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April 20, 2004. Material about magic cubes continues to appear, but at a slower rate. I am
including some material here on other magic objects.
Also links to new pages.
Magic rectangles are similar to magic squares except (as the name suggests) they are bigger in one dimension then the other. All rows sum to a magic constant, and all columns sum to a different magic constant. Just as with magic squares, the concept can be carried over to higher dimensions. Obviously, diagonals are not required to be magic, because they cannot extend from corner to corner in an object with different dimensions.
The term Magic Rectangle has been in use at
least since 1908. [1]
The term cuboid is used by Abhinav Soni, but is defined in a
popular mathematics dictionary[2]
Aale de Winkel uses the term Magic Beam and Magic Hyperbeam
for 3 and higher dimension objects of this type. [3]
(Even, even) orders are relatively simple to construct. (Odd, odd) are more difficult. (Even, odd) normal magic rectangles are impossible.
| Here are two simple examples of magic rectangles from Marián Trenkler. [4] | 1
7 6 4 and 01
10 14 09 06 8 2 3 5 15 02 07 11 05 08 12 03 04 13 |
If the even, even order rectangle desired has 1 dimension an even multiple of the other, there is a very simple method available. Using a magic square of order equal to the smaller dimension for a pattern, distribute the numbers from 1 to m among the m cells of the rectangle.
Example:
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As this figure consists of 4 order-4 magic squares, they may be arranged to form an order-8 magic square. (I used this same idea to construct a pyramid of 16 order-4 squares for the frontispiece of the Magic Square Lexicon. [5]
| Here is an Order-8 magic
square with inlaid 4x6 magic rectangle. This was constructed by then 90 year old E. W. Shineman.,Jr. © 2005 (used by permission). The 8x8 is a pandiagonal magic square S=260 The 4x6 is a magic rectangle S=195 and 130; (195/6=32.5*4=130). All sets of 8 numbers of like color sum to 260 |
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As I mentioned above, magic
rectangles can be extended to the third (or higher) dimension.
Here is a 3x3x11 constructed by Abhinav Soni [6]
A 3x3x11 magic cuboid. Rows sum to 550, columns and pillars sum to 150.
The text listing:
Front face Middle plane Back face
88 95 82 84 69 08 48 19 26 14 17 34 43 66 57 54 79 04 40 53 64 56 28 12 02 09 27 63 98 91 71 72 77
61 45 35 42 60 96 32 58 38 39 44 22 29 16 18 03 41 81 85 92 80 83 67 76 99 90 87 13 37 07 20 31 23
01 10 33 24 21 46 70 73 86 97 89 94 78 68 75 93 30 65 25 05 06 11 55 62 49 51 36 74 15 52 59 47 50
[1] Andrews, W. S., Magic Squares & Cubes, Open Court,
1908, page 170. Also the same page in Edition 2,
1960
[2] Harper-Collins Mathematical Dictionary
[3] Aale de Winkel, The Magic Encyclopedia, at
http://www.magichypercubes.com/Encyclopedia/index.html
[4] Marián Trenkler, The
Mathematical Gazette, March,1999
[5] The Lexicon is available at
http://www.magic-squares.net/BookSale.htm
[6] Soni no longer has a web page (Nov./09).
Mitsutoshi Nakamura also has a good site on magic rectangles, at
http://homepage2.nifty.com/googol/magcube/en/rectangles.htm
Order-10 Pantriagonal Cube
Recently I was reminded that in November, 2005, Walter Trump advised me of a transformation from single even to pantriagonal magic cubes. He reported that he first saw it (as applied to squares) on Mutsumi Suzuki’s now defunct Web site.
This is the procedure for an order 10 associated simple magic cube
Starting with the X planes, exchange plane x=6 with x=10 and plane x=7 with
plane x=9.
With this new cube, exchange plane y=6 with y=10 and plane y=7 with plane y=9.
Finally, with this new cube, exchange plane z=6 with z=10 and plane z=7 with
plane z=9.
The resulting cube is a non-associated pantriagonal magic cube.
I tested this procedure out on the on the Planck order-10 associated cube. The following is the resulting pantriagonal cube. Being pantriagonal, this cube can be changed to another pantriagonal magic cube simply by moving any outside plane to the opposite side of the cube. However, it is no longer associated!
