Semi-Magic Knight's Tours

Part 3: Semi-Magic Tours of Knight on 8×N and Larger Boards

By George Jelliss (August 2003). Further examples and some diagrams added 2012/2013.

8×10 Board

I constructed nine 8×10 tours (three typical examples shown here) based on extending the braid in the Beverley type tours on the 8×8 to cover the extra two files. They all add to 324 in the files, as required in a magic tour, but the ranks sum to two alternating values. The first has the sums 417 and 393, the second 445 and 365, the third 415 and 395.

 ```22 43 80 17 76 15 74 13 28 49 79 18 21 44 25 46 27 48 73 12 20 23 42 77 16 75 14 71 50 29 41 78 19 24 45 26 47 30 11 72 62 3 40 57 36 55 34 51 70 9 39 58 61 4 65 6 67 10 31 52 60 63 2 37 56 35 54 33 8 69 1 38 59 64 5 66 7 68 53 32 ``` ```22 43 80 17 76 45 26 15 74 47 79 18 21 44 25 16 75 46 27 14 20 23 42 77 50 71 12 29 48 73 41 78 19 24 11 30 49 72 13 28 62 3 40 57 70 51 32 9 68 53 39 58 61 4 31 10 69 52 33 8 60 63 2 37 56 65 6 35 54 67 1 38 59 64 5 36 55 66 7 34 ``` ```22 43 80 17 76 15 74 47 28 13 79 18 21 44 25 46 27 14 49 72 20 23 42 77 16 75 48 73 12 29 41 78 19 24 45 26 11 30 71 50 62 3 40 57 36 55 70 51 10 31 39 58 61 4 65 6 33 8 69 52 60 63 2 37 56 35 54 67 32 9 1 38 59 64 5 66 7 34 53 68 ```

The next example is constructed by the ‘lozenge’ method that I found for 12×12 magic tours, but due to the limitations of this board the result is only quasi-magic. The 10-cell lines add to 405 in five pairs adding to 81. The 8-cell lines add to 364 and 284 (the magic constant would be 324).

The final two examples, found among my notes and added here January 2013, have file totals 324. The rank totals are 445, 365 in one and 361, 449 innthe other.

 ```65 70 67 60 63 18 21 14 11 16 68 57 64 19 22 59 62 17 24 13 71 66 69 58 61 20 23 12 15 10 56 29 6 73 54 27 8 75 52 25 5 72 55 28 7 74 53 26 9 76 30 35 32 3 38 41 80 49 46 51 33 4 37 42 79 2 39 44 77 48 36 31 34 1 40 43 79 47 50 45 ``` ```38 47 74 03 72 49 28 13 62 59 75 02 39 48 29 12 61 58 27 14 46 37 04 73 50 71 26 15 60 63 01 76 45 40 11 30 57 64 25 16 44 05 36 77 70 51 24 17 56 65 35 80 41 08 31 10 55 66 21 18 06 43 78 33 52 69 20 23 54 67 79 34 07 42 09 32 53 68 19 22 ``` ```39 18 43 62 15 22 47 78 13 24 42 63 40 17 46 79 14 23 48 77 19 38 61 44 21 16 75 50 25 12 64 41 20 37 80 45 26 11 76 49 01 36 65 60 05 32 51 74 09 28 66 59 04 33 56 69 10 27 52 73 35 02 57 68 31 06 71 54 29 08 58 67 34 03 70 55 30 07 72 53 ```

The first four results above first appeared in The Games and Puzzles Journal, issue 26, April 2003.

8×14 Board

This example was constructed (by the lozenge method) 25 December 2012. The files add to 452 (formed of four pairs adding to 113 except in the 5th and 7th files where 1 - 3 and 59 - 57 are transposed). The ranks however add to four different totals 875, 707, (3 times each) and 819, 763 (once each).

``` 36  47  70  73  34 109  52  89  32 107  12  15  94 105
71  74  35  46  51  58  33 108  27  90  95 106  11  14
48  37  72  69 110  53 112  29  88  31  16  13 104  93
75  64  45  50  59 111  54  87  30  85  22  17  92 103
65  76  41  44   1  60   3  84  25  82  97 100   9  20
42  39  78  67  62  55  80   5  86  23  18   7 102  99
77  66  43  40  79   4  61  24  81   6 101  98  19   8
```
The structure consists of two central 3×8 areas containing the lozenges, with a 4×8 area added at right and left. For a magic rectangle of this size the rank sum should be 791 = 7×113.

