954 70 933 91 956 68 931 93 958 66 929 95 960
337 690 336 687 339 692 334 685 341 694 332 683 343 696 330 681 345 698 328 679 347 700 326 677 349 702 324 675 351 704 322 673
912 111 913 114 910 109 915 116 908 107 917 118 906 105 919 120 904 103 921 122 902 101 923 124 900 99 925 126 898 97 927 128
657 370 656 367 659 372 654 365 661 374 652 363 663 376 650 361 665 378 648 359 667 380 646 357 669 382 644 355 671 384 642 353
880 143 881 146 878 141 883 148 876 139 885 150 874 137 887 152 872 135 889 154 870 133 891 156 868 131 893 158 866 129 895 160
625 402 624 399 627 404 622 397 629 406 620 395 631 408 618 393 633 410 616 391 635 412 614 389 637 414 612 387 639 416 610 385
176 847 177 850 174 845 179 852 172 843 181 854 170 841 183 856 168 839 185 858 166 837 187 860 164 835 189 862 162 833 191 864
433 594 432 591 435 596 430 589 437 598 428 587 439 600 426 585 441 602 424 583 443 604 422 581 445 606 420 579 447 608 418 577
208 815 209 818 206 813 211 820 204 811 213 822 202 809 215 824 200 807 217 826 198 805 219 828 196 803 221 830 194 801 223 832
465 562 464 559 467 564 462 557 469 566 460 555 471 568 458 553 473 570 456 551 475 572 454 549 477 574 452 547 479 576 450 545
784 239 785 242 782 237 787 244 780 235 789 246 778 233 791 248 776 231 793 250 774 229 795 252 772 227 797 254 770 225 799 256
529 498 528 495 531 500 526 493 533 502 524 491 535 504 522 489 537 506 520 487 539 508 518 485 541 510 516 483 543 512 514 481
496 527 497 530 494 525 499 532 492 523 501 534 490 521 503 536 488 519 505 538 486 517 507 540 484 515 509 542 482 513 511 544
241 786 240 783 243 788 238 781 245 790 236 779 247 792 234 777 249 794 232 775 251 796 230 773 253 798 228 771 255 800 226 769
560 463 561 466 558 461 563 468 556 459 565 470 554 457 567 472 552 455 569 474 550 453 571 476 548 451 573 478 546 449 575 480
817 210 816 207 819 212 814 205 821 214 812 203 823 216 810 201 825 218 808 199 827 220 806 197 829 222 804 195 831 224 802 193
592 431 593 434 590 429 595 436 588 427 597 438 586 425 599 440 584 423 601 442 582 421 603 444 580 419 605 446 578 417 607 448
849 178 848 175 851 180 846 173 853 182 844 171 855 184 842 169 857 186 840 167 859 188 838 165 861 190 836 163 863 192 834 161
400 623 401 626 398 621 403 628 396 619 405 630 394 617 407 632 392 615 409 634 390 613 411 636 388 611 413 638 386 609 415 640
145 882 144 879 147 884 142 877 149 886 140 875 151 888 138 873 153 890 136 871 155 892 134 869 157 894 132 867 159 896 130 865
368 655 369 658 366 653 371 660 364 651 373 662 362 649 375 664 360 647 377 666 358 645 379 668 356 643 381 670 354 641 383 672
113 914 112 911 115 916 110 909 117 918 108 907 119 920 106 905 121 922 104 903 123 924 102 901 125 926 100 899 127 928 98 897
688 335 689 338 686 333 691 340 684 331 693 342 682 329 695 344 680 327 697 346 678 325 699 348 676 323 701 350 674 321 703 352
945 82 944 79 947 84 942 77 949 86 940 75 951 88 938 73 953 90 936 71 955 92 934 69 957 94 932 67 959 96 930 65
720 303 721 306 718 301 723 308 716 299 725 310 714 297 727 312 712 295 729 314 710 293 731 316 708 291 733 318 706 289 735 320
977 50 976 47 979 52 974 45 981 54 972 43 983 56 970 41 985 58 968 39 987 60 966 37 989 62 964 35 991 64 962 33
272 751 273 754 270 749 275 756 268 747 277 758 266 745 279 760 264 743 281 762 262 741 283 764 260 739 285 766 258 737 287 768
17 1010 16 1007 19 1012 14 1005 21 1014 12 1003 23 1016 10 1001 25 1018 8 999 27 1020 6 997 29 1022 4 995 31 1024 2 993

 

In the above square all rows, columns, full diagonals, broken diagonals, and bent diagonals total 16400. All half rows and half columns starting at an outside edge total 8200, half row total. All 2x2’s total 2050. It has a consecutive run of numbers from one to 1024. These facts make it a pandiagonal magic square with Franklin square features. As far as I know the only other person to produce a square like this was Franklin-his improved 16x16.

This square also has other features that are not in Franklin’s square.

Any quarter diagonal (eight cells) down to the right starting at the intersection of any odd row and any odd column will total 4100. Any quarter diagonal down to the left starting at the intersection of any odd row and any even column also totals 4100.

Any imbedded square of an order that is a multiple of eight with its upper left corner at the intersection of row/column one, five, nine, 13, 17, 21, 25, or 29, with wraparound, is a pandiagonal magic square with all the Franklin features including bent diagonals. If a bent diagonal runs off the edge of an imbedded square and is wrapped around into the imbedded square, wrapped around into the parent square, or allowed to run into the adjacent square it will have the proper total. In fact if it is wrapped around into any odd column it will have the proper total.

I’ve added a symmetry feature as well. If a line is drawn between rows 16 and 17 it becomes an axis of symmetry. Any two cells symmetrical about this axis will total 1025. If 32 cells are chosen that are in a symmetrical pattern they will total 16400, row total.

  • All of these symmetrical patterns have 32 cells. They all total 16400.
  • The red cells in the corner patterns
  • The blue cells in the corner patterns
  • The red cells in the square
  • The blue cells in the square
  • The red “B”
  • The blue “B”

I chose blue and red on a white background to honor Franklin as a Founding Father. Since this square was produced using an adaptation of Franklin’s method the “B in a square” denotes “Ben’s square”.

In my opinion this is the most complex magic square with the most structural features ever produced.

 

 
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