Horizontal plane 10 – top Horizontal plane 9 _ top-1 1000 999 903 94 6 991 992 8 7 5 191 109 898 897 805 110 102 893 894 106 990 912 83 17 986 981 19 18 14 985 120 889 888 814 185 111 882 883 117 116 921 72 28 977 976 30 29 23 974 975 880 879 823 174 126 871 872 128 127 125 61 39 968 967 935 40 32 963 964 36 870 832 163 137 866 861 139 138 134 865 50 959 958 944 55 41 952 953 47 46 841 152 148 857 856 150 149 143 854 855 910 909 93 4 95 901 902 998 97 96 101 192 808 807 195 200 199 803 804 896 920 82 13 84 916 911 989 88 87 915 181 819 818 184 115 190 812 813 887 186 71 22 73 927 926 980 79 78 924 925 830 829 173 124 175 821 822 878 177 176 31 62 938 937 65 70 69 933 934 966 840 162 133 164 836 831 869 168 167 835 51 949 948 54 45 60 942 943 957 56 151 142 153 847 846 860 159 158 844 845 Horizontal plane 8_ top-2 Horizontal plane 7 _ top-3 800 702 293 207 796 791 209 208 204 795 310 699 698 604 395 301 692 693 307 306 711 282 218 787 786 220 219 213 784 785 690 689 613 384 316 681 682 318 317 315 271 229 778 777 725 230 222 773 774 226 680 622 373 327 676 671 329 328 324 675 240 769 768 734 265 231 762 763 237 236 631 362 338 667 666 340 339 333 664 665 760 759 743 254 246 751 752 248 247 245 351 349 658 657 645 350 342 653 654 346 710 292 203 294 706 701 799 298 297 705 391 609 608 394 305 400 602 603 697 396 281 212 283 717 716 790 289 288 714 715 620 619 383 314 385 611 612 688 387 386 221 272 728 727 275 280 279 723 724 776 630 372 323 374 626 621 679 378 377 625 261 739 738 264 235 270 732 733 767 266 361 332 363 637 636 670 369 368 634 635 750 749 253 244 255 741 742 758 257 256 341 352 648 647 355 360 359 643 644 656 Horizontal plane 6 _ top-4 Horizontal plane 5 _ bottom+4 501 492 408 597 596 410 409 403 594 595 100 99 3 904 905 91 92 908 997 906 481 419 588 587 515 420 412 583 584 416 90 12 913 914 86 81 919 988 917 85 430 579 578 524 475 421 572 573 427 426 21 922 923 77 76 930 979 928 74 75 570 569 533 464 436 561 562 438 437 435 931 932 68 67 35 970 939 63 64 936 560 542 453 447 556 551 449 448 444 555 941 59 58 44 945 950 52 53 947 956 491 402 493 507 506 600 499 498 504 505 10 9 993 994 996 1 2 98 907 995 411 482 518 517 485 490 489 513 514 586 20 982 983 987 16 11 89 918 984 15 471 529 528 474 425 480 522 523 577 476 971 972 978 27 26 80 929 973 24 25 540 539 463 434 465 531 532 568 467 466 961 969 38 37 965 940 962 33 34 66 550 452 443 454 546 541 559 458 457 545 960 49 48 954 955 951 42 43 57 946 Horizontal plane 4 _ bottom+3 Horizontal plane 3 _ bottom+2 801 802 198 197 105 900 809 193 194 806 300 202 703 704 296 291 709 798 707 295 811 189 188 114 815 820 182 183 817 886 211 712 713 287 286 720 789 718 284 285 180 179 123 824 825 171 172 828 877 826 721 722 278 277 225 780 729 273 274 726 170 132 833 834 166 161 839 868 837 165 731 269 268 234 735 740 262 263 737 766 141 842 843 157 156 850 859 848 154 155 260 259 243 744 745 251 252 748 757 746 891 899 108 107 895 810 892 103 104 196 210 792 793 797 206 201 299 708 794 205 890 119 118 884 885 881 112 113 187 816 781 782 788 217 216 290 719 783 214 215 130 129 873 874 876 121 122 178 827 875 771 779 228 227 775 730 772 223 224 276 140 862 863 867 136 131 169 838 864 135 770 239 238 764 765 761 232 233 267 736 851 852 858 147 146 160 849 853 144 145 250 249 753 754 756 241 242 258 747 755 Horizontal plane 2 _ bottom+1 Horizontal plane 1 _ bottom 601 399 398 304 605 610 392 393 607 696 401 502 503 497 496 510 599 508 494 495 390 389 313 614 615 381 382 618 687 616 511 512 488 487 415 590 519 483 484 516 380 322 623 624 376 371 629 678 627 375 521 479 478 424 525 530 472 473 527 576 331 632 633 367 366 640 669 638 364 365 470 469 433 534 535 461 462 538 567 536 641 642 358 357 345 660 649 353 354 646 460 442 543 544 456 451 549 558 547 455 700 309 308 694 695 691 302 303 397 606 591 592 598 407 406 500 509 593 404 405 320 319 683 684 686 311 312 388 617 685 581 589 418 417 585 520 582 413 414 486 330 672 673 677 326 321 379 628 674 325 580 429 428 574 575 571 422 423 477 526 661 662 668 337 336 370 639 663 334 335 440 439 563 564 566 431 432 468 537 565 651 659 348 347 655 650 652 343 344 356 450 552 553 557 446 441 459 548 554 445
At first glance this is an ordinary simple magic associated order-5 magic cube. Because it is associated, the center plane in each of the 3 orientations is an associated magic square. The cube contains no other magic squares.