8×18 Board

These two new examples were constructed by splitting the 8×8 magic tour 00b in two and joining up the loose ends by four paths in direct quaternary symmetry.

 ``` 42 31 110 107 26 47 128 87 24 85 22 53 134 83 74 7 66 143 111 108 43 30 129 88 25 48 127 54 133 84 21 52 65 142 75 6 32 41 106 109 46 27 130 55 86 23 126 51 62 135 8 73 144 67 105 112 29 44 117 100 89 14 49 132 95 20 9 64 141 68 5 76 40 33 116 101 28 45 56 131 96 13 50 125 136 81 72 77 140 1 113 104 39 36 99 118 15 90 59 122 19 94 63 10 137 4 69 78 34 37 102 115 16 57 120 97 18 91 12 61 124 93 80 71 2 139 103 114 35 38 119 98 17 58 121 60 123 92 11 62 3 138 79 70 ``` ``` 42 31 110 107 26 53 98 123 126 55 58 11 128 83 74 7 66 143 111 108 43 30 99 124 25 54 97 12 127 84 57 10 65 142 75 6 32 41 106 109 52 27 96 125 122 59 56 13 82 129 8 73 144 67 105 112 29 44 117 100 51 24 95 14 85 130 9 64 141 68 5 76 40 33 116 101 28 45 94 121 50 131 60 15 136 81 72 77 140 1 113 104 39 36 93 118 49 20 23 86 89 132 63 16 137 4 69 78 34 37 102 115 46 21 120 91 48 133 18 61 88 135 80 71 2 139 103 114 35 38 119 92 47 22 19 90 87 134 17 62 3 138 79 70 ```

10×10 Board

Here is Tom Marlow's 10×10 semi-magic tour with oblique quaternary symmetry (i.e. 90 degree rotational symmetry) which appeared in The Games and Puzzles Journal, issue 25, March 2003, where a geometrical diagram is also given. The yellow coloured squares are those containing numbers of the forms 4n + 2 or 4n + 3: there must be an even number of these in a line for it to be magic.

 77 58 45 56 41 60 47 50 19 52 44 73 76 59 46 55 18 53 48 33 75 78 57 42 61 40 49 32 51 20 72 43 74 79 64 37 54 17 34 31 85 80 65 62 13 88 39 36 21 16 66 71 86 89 38 63 12 15 30 35 81 84 67 4 87 14 29 24 93 22 70 1 82 99 90 11 92 7 28 25 83 98 3 68 5 96 9 26 23 94 2 69 100 97 10 91 6 95 8 27

18×18 Board

Here is an 18×18 semimagic tour by Awani Kumar sent to me on 19 June 2013. It is based on 6×6 tours joined together.

```001 314 323 012 009 316 025 300 029 296 023 302 039 286 047 278 041 284
312 013 002 315 322 011 030 297 024 301 028 295 046 277 040 285 048 279
003 324 313 010 317 008 299 026 031 294 303 022 287 038 273 042 283 052
014 311 004 321 308 017 032 293 298 027 290 035 274 045 276 051 280 049
005 320 309 016 007 318 019 306 291 034 021 304 037 288 043 272 053 282
310 015 006 319 018 307 292 033 020 305 036 289 044 275 054 281 050 271
129 196 185 140 131 194 163 174 161 152 149 176 055 058 061 270 267 264
184 139 130 195 186 141 172 151 164 175 160 153 060 067 056 265 258 269
197 128 183 142 193 132 165 162 173 150 177 148 057 062 059 268 263 266
138 135 190 187 182 143 168 171 166 159 154 157 068 071 066 259 254 257
127 198 137 192 133 188 145 180 169 156 147 178 063 260 255 070 065 262
136 191 134 189 144 181 170 167 146 179 158 155 072 069 064 261 256 253
199 126 209 116 123 202 217 108 227 098 105 220 235 090 251 074 241 084
210 115 124 201 216 109 228 097 106 219 234 091 250 073 242 083 252 075
125 200 117 208 203 122 107 218 099 226 221 104 243 236 089 240 085 082
114 211 204 121 110 215 096 229 222 103 092 233 246 249 244 081 076 079
205 118 213 112 207 120 223 100 231 094 225 102 237 088 247 078 239 086
212 113 206 119 214 111 230 095 224 101 232 093 248 245 238 087 080 077
```

All ranks add to the magic number 2925.
File sums vary, e.g.: first file 2621, second file 3167
Regrettably I have not had time to check all the totals properly.

30×30 Board

Awani Kumar has also sent this even larger 30×30 semi-magic tour, on 21 June 2013. It is formed from the 18×18 example by surrounding it with 6×6 tours.

