Magic Cubes - Order 12
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Magic Cubes - Order 12
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Not shown on this page is my order 12 composition magic cube, which appears on the Composition Cubes page. Three more inlaid magic cubes and an inlaid tesseract appear on my Inlaid Cubes page.
Order 12 cubes on other pages
Poyo Simple Magic cubePoyo’s associated simple magic cube was obtained
from Suzuki's original web page. [1]
I will show the top horizontal plane only. 1728 145 289 1296 1152 721 865 720 576 1297 1441 144 13 1572 1428 445 589 996 852 1021 1165 420 276 1597 25 1560 1416 457 601 984 840 1033 1177 408 264 1609 1692 181 325 1260 1116 757 901 684 540 1333 1477 108 1680 193 337 1248 1104 769 913 672 528 1345 1489 96 61 1524 1380 493 637 948 804 1069 1213 372 228 1645 73 1512 1368 505 649 936 792 1081 1225 360 216 1657 1644 229 373 1212 1068 805 949 636 492 1381 1525 60 1632 241 385 1200 1056 817 961 624 480 1393 1537 48 109 1476 1332 541 685 900 756 1117 1261 324 180 1693 121 1464 1320 553 697 888 744 1129 1273 312 168 1705 1596 277 421 1164 1020 853 997 588 444 1429 1573 12 [1] Matsumi Suzuki's math pages are now available at MathForum
de Winkel's Pantriagonal type magic cube This order 12 magic cube is not associated, and does
not contain any magic squares. However, it is pantriagonal, which means
that all broken triagonal lines (as well as the four main triagonals) sum
correctly. The above characteristics seem to indicate that this
could be considered a Most-perfect magic
cube. However, this cube is not the highest possible order cube, which is
also a requirement, just as the pantriagonal magic square is the highest
possible class of magic square. Horizontal plane 1 - Top 1 1716 25 1692 49 1668 133 1608 109 1632 85 1656 1727 14 1703 38 1679 62 1595 122 1619 98 1643 74 3 1714 27 1690 51 1666 135 1606 111 1630 87 1654 1725 16 1701 40 1677 64 1593 124 1617 100 1641 76 5 1712 29 1688 53 1664 137 1604 113 1628 89 1652 1723 18 1699 42 1675 66 1591 126 1615 102 1639 78 12 1705 36 1681 60 1657 144 1597 120 1621 96 1645 1718 23 1694 47 1670 71 1586 131 1610 107 1634 83 10 1707 34 1683 58 1659 142 1599 118 1623 94 1647 1720 21 1696 45 1672 69 1588 129 1612 105 1636 81 8 1709 32 1685 56 1661 140 1601 116 1625 92 1649 1722 19 1698 43 1674 67 1590 127 1614 103 1638 79 Horizontal plane 2 1584 157 1560 181 1536 205 1452 265 1476 241 1500 217 146 1571 170 1547 194 1523 278 1463 254 1487 230 1511 1582 159 1558 183 1534 207 1450 267 1474 243 1498 219 148 1569 172 1545 196 1521 280 1461 256 1485 232 1509 1580 161 1556 185 1532 209 1448 269 1472 245 1496 221 150 1567 174 1543 198 1519 282 1459 258 1483 234 1507 1573 168 1549 192 1525 216 1441 276 1465 252 1489 228 155 1562 179 1538 203 1514 287 1454 263 1478 239 1502 1575 166 1551 190 1527 214 1443 274 1467 250 1491 226 153 1564 177 1540 201 1516 285 1456 261 1480 237 1504 1577 164 1553 188 1529 212 1445 272 1469 248 1493 224 151 1566 175 1542 199 1518 283 1458 259 1482 235 1506 Horizontal plane 3 289 1428 313 1404 337 1380 421 1320 397 1344 373 1368 1439 302 1415 326 1391 350 1307 410 1331 386 1355 362 291 1426 315 1402 339 1378 423 1318 399 1342 375 1366 1437 304 1413 328 1389 352 1305 412 1329 388 1353 364 293 1424 317 1400 341 1376 425 1316 401 1340 377 1364 1435 306 1411 330 1387 354 1303 414 1327 390 1351 366 300 1417 324 1393 348 1369 432 1309 408 1333 384 1357 1430 311 1406 335 1382 359 1298 419 1322 395 1346 371 298 1419 322 1395 346 