However, on closer inspection, all possible lines of 5 numbers sum to a multiple of 5. This includes the straight (1 segment) lines, and also the 2-segment and 3-segment lines.
So this is a Perfect magic cube (Modulo 5).
75 1-agonals, 150 2-agonals, and 100 3-agonals all sum to 0 mod 5.
Because it is perfect mod 5, this cube also contains 45 order-5 pandiagonal
magic squares (mod 5).
Walter
Trump sent me this cube via email on Sept. 3, 2005. He easily constructed it
using a magic cube generating program written by Peter Bartsch.
Horizontal plane 5 – top Horizontal plane 4 Horizontal plane 3
94 23 52 106 40 111 45 99 3 57 8 62 116 50 79
55 109 38 92 21 97 1 60 114 43 119 48 77 6 65
36 95 24 53 107 58 112 41 100 4 80 9 63 117 46
22 51 110 39 93 44 98 2 56 115 61 120 49 78 7
108 37 91 25 54 5 59 113 42 96 47 76 10 64 118
Horizontal plane 2 Horizontal plane 1 - bottom
30 84 13 67 121 72 101 35 89 18
11 70 124 28 82 33 87 16 75 104
122 26 85 14 68 19 73 102 31 90
83 12 66 125 29 105 34 88 17 71
69 123 27 81 15 86 20 74 103 32
Magic Knight Tour - 1 ..... on order-8 cube surfaces.
| On my
unusual cubes page I show an illustration of an 8x8x8 cube with a
chess knight tour on each of the six surfaces. It was constructed by H. E. Dudeney and published in 1917. If each move of the knight is numbered, rows and columns sum to various values. However, he does mention that knight tours of 8x8 (chess)boards are possible with all rows and columns summing to the same constant value. However, at that time no KT had been discovered where the diagonals also summed correctly. It was proven in August of 2003 that it is impossible for a knight tour
on an 8x8 board to have row and columns, and also the 2 main diagonals all
sum to the same value. Awani Kumar is an active investigator of Knight Tours. In May 2007, he
improved on Dudeney's cube faces tour. More on Knight Tours and links to other sites are on my Knight Tours page. |
Awani Kumar's MKT on the six faces of an |
Listing of Knight moves for the above cube surface tours.
Rows & columns = 1540, diagonals = 1494 & 1592 diagonals = 1504 & 1496 203 182 207 186 209 164 223 166 55 38 347 330 35 58 351 326 206 185 204 183 222 167 162 211 346 331 56 37 350 327 34 59 181 202 187 208 163 210 165 224 39 54 329 348 57 36 325 352 188 205 184 201 168 221 212 161 332 345 40 53 328 349 60 33 175 180 189 220 193 200 169 214 41 52 333 344 45 62 339 324 190 219 174 177 172 213 196 199 334 343 44 49 340 323 46 61 179 176 217 192 197 194 215 170 51 42 341 336 63 48 321 338 218 191 178 173 216 171 198 195 342 335 50 43 322 337 64 47 Rows & columns = 1540, diagonals = 1866 & 1870 diagonals = 1504 & 1496 359 10 25 376 361 8 23 378 246 239 146 139 226 241 160 143 26 375 360 9 24 377 6 363 147 138 245 240 159 144 225 242 373 358 11 28 7 362 379 22 238 247 140 145 244 227 142 157 12 27 374 357 380 21 364 5 137 148 237 248 141 158 243 228 353 372 29 16 365 4 381 20 236 249 136 149 232 253 156 129 32 13 356 369 384 17 366 3 135 150 233 252 153 132 229 256 371 354 15 30 1 368 19 382 250 235 152 133 254 231 130 155 14 31 370 355 18 383 2 367 151 134 251 234 131 154 255 230 Rows & columns = 1540, diagonals = 1504 & 1496 diagonals = 1504 & 1496 294 319 90 67 298 315 70 87 262 123 288 97 260 125 274 111 91 66 295 318 69 88 299 314 287 98 261 124 275 110 257 128 320 293 68 89 316 297 86 71 122 263 100 285 126 259 112 273 65 92 317 296 85 72 313 300 99 286 121 264 109 276 127 258 292 307 94 77 312 301 84 73 266 101 284 117 280 113 272 107 93 78 291 308 81 76 311 302 283 120 265 104 269 108 277 114 306 289 80 95 304 309 74 83 102 267 118 281 116 279 106 271 79 96 305 290 75 82 303 310 119 282 103 268 105 270 115 278
Magic Knight Tour - 2 ..... in an Order-8 cube
On April 28, 2007, Awani Kumar announced via email, the first solution to the magic Knight Tour of an order-8 cube.
When the steps a chess knight takes while traveling from cell to cell through the cube are numbered
This cube consists of the numbered steps of a knight as it travels through the 512 cells of the cube. All rows, columns and pillars sum correctly to 2052. Only two of the four triagonals sum to this value. The other two sum to 2500 and 2020. Features of this cube were confirmed by Dan Thomasson email of April 30,2007.
However, Knight Tour fans do not require that the n-agonals sum correctly for a tour to be considered magic. Magic square and cube fans, of course, would consider this cube to be only semi-magic.
There are no magic squares in this cube, because none of the planar diagonals sum correctly (not a requirement).
The tour is re-entrant, meaning that the last cell visited is exactly one knight move away from the first cell. Therefore the tour may be started on any cell, and will successfully visit all 512 cells in the cube. Of course, because the numbers are now in different positions, the tour will no longer be magic.
Kumar reports however, that the tour will be magic if started at position 257 instead of position 1.
More on Knight Tours, and links to other sites are on my
Knight Tours page.
An 0rder-4 cube Magic Knight Tour is shown on my Unusual cubes pages.
Addendum: Listed here is the cube mentioned above.
However, it is NOT magic because only 2 of the four triagonals sum correctly!
On May 8, 2007, Kumar announced a new magic knight tour order-8 cube that had
all 4 triagonals correct. This cube is simple magic, not associated, and
includes no order 8 magic squares, but consists of a reentry knight tour!
I list it on my Knight Tours
page.
Listing for the order-8 Knight Tour cube.
Planes 1 & 2 19 482 509 16 461 480 35 50 510 15 20 481 472 453 58 43 490 27 8 501 36 49 462 479 7 502 489 28 57 44 471 454 511 14 17 484 465 452 63 46 18 483 512 13 460 473 38 55 6 503 492 25 64 45 466 451 491 26 5 504 37 56 459 474 117 100 411 398 429 448 67 82 414 395 116 101 440 421 90 75 410 399 120 97 68 81 430 447 113 104 415 394 89 76 439 422 387 118 109 412 433 420 95 78 108 413 390 115 428 441 70 87 112 409 386 119 96 77 434 419 391 114 105 416 69 88 427 442
Planes 3 & 4 495 30 1 500 33 52 463 478 2 499 496 29 60 41 470 455 22 487 508 9 464 477 34 51 507 10 21 488 469 456 59 42 3 498 493 32 61 48 467 450 494 31 4 497 40 53 458 475 506 11 24 485 468 449 62 47 23 486 505 12 457 476 39 54 99 406 397 124 65 84 431 446 396 125 102 403 92 73 438 423 400 121 98 407 432 445 66 83 103 402 393 128 437 424 91 74 405 388 123 110 93 80 435 418 126 107 404 389 72 85 426 443 122 111 408 385 436 417 94 79 401 392 127 106 425 444 71 86
Planes 5 & 6 231 252 285 258 219 296 197 314 276 271 234 245 306 221 304 195 286 257 232 251 294 201 316 215 233 246 275 272 207 308 209 302 249 230 259 288 311 220 297 198 270 273 248 235 222 289 196 319 260 287 250 229 202 309 216 299 247 236 269 274 291 208 317 210 145 136 383 362 165 334 339 188 378 367 152 129 330 161 192 343 382 363 148 133 340 187 326 173 149 132 379 366 191 344 169 322 359 146 137 384 331 164 189 342 144 377 354 151 168 335 338 185 140 381 358 147 190 341 172 323 355 150 141 380 337 186 327 176
Planes 7 & 8 253 226 263 284 295 204 313 214 266 277 244 239 206 305 212 303 264 283 254 225 218 293 200 315 243 240 265 278 307 224 301 194 227 256 281 262 203 312 213 298 280 267 238 241 290 205 320 211 282 261 228 255 310 217 300 199 237 242 279 268 223 292 193 318 135 370 361 160 179 348 325 174 368 153 130 375 352 183 170 321 364 157 134 371 166 333 180 347 131 374 365 156 329 162 351 184 369 360 159 138 349 182 171 324 154 143 376 353 178 345 328 175 158 139 372 357 332 163 350 181 373 356 155 142 167 336 177 346
Order-12 Magic Knight Tour Cube
On June 19 2007, Awani Kumar announced via email, the first solution to the magic Knight Tour of an order-12 cube.
This is a simple magic cube with all rows, columns, pillars, and the 4 main triagonals summing correctly to 10374. It is classed as simple magic, not associated, with no included order 12 magic squares. The knight tour is not quite re-entrant.
The Knight Tours of both this cube and the magic order-8 cube (May 8) mentioned above were confirmed correct by Guenter Stertenbrink.
Cube_12-Kumar-KT.xls 1 1136 601 578 1143 626 1117 624 1091 1088 649 658 1063 583 1138 1129 608 1119 628 1089 622 659 1062 1085 652 579 1142 1133 604 1099 616 1109 634 663 1058 1081 656 1132 605 582 1139 614 1097 636 1111 1084 653 662 1059 602 591 1144 1121 1118 625 1092 623 650 671 1064 1073 1137 1128 607 586 627 1120 621 1090 1061 1076 651 670 1141 1124 603 590 615 1100 633 1110 1057 1080 655 666 606 587 1140 1125 1098 613 1112 635 654 667 1060 1077 758 987 970 743 996 1015 730 717 1052 685 694 1027 975 738 755 990 735 716 997 1010 695 1026 1049 688 985 744 757 972 1017 1006 707 728 691 1030 1053 684 756 973 992 737 710 721 1024 1003 1056 681 690 1031 748 965 984 761 734 713 1000 1011 686 699 1028 1045 977 768 749 964 993 1014 731 720 1025 1048 687 698 967 762 747 982 711 724 1021 1002 1029 1044 683 702 750 979 962 767 1020 1007 706 725 682 703 1032 1041 942 773 796 947 908 919 818 813 876 887 850 845 769 938 951 800 831 804 901 922 863 836 869 890 795 948 781 934 817 814 907 920 849 846 875 888 952 799 930 777 902 921 832 803 870 889 864 835 772 939 950 797 918 905 816 819 886 873 848 851 943 776 793 946 801 830 923 904 833 862 891 872 949 798 931 780 815 820 917 906 847 852 885 874 794 945 784 935 924 903 802 829 892 871 834 861 2 1146 1135 600 577 1102 609 1108 639 1066 1087 648 657 593 584 1151 1130 611 1104 637 1106 645 660 1067 1086 597 580 1147 1134 631 1116 617 1094 641 664 1071 1082 1150 1131 596 581 1114 629 1096 619 1070 1083 644 661 592 1145 1122 599 610 1101 640 1107 672 1065 1074 647 1127 594 585 1152 1103 612 1105 638 1075 646 669 1068 1123 598 589 1148 1115 632 1093 618 1079 642 665 1072 588 1149 1126 595 630 1113 620 1095 668 1069 1078 643 986 759 742 971 1018 1005 708 727 1038 1051 676 693 739 974 991 754 709 722 1023 1004 673 696 1039 1050 741 988 969 760 995 1016 729 718 677 692 1035 1054 976 753 740 989 736 715 998 1009 1034 1055 680 689 968 745 764 981 712 723 1022 1001 700 1037 1046 675 765 980 961 752 1019 1008 705 726 1047 674 697 1040 763 966 983 746 733 714 999 1012 1043 678 701 1036 978 751 766 963 994 1013 732 719 704 1033 1042 679 956 787 782 933 914 909 812 823 882 877 844 855 791 960 929 778 805 826 927 900 837 858 895 868 941 774 955 788 811 824 913 910 843 856 881 878 770 937 792 959 928 899 806 825 896 867 838 857 790 957 932 779 912 915 822 809 880 883 854 841 953 786 783 936 827 808 897 926 859 840 865 894 771 940 789 958 821 810 911 916 853 842 879 884 944 775 954 785 898 925 828 807 866 893 860 839 3 1164 573 550 1171 1190 537 1212 519 1242 503 1222 491 547 1174 1165 572 539 1208 517 1194 1219 494 1247 498 551 1170 1161 576 527 1188 529 1214 1227 486 1239 506 1168 569 546 1175 1202 525 1200 531 1234 511 1230 483 574 555 1172 1157 538 1205 520 1195 504 1241 492 1221 1173 1156 571 558 1191 540 1209 518 493 1220 497 1248 1169 1160 575 554 1203 528 1197 530 485 1228 505 1240 570 559 1176 1153 526 1185 532 1215 512 1233 484 1229 1342 401 1316 399 1308 1287 418 445 468 1277 464 1249 391 1340 409 1318 431 436 1301 1290 1253 460 469 1276 403 1328 397 1330 417 446 1307 1288 449 1264 477 1268 1322 389 1336 411 1302 1289 432 435 476 1269 1260 453 386 1341 416 1315 1286 1305 448 419 1278 467 1250 463 1339 408 1317 394 433 430 1291 1304 459 1254 1275 470 1327 388 1329 414 447 420 1285 1306 1263 450 1267 478 406 1321 396 1335 1292 1303 434 429 1270 475 454 1259 362 1345 1376 375 348 327 1378 1405 316 301 1430 1411 1349 366 371 1372 1391 1396 341 330 1427 1414 317 300 383 1368 361 1346 1377 1406 347 328 1431 1410 313 304 1364 379 1350 365 342 329 1392 1395 320 297 1426 1415 1352 367 370 1369 326 345 1408 1379 302 1435 1412 309 363 1348 1373 374 1393 1390 331 344 1413 308 299 1438 1361 378 1351 368 1407 1380 325 346 1409 312 303 1434 382 1365 364 1347 332 343 1394 1389 298 1439 1416 305 4 1182 1163 564 549 522 1189 536 1211 502 1243 490 1223 565 548 1179 166 1207 524 1193 534 495 1218 499 1246 561 552 1183 1162 1187 544 1213 514 487 1226 507 1238 1178 1167 568 545 542 1201 516 1199 510 1235 482 1231 556 1181 1158 563 1206 521 1196 535 1244 501 1224 489 1155 566 557 1180 523 1192 533 1210 1217 496 1245 500 1159 562 553 1184 543 1204 513 1198 1225 488 1237 508 560 1177 1154 567 1186 541 1216 515 1236 509 1232 481 402 1325 400 1331 1282 1309 444 423 1280 465 1252 461 1323 392 1333 410 437 426 1295 1300 457 1256 1273 472 1343 404 1313 398 443 424 1281 1310 1261 452 1265 480 390 1337 412 1319 1296 1299 438 425 1272 473 456 1257 1326 385 1332 415 1312 1283 422 441 466 1279 462 1251 407 1324 393 334 427 440 1297 1294 1255 458 471 1274 387 1344 413 1314 421 442 1311 1284 451 1262 479 1266 1338 405 1320 395 1298 1293 428 439 474 1271 1258 455 384 1367 1354 353 322 349 1404 1383 1422 315 292 1429 1363 380 357 1358 1397 1386 335 340 293 1428 1419 318 1353 354 1375 376 1403 1384 321 350 289 1432 1423 314 358 1357 372 1371 336 339 1398 1385 1418 319 296 1425 1362 377 360 1359 352 323 1382 1401 1436 1421 310 291 381 1366 1355 356 1387 1400 337 334 307 294 1437 1420 359 1360 369 1370 1381 1402 351 324 311 290 1433 1424 1356 355 1374 373 338 333 1388 1399 1440 1417 306 295 5 18 1725 16 1699 1666 1693 60 39 1664 1641 82 71 1727 4 1697 30 59 40 1665 1694 83 70 1661 1644 1707 24 1717 10 53 42 1679 1684 87 66 1657 1648 6 1705 28 1719 1680 1683 54 41 1660 1645 86 67 1726 1 1700 31 1692 1671 34 61 74 1663 1640 81 19 1728 13 1698 33 62 1691 1672 1637 84 75 1662 7 1708 25 1718 47 52 1685 1674 1633 88 79 1658 1706 21 1720 11 1686 1673 48 51 78 1659 1636 85 182 1563 1546 167 1572 1591 154 141 1628 109 118 1603 1551 162 179 1566 159 140 1573 1586 119 1602 1625 112 1561 168 181 1548 1593 1582 131 152 115 1606 1629 108 180 1549 1568 161 134 145 1600 1579 1632 105 114 1607 172 1541 1560 185 158 137 1576 1587 110 123 1604 1621 1553 192 173 1540 1569 1590 155 144 1601 1624 111 122 1543 186 171 1558 135 148 1597 1578 1605 1620 107 126 174 1555 1538 191 1596 1583 130 149 106 127 1608 1617 1526 1511 220 201 250 237 1476 1495 288 1449 1458 263 1519 1534 193 212 1477 1490 255 236 1463 258 281 1456 219 202 1525 1512 227 248 1497 1486 1459 262 285 1452 194 211 1520 1533 1504 1483 230 241 284 1453 1462 259 1514 1531 200 213 1480 1491 254 233 266 287 1448 1457 1523 1506 221 208 251 240 1473 1494 1441 1464 271 282 199 214 1513 1532 1501 1482 231 244 1445 1460 267 286 222 207 1524 1505 226 245 1500 1487 270 283 1444 1461 6 1710 17 1716 15 1696 1667 38 57 1642 95 72 1649 3 1712 29 1714 37 58 1695 1668 69 1652 1643 94 23 1724 9 1702 43 56 1681 1678 65 1656 1647 90 1722 5 1704 27 1682 1677 44 55 1646 91 68 1653 2 1709 32 1715 1670 1689 64 35 96 73 1650 1639 1711 20 1713 14 63 36 1669 1690 1651 1638 93 76 1723 8 1701 26 49 46 1675 1688 1655 1634 89 80 22 1721 12 1703 1676 1687 50 45 92 77 1654 1635 1562 183 166 1547 1594 1581 132 151 1614 1627 100 117 163 1550 1567 178 133 146 1599 1580 97 120 1615 1626 165 1564 1545 184 1571 1592 153 142 101 116 1611 1630 1552 177 164 1565 160 139 1574 1585 1610 1631 104 113 1544 169 188 1557 136 147 1598 1577 124 1613 1622 99 189 1556 1537 176 1595 1584 129 150 1623 98 121 1616 187 1542 1559 170 157 138 1575 1588 1619 102 125 1612 1554 175 190 1539 1570 1589 156 143 128 1609 1618 103 218 203 1528 1509 228 247 1498 1485 1450 1471 264 273 195 210 1517 1536 1503 1484 229 242 257 280 1455 1466 1527 1510 217 204 249 238 1475 1496 261 276 1451 1470 1518 1535 196 209 1478 1489 256 235 1454 1467 260 277 198 215 1516 1529 1502 1481 232 243 1472 265 274 1447 223 206 1521 1508 225 246 1499 1488 279 1442 1465 272 1515 1530 197 216 1479 1492 253 234 275 1446 1469 268 1522 1507 224 205 252 239 1474 1493 1468 269 278 1443
Edward's
Multiply Square
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Symmetric Magic Diamonds
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Ted Harper sent me these patterns in July, 2005. This is an order-8 simple
magic square , numbered from 0 to 63 so S = 252.