```001 890 899 012 009 892 021 880 869 032 023 878 039 862 051 850 041 860 057 844 069 832 059 842 817 814 823 078 087 084
888 013 002 891 898 011 868 031 022 879 870 033 852 049 040 861 052 849 068 833 058 843 070 831 822 077 816 085 824 079
003 900 889 010 893 008 881 020 867 034 877 024 863 038 851 050 859 042 845 056 067 834 841 060 815 818 813 088 083 086
014 887 004 897 884 017 030 027 874 871 866 035 856 853 048 045 848 053 066 063 838 071 830 835 076 821 074 827 080 825
005 896 885 016 007 894 019 882 029 876 025 872 037 864 855 858 043 046 055 846 065 836 061 840 073 828 819 812 089 082
886 015 006 895 018 883 028 875 026 873 036 865 854 857 044 047 054 847 064 837 062 839 072 829 820 075 090 081 826 811
275 278 285 616 623 626 291 610 599 302 293 608 309 592 317 584 311 590 327 574 335 566 329 572 797 802 799 102 099 104
284 617 276 625 286 615 600 301 292 609 598 303 318 585 310 591 316 583 334 565 328 573 336 567 800 091 796 105 810 101
277 274 279 622 627 624 611 290 597 304 607 294 593 308 319 582 589 312 575 326 561 330 571 340 803 798 801 100 103 098
280 283 618 287 614 621 604 601 300 297 596 305 320 581 586 315 578 323 562 333 564 339 568 337 092 795 094 807 106 809
273 630 281 620 271 628 289 612 603 606 295 298 307 594 579 322 313 588 325 576 331 560 341 570 095 804 107 794 097 806
282 619 272 629 288 613 602 605 296 299 306 595 580 321 314 587 324 577 332 563 342 569 338 559 108 093 096 805 808 793
631 646 643 258 255 270 417 484 425 476 419 482 451 462 449 440 437 464 545 550 547 354 351 356 779 784 781 120 117 122
644 259 632 269 642 257 426 477 418 483 424 475 460 439 452 463 448 441 548 343 544 357 558 353 782 109 778 123 792 119
647 268 645 256 633 254 485 416 427 474 481 420 453 450 461 438 465 436 551 546 549 352 355 350 785 780 783 118 121 116
260 265 262 639 636 641 428 473 478 423 470 431 456 459 454 447 442 445 344 543 346 555 358 557 110 777 112 789 124 791
263 648 267 634 253 638 415 486 471 430 421 480 433 468 457 444 435 466 347 552 359 542 349 554 113 786 125 776 115 788
266 261 264 637 640 635 472 429 422 479 432 469 458 455 434 467 446 443 360 345 348 553 556 541 126 111 114 787 790 775
649 664 661 240 237 252 487 414 497 404 411 490 505 396 515 386 393 508 523 378 539 362 529 372 761 766 763 138 135 140
662 241 650 251 660 239 498 403 412 489 504 397 516 385 394 507 522 379 538 361 530 371 540 363 764 127 760 141 774 137
665 250 663 238 651 236 413 488 405 496 491 410 395 506 387 514 509 392 531 524 377 528 373 370 767 762 765 136 139 134
242 247 244 657 654 659 402 499 492 409 398 503 384 517 510 391 380 521 534 537 532 369 364 367 128 759 130 771 142 773
245 666 249 652 235 656 493 406 501 400 495 408 511 388 519 382 513 390 525 376 535 366 527 374 131 768 143 758 133 770
248 243 246 655 658 653 500 401 494 407 502 399 518 383 512 389 520 381 536 533 526 375 368 365 144 129 132 769 772 757
667 670 677 224 231 234 685 216 209 206 695 692 703 198 713 188 195 706 721 180 173 170 731 728 739 162 755 146 741 160
678 223 668 233 684 217 208 205 696 693 702 199 714 187 196 705 720 181 172 169 732 729 738 163 754 145 740 161 756 147
669 676 671 230 225 232 215 686 207 210 691 694 197 704 189 712 707 194 179 722 171 174 727 730 155 746 151 750 159 742
222 679 228 673 218 683 204 697 688 213 200 701 186 715 708 193 182 719 168 733 724 177 164 737 152 753 154 747 148 749
675 672 681 220 229 226 687 214 699 202 211 690 709 190 717 184 711 192 723 178 735 166 175 726 745 156 751 150 743 158
680 221 674 227 682 219 698 203 212 689 700 201 716 185 710 191 718 183 734 167 176 725 736 165 752 153 744 157 748 149
```

All ranks add to the magic number 13515.
Here again I have not been able to check all the totals properly.

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