1371 430 1311 406 1335 382 1359 1432 309 1408 333 1384 357 1300 417 1324 393 1348 369 296 1421 320 1397 344 1373 428 1313 404 1337 380 1361 1434 307 1410 331 1386 355 1302 415 1326 391 1350 367 Horizontal plane 4 1296 445 1272 469 1248 493 1164 553 1188 529 1212 505 434 1283 458 1259 482 1235 566 1175 542 1199 518 1223 1294 447 1270 471 1246 495 1162 555 1186 531 1210 507 436 1281 460 1257 484 1233 568 1173 544 1197 520 1221 1292 449 1268 473 1244 497 1160 557 1184 533 1208 509 438 1279 462 1255 486 1231 570 1171 546 1195 522 1219 1285 456 1261 480 1237 504 1153 564 1177 540 1201 516 443 1274 467 1250 491 1226 575 1166 551 1190 527 1214 1287 454 1263 478 1239 502 1155 562 1179 538 1203 514 441 1276 465 1252 489 1228 573 1168 549 1192 525 1216 1289 452 1265 476 1241 500 1157 560 1181 536 1205 512 439 1278 463 1254 487 1230 571 1170 547 1194 523 1218 Horizontal plane 5 577 1140 601 1116 625 1092 709 1032 685 1056 661 1080 1151 590 1127 614 1103 638 1019 698 1043 674 1067 650 579 1138 603 1114 627 1090 711 1030 687 1054 663 1078 1149 592 1125 616 1101 640 1017 700 1041 676 1065 652 581 1136 605 1112 629 1088 713 1028 689 1052 665 1076 1147 594 1123 618 1099 642 1015 702 1039 678 1063 654 588 1129 612 1105 636 1081 720 1021 696 1045 672 1069 1142 599 1118 623 1094 647 1010 707 1034 683 1058 659 586 1131 610 1107 634 1083 718 1023 694 1047 670 1071 1144 597 1120 621 1096 645 1012 705 1036 681 1060 657 584 1133 608 1109 632 1085 716 1025 692 1049 668 1073 1146 595 1122 619 1098 643 1014 703 1038 679 1062 655 Horizontal plane 6 1008 733 984 757 960 781 876 841 900 817 924 793 722 995 746 971 770 947 854 887 830 911 806 935 1006 735 982 759 958 783 874 843 898 819 922 795 724 993 748 969 772 945 856 885 832 909 808 933 1004 737 980 761 956 785 872 845 896 821 920 797 726 991 750 967 774 943 858 883 834 907 810 931 997 744 973 768 949 792 865 852 889 828 913 804 731 986 755 962 779 938 863 878 839 902 815 926 999 742 975 766 951 790 867 850 891 826 915 802 729 988 753 964 777 940 861 880 837 904 813 928 1001 740 977 764 953 788 869 848 893 824 917 800 727 990 751 966 775 942 859 882 835 906 811 930 Horizontal plane 7 1585 132 1609 108 1633 84 1717 24 1693 48 1669 72 143 1598 119 1622 95 1646 11 1706 35 1682 59 1658 1587 130 1611 106 1635 82 1719 22 1695 46 1671 70 141 1600 117 1624 93 1648 9 1708 33 1684 57 1660 1589 128 1613 104 1637 80 1721 20 1697 44 1673 68 139 1602 115 1626 91 1650 7 1710 31 1686 55 1662 1596 121 1620 97 1644 73 1728 13 1704 37 1680 61 134 1607 110 1631 86 1655 2 1715 26 1691 50 1667 1594 123 1618 99 1642 75 1726 15 1702 39 1678 63 136 1605 112 1629 88 1653 4 1713 28 1689 52 1665 1592 125 1616 101 1640 77 1724 17 1700 41 1676 65 138 1603 114 1627 90 1651 6 1711 30 1687 54 1663 Horizontal plane 8 288 1453 264 1477 240 1501 156 1561 180 1537 204 1513 1442 275 1466 251 1490 227 1574 167 1550 191 1526 215 286 1455 262 1479 238 1503 154 1563 178 1539 202 1515 1444 273 1468 249 1492 225 1576 165 1552 189 1528 213 284 1457 260 1481 236 1505 152 1565 176 1541 200 1517 1446 271 1470 247 1494 223 1578 163 1554 187 1530 211 277 1464 253 1488 229 1512 145 1572 169 1548 193 1524 1451 266 1475 242 1499 218 1583 158 1559 182 1535 206 279 1462 255 1486 231 1510 147 1570 171 1546 195 1522 1449 268 1473 244 1497 220 1581 160 1557 184 1533 208 281 1460 257 1484 233 1508 149 1568 173 1544 197 1520 1447 270 1471 246 1495 222 1579 162 1555 186 1531 210 Horizontal plane 9 1297 420 1321 396 1345 372 1429 312 1405 336 1381 360 431 1310 407 1334 383 