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 |
| This is another simple magic square. This time, the integers have been converted to the binary number system in the diamond. Again consider just the digits when checking out the symmetry.
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Durupt Bent-diagonal Order-8 Cube
In 2005, Arséne Durupt (France) constructed an order 8 pantriagonal magic cube with some nice additional features. At that time he was 83 years old and did this without the help of a computer.
Features:
·
Sum of the two planar diagonals in
each plane is 4104 ( which = 2S)
(No planes have correctly summing diagonals, so there are no magic squares.)
·
Sums of the corners of all orders
2, 4, 6 and 8 cubes (within this main cube) is equal to S
This is with the corner of the sub-cube on any cell and includes wrap-around.
· Because corners of all 2x2 squares sum to ½ S, this cube is classed as compact.
·
Horizontal Planar Bent Diagonals (V
shaped):
on all horizontal planes, and vertical planes parallel to the front of the cube
starting on columns 1 and 5,
on all vertical planes parallel to the sides, starting on columns 3 and 7.
There are no planes that have all vertical bent diagonals starting on any
particular row or column.
I show a cube
with similar (but slightly improved features)
here. It was constructed by Abhinav Soni in 2005 (using a computer).
I show a cube by John Hendricks here
that contains 28 order-4 magic cubes, so contains many bent triagonals!
Durupt’s cube.
Horizontal plane 1 2 512 2 510 4 509 3 511 1 65 447 67 445 68 446 66 448 273 239 275 237 276 238 274 240 176 338 174 340 173 339 175 337 496 18 494 20 493 19 495 17 81 431 83 429 84 430 82 432 257 255 259 253 260 254 258 256 192 322 190 324 189 323 191 321 208 306 206 308 205 307 207 305 369 143 371 141 372 142 370 144 33 479 35 477 36 478 34 480 416 98 414 100 413 99 415 97 224 290 222 292 221 291 223 289 353 159 355 157 356 158 354 160 49 463 51 461 52 462 50 464 400 114 398 116 397 115 399 113 Horizontal plane 3 4 320 194 318 196 317 195 319 193 129 383 131 381 132 382 130 384 465 47 467 45 468 46 466 48 112 402 110 404 109 403 111 401 304 210 302 212 301 211 303 209 145 367 147 365 148 366 146 368 449 63 451 61 452 62 450 64 128 386 126 388 125 387 127 385 16 498 14 500 13 499 15 497 433 79 435 77 436 78 434 80 225 287 227 285 228 286 226 288 352 162 350 164 349 163 351 161 32 482 30 484 29 483 31 481 417 95 419 93 420 94 418 96 241 271 243 269 244 270 242 272 336 178 334 180 333 179 335 177 Horizontal plane 5 6 508 6 506 8 505 7 507 5 69 443 71 441 72 442 70 444 277 235 279 233 280 234 278 236 172 342 170 344 169 343 171 341 492 22 490 24 489 23 491 21 85 427 87 425 88 426 86 428 261 251 263 249 264 250 262 252 188 326 186 328 185 327 187 325 204 310 202 312 201 311 203 309 373 139 375 137 376 138 374 140 37 475 39 473 40 474 38 476 412 102 410 104 409 103 411 101 220 294 218 296 217 295 219 293 357 155 359 153 360 154 358 156 53 459 55 457 56 458 54 460 396 118 394 120 393 119 395 117 Horizontal plane 7 8 316 198 314 200 313 199 315 197 133 379 135 377 136 378 134 380 469 43 471 41 472 42 470 44 108 406 106 408 105 407 107 405 300 214 298 216 297 215 299 213 149 363 151 361 152 362 150 364 453 59 455 57 456 58 454 60 124 390 122 392 121 391 123 389 12 502 10 504 9 503 11 501 437 75 439 73 440 74 438 76 229 283 231 281 232 282 230 284 348 166 346 168 345 167 347 165 28 486 26 488 25 487 27 485 421 91 423 89 424 90 422 92 245 267 247 265 248 266 246 268 332 182 330 184 329 183 331 181
Fillion Bent-diagonal Order-16 Cube
On June 10, 2007 I received an email from Jacques Fillion mentioning that he had constructed an order 16 cube with bent diagonals.
