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On September 27, 2008, Awani Kumar announced the discovery of the first Magic Knight Tour of a tesseract figure.
ADDENDUM; By November 21, 2008 Awani Kumar has enumerated 208
magic tours of knights in 4x4x4x4 hyperspace.
Sum of all the rows, columns, pillars and files is 514. See More
on Knight-tours links at end of this page.
| This figure shows the 4-dimensional
tesseract projected as 2-dimensions on a flat surface (this screen). In the
following discussion, I will refer to the 4 dimensions as 4 directions. To save space, distances are not the same in all directions, so the length of the L shaped steps will not be in consistent proportions. All 64 rows, columns, pillars, and files of this figure sum to the same constant, 514. However, this tesseract is not magic because none of the eight
quadragonals sum to the constant 514. Their sums range from 472 to 556 but
the total of these sums is 4072. I am surprised that this total is not 8
times 514! The corners (which are the ends of the quadragonals) are obvious in this diagram. In the text listing following, I have indicated them by underlining the numbers. For those having difficulty seeing the quadragonals,
001 063 208 242 204 059 005 224 233 215 040 026 036 211 237 056 195 253 014 052 010 249 199 030 043 021 230 220 226 017 047 246 Sums are: 472 552 492 540 476 536 448 556 Unlike magic squares, cubes, tesseracts, etc., the space diagonals are not required to be magic, for a rectilinear figure to be a magic Knight Tour. In studies of Knight tours, the requirement is that all integers in the figure are accessed by a series of identical steps, usually 2 positions in 1 direction, and 1 position in another direction. This example is such a case, with each move utilizing steps in two of the four available directions. To make this easier to see, I have indicated the numbers 1 to 5 in red,
and the numbers 6 to 10 in green. |
 |
This particular knight tour is re-entrant. The first number in the series is exactly one move away from the last number, so on completion of the tour, the next move could be back to the beginning! It is also a four-fold cyclic magic tour. That is, it remains magic when the tour starts from 65, 129 and 193. Awani Kumar has compiled over 50 such tours.
I have not personally checked the orthogonal line sums of this tesseract, but they have been independently confirmed correct by Guenter Stertenbrink.
The text representation of the
above tesseract.
This is the form in which Awani Kumar announced this discovery in an
email on September 27, 2008.
001 080 191 242 112 033 210 159 177 256 015 066 224 145 098 047 192 241 002 079 209 160 111 034 016 065 178 255 097 048 223 146 113 064 207 130 032 081 162 239 193 144 127 050 176 225 018 095 208 129 114 063 161 240 031 082 128 049 194 143 017 096 175 226 120 057 202 135 025 088 167 234 200 137 122 055 169 232 023 090 201 136 119 058 168 233 026 087 121 056 199 138 024 089 170 231 008 073 186 247 105 040 215 154 184 249 010 071 217 152 103 042 185 248 007 074 216 153 106 039 009 072 183 250 104 041 218 151 189 244 003 078 212 157 110 035 013 068 179 254 100 045 222 147 004 077 190 243 109 036 211 158 180 253 014 067 221 148 099 046 205 132 115 062 164 237 030 083 125 052 195 142 020 093 174 227 116 061 206 131 029 084 163 238 196 141 126 051 173 228 019 094 204 133 118 059 165 236 027 086 124 053 198 139 021 092 171 230 117 060 203 134 028 085 166 235 197 140 123 054 172 229 022 091 188 245 006 075 213 156 107 038 012 069 182 251 101 044 219 150 005 076 187 246 108 037 214 155 181 252 011 070 220 149 102 043
A 5x5x5x5x5 Magic Knight Tour (added Oct.18, 2008)
Less then 24 hours after I posted this page, Awani Kumar announced to the Internet community by email that he had successfully constructed a dimension 5 magic knight tour. He said
|
I am happy to inform you that magic knight tour has been extended further into five dimensional hyperspace. A magic tour of knight has been constructed in 4x4x4x4x4 hyperspace. Sum of all the rows, columns, pillars, files and poles is 2050. Guenter Stertenbrink has checked it. Thank you very much Guenter, for checking and putting it on the computing magic knight tours web site. |
As with the above tesseract, this figure is not considered magic because the 16 quintagonals do not sum correctly. I have again underlined the 32 corners of the figure, but leave it to the reader to figure out the quintagonals (if he is curious).