1358 299 1418 323 1394 347 1370 1299 418 1323 394 1347 370 1431 310 1407 334 1383 358 429 1312 405 1336 381 1360 297 1420 321 1396 345 1372 1301 416 1325 392 1349 368 1433 308 1409 332 1385 356 427 1314 403 1338 379 1362 295 1422 319 1398 343 1374 1308 409 1332 385 1356 361 1440 301 1416 325 1392 349 422 1319 398 1343 374 1367 290 1427 314 1403 338 1379 1306 411 1330 387 1354 363 1438 303 1414 327 1390 351 424 1317 400 1341 376 1365 292 1425 316 1401 340 1377 1304 413 1328 389 1352 365 1436 305 1412 329 1388 353 426 1315 402 1339 378 1363 294 1423 318 1399 342 1375 Horizontal plane 10 576 1165 552 1189 528 1213 444 1273 468 1249 492 1225 1154 563 1178 539 1202 515 1286 455 1262 479 1238 503 574 1167 550 1191 526 1215 442 1275 466 1251 490 1227 1156 561 1180 537 1204 513 1288 453 1264 477 1240 501 572 1169 548 1193 524 1217 440 1277 464 1253 488 1229 1158 559 1182 535 1206 511 1290 451 1266 475 1242 499 565 1176 541 1200 517 1224 433 1284 457 1260 481 1236 1163 554 1187 530 1211 506 1295 446 1271 470 1247 494 567 1174 543 1198 519 1222 435 1282 459 1258 483 1234 1161 556 1185 532 1209 508 1293 448 1269 472 1245 496 569 1172 545 1196 521 1220 437 1280 461 1256 485 1232 1159 558 1183 534 1207 510 1291 450 1267 474 1243 498 Horizontal plane 11 1009 708 1033 684 1057 660 1141 600 1117 624 1093 648 719 1022 695 1046 671 1070 587 1130 611 1106 635 1082 1011 706 1035 682 1059 658 1143 598 1119 622 1095 646 717 1024 693 1048 669 1072 585 1132 609 1108 633 1084 1013 704 1037 680 1061 656 1145 596 1121 620 1097 644 715 1026 691 1050 667 1074 583 1134 607 1110 631 1086 1020 697 1044 673 1068 649 1152 589 1128 613 1104 637 710 1031 686 1055 662 1079 578 1139 602 1115 626 1091 1018 699 1042 675 1066 651 1150 591 1126 615 1102 639 712 1029 688 1053 664 1077 580 1137 604 1113 628 1089 1016 701 1040 677 1064 653 1148 593 1124 617 1100 641 714 1027 690 1051 666 1075 582 1135 606 1111 630 1087 Horizontal plane 12 - Bottom 864 877 840 901 816 925 732 985 756 961 780 937 866 851 890 827 914 803 998 743 974 767 950 791 862 879 838 903 814 927 730 987 754 963 778 939 868 849 892 825 916 801 1000 741 976 765 952 789 860 881 836 905 812 929 728 989 752 965 776 941 870 847 894 823 918 799 1002 739 978 763 954 787 853 888 829 912 805 936 721 996 745 972 769 948 875 842 899 818 923 794 1007 734 983 758 959 782 855 886 831 910 807 934 723 994 747 970 771 946 873 844 897 820 921 796 1005 736 981 760 957 784 857 884 833 908 809 932 725 992 749 968 773 944 871 846 895 822 919 798 1003 738 979 762 955 786
Benson & Jacoby's Diagonal type magic cube This cube appears in Magic Cubes New Recreations [1] where the authors refer to it as ‘perfect’ because all planar squares are magic. By the new Hendricks consistent magic cube definitions, it should be called a diagonal type cube because both main diagonals of all 3m planar squares are correct. C. Boyer refers to this type as perfect.(For a short time I called this type a myers cube, after R.L.Myers who constructed an order 8 cube of this type). So this cube has 3m+6 simple magic squares. An additional feature (not part of the 'diagonal' definition) is that the 8 corners of all 123 (including wrap-around) order 7 sub-cubes sum to 6916 which is 8/12 of the constant of 10374 for this order 12 cube. This cube is not associated. All pantriagonals in one of the four directions sum correctly to S. [1] Benson & Jacoby, Magic Cubes: New Recreations, Dover Publ., 1981, pp.103-115 (fig. 11-4 for the top Horizontal Plane) Horizontal plane 