This cube has the following features:
Horizontal Bent Diagonals (i.e. 2 sections of 8 cells at 90 degrees).
Vertical Bent diagonals
Good work Jacques!
Recent
Publications and Postings
Recent publishings by this editor, in The Journal of Recreational Mathematics
|
Problem 2617: Magic Cube of Primes |
Harvey D. Heinz |
JRM:31:4:2002-2003: 298 |
|
Problem 2584: Prime (Magical) Square (Solution) |
Harvey D. Heinz |
JRM:32:1:2003-2004: 30-36 |
|
Problem 2617: Magic Cube of Primes (Solution) |
Allen Wm. Johnson, Jr. |
JRM:32:4:2003-2004: 338-9 |
|
A Unified Classification System for Magic Hypercubes |
H. Heinz and J. Hendricks |
JRM:32:1:2003-2004: 30-36 |
|
The First (?) Magic Cube |
Harvey D. Heinz |
JRM:33:2:2004-2005: 111-115 |
|
The First (?) Perfect Magic Cubes |
Harvey D. Heinz |
JRM:33:2:2004-2005: 116-119 |
| Hypercube Classes - An Update | Harvey D. Heinz | JRM:35:1:2006: 5-10 |
| Magic Tesseract Classes | Harvey D. Heinz | JRM:35:1:2006: 11-14 |
| Magic 9x5 Hexagrams | April 4, 2007 | This new page shows 1
solution for each S of 46 to 54 of the 5 numbers in each of 9 lines magic
hexagram. |
| Most-perfect Bent-diagonal Magic Squares | April 4, 2007 | This new page starts with a
discussion of "the ‘most magical magic square in 5,000 years’. (It's not!) Then is shown order-12 and order-16 Most-perfect Bent diagonal magic squares. |
| Knight Tours | April 14, 2007 | . A little bit about another old recreation, touring the
chessboard with a knight, then arranging the tour so the numbered steps
form a magic square. |
| Compact Magic Squares | May 18, 2007 | A proof and demonstration that all sub-arrays with even
dimensions are pan-magic in a compact magic square. |
| Addendum to Order 6 stars | May 17,2007 | A short article on the first solvers of the magic hexagon (6-pointed star). |
| Magic Square Update | Sept. 29/2009 | 3 new types. Normal squares just a subset. Postage stamp
m.s., etc. |
For their contributions to these five latest pages, I wish to
thank the following (in no particular order).
Christian Boyer, Ted Harper, Awani Kumar, Donald
Morris, Abhinav Soni, Walter Trump, and Aale de Winkel.
And thanks to the following for recent correspondence pointing
out typos (or other errors) on my pages.
Jp Gesukens, Frans Lelieveld, and Yu Jianbin.
New Tesseract Pages
On November 22, 2007, I posted 11 pages devoted to magic
Tesseracts. Because of space restraints, I again had to open another site.
However, all 3 sites on this subject; My Geocities
Magic-squares.net site, My Shaw cubes site, and
now my Shaw Tesseract site are linked by buttons in the button bars of most
pages. For those of you with high speed Internet (most I hope) the delay in
switching sites should be almost non-existent.
I apologize for any inconvenience. My hope is to eventually have all these sites
consolidated into one big site.
The new site is http://members.shaw.ca/tesseracts/
Nasik (Perfect) Hypercube Generators
Recently, Dwane Campbell posted a new site that discusses Perfect magic hypercubes.
He uses Hendricks definition of perfect, where all possible lines must sum correctly. To make certain that he s understood correctly, he also uses Nakamura's definitions of pan-2,3-agonal for cubes, pan-2,3,4-agonal for tesseracts, pan-2,3,4,5-agonal for dimension 5 perfect hypercubes, etc. He finalizes the definition with my new suggestion of the term Nasik.
Campbell discusses a basic method of constructing this type of magic hypercube. He supplies references, examples, and some definitions.
He concludes with 3 perfect magic hypercube generators and an order-8 magic cube tester spreadsheet that may be downloaded. The tester is quite comprehensive, although applicable only to order-8 cubes.
Dwane Campbell's website is at http://home.earthlink.net/~dwanecampbel/
Please send me Feedback about my Web
site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
February 05, 2010
Copyright © 2005 by Harvey D. Heinz