5x5x5x5x5 decimal: 1 320 767 962 448 129 834 639 705 1024 63 258 896 577 386 191 768 961 2 319 833 640 447 130 64 257 706 1023 385 192 895 578 449 256 831 514 128 321 642 959 769 576 511 194 704 897 66 383 832 513 450 255 641 960 127 322 512 193 770 575 65 384 703 898 480 225 802 543 97 352 671 930 800 545 482 223 673 928 95 354 801 544 479 226 672 929 98 351 481 224 799 546 96 353 674 927 32 289 738 991 417 160 863 610 736 993 34 287 865 608 415 162 737 992 31 290 864 609 418 159 33 288 735 994 416 161 866 607 753 976 15 306 848 625 434 143 49 272 719 1010 400 177 882 591 16 305 754 975 433 144 847 626 720 1009 50 271 881 592 399 178 817 528 463 242 656 945 114 335 497 208 783 562 80 369 690 911 464 241 818 527 113 336 655 946 784 561 498 207 689 912 79 370 816 529 466 239 657 944 111 338 496 209 786 559 81 368 687 914 465 240 815 530 112 337 658 943 785 560 495 210 688 913 82 367 752 977 18 303 849 624 431 146 48 273 722 1007 401 176 879 594 17 304 751 978 432 145 850 623 721 1008 47 274 880 593 402 175 ------ ------ 456 249 826 519 121 328 647 954 776 569 506 199 697 904 71 378 825 520 455 250 648 953 122 327 505 200 775 570 72 377 698 903 8 313 762 967 441 136 839 634 712 1017 58 263 889 584 391 186 761 968 7 314 840 633 442 135 57 264 711 1018 392 185 890 583 25 296 743 986 424 153 858 615 729 1000 39 282 872 601 410 167 744 985 26 295 857 616 423 154 40 281 730 999 409 168 871 602 473 232 807 538 104 345 666 935 793 552 487 218 680 921 90 359 808 537 474 231 665 936 103 346 488 217 794 551 89 360 679 922 824 521 458 247 649 952 119 330 504 201 778 567 73 376 695 906 457 248 823 522 120 329 650 951 777 568 503 202 696 905 74 375 760 969 10 311 841 632 439 138 56 265 714 1015 393 184 887 586 9 312 759 970 440 137 842 631 713 1016 55 266 888 585 394 183 745 984 23 298 856 617 426 151 41 280 727 1002 408 169 874 599 24 297 746 983 425 152 855 618 728 1001 42 279 873 600 407 170 809 536 471 234 664 937 106 343 489 216 791 554 88 361 682 919 472 233 810 535 105 344 663 938 792 553 490 215 681 920 87 362 ------ ------ 765 964 3 318 836 637 446 131 61 260 707 1022 388 189 894 579 4 317 766 963 445 132 835 638 708 1021 62 259 893 580 387 190 829 516 451 254 644 957 126 323 509 196 771 574 68 381 702 899 452 253 830 515 125 324 643 958 772 573 510 195 701 900 67 382 804 541 478 227 669 932 99 350 484 221 798 547 93 356 675 926 477 228 803 542 100 349 670 931 797 548 483 222 676 925 94 355 740 989 30 291 861 612 419 158 36 285 734 995 413 164 867 606 29 292 739 990 420 157 862 611 733 996 35 286 868 605 414 163 13 308 755 974 436 141 846 627 717 1012 51 270 884 589 398 179 756 973 14 307 845 628 435 142 52 269 718 1011 397 180 883 590 461 244 819 526 116 333 654 947 781 564 499 206 692 909 78 371 820 525 462 243 653 948 115 334 500 205 782 563 77 372 691 910 468 237 814 531 109 340 659 942 788 557 494 211 685 916 83 366 813 532 467 238 660 941 110 339 493 212 787 558 84 365 686 915 20 301 750 979 429 148 851 622 724 1005 46 275 877 596 403 174 749 980 19 302 852 621 430 147 45 276 723 1006 404 173 878 595 ------ ------ 828 517 454 251 645 956 123 326 508 197 774 571 69 380 699 902 453 252 827 518 124 325 646 955 773 572 507 198 700 901 70 379 764 965 6 315 837 636 443 134 60 261 710 1019 389 188 891 582 5 316 763 966 444 133 838 635 709 1020 59 262 892 581 390 187 741 988 27 294 860 613 422 155 37 284 731 998 412 165 870 603 28 293 742 987 421 156 859 614 732 997 38 283 869 604 411 166 805 540 475 230 668 933 102 347 485 220 795 550 92 357 678 923 476 229 806 539 101 348 667 934 796 549 486 219 677 924 91 358 460 245 822 523 117 332 651 950 780 565 502 203 693 908 75 374 821 524 459 246 652 949 118 331 501 204 779 566 76 373 694 907 12 309 758 971 437 140 843 630 716 1013 54 267 885 588 395 182 757 972 11 310 844 629 438 139 53 268 715 1014 396 181 886 587 21 300 747 982 428 149 854 619 725 1004 43 278 876 597 406 171 748 981 22 299 853 620 427 150 44 277 726 1003 405 172 875 598 469 236 811 534 108 341 662 939 789 556 491 214 684 917 86 363 812 533 470 235 661 940 107 342 492 213 790 555 85 364 683 918
More on Magic Knight Tours.
Much more information on Magic Knight Tours may be seen on my Knight Tours page, including the listing for a Kumar 3x3x3 magic knight tour.
Awani Kumar’s 3-D knight tours are at http://www.gpj.connectfree.co.uk/gpj43.htm
Guenter Stertenbrink and Jean Charles Meyrignac created a website in 2003 that chronicles magic knight tour events and updates at http://magictour.free.fr/.
More information on magic tesseracts are available at other pages on this site. See especially my Overview page.
Please send me Feedback about my Web
site!
Harvey Heinz harveyheinz@shaw.ca
This page originally posted October 17, 2008
This page last updated
October 14, 2009
Copyright © 2008 by Harvey D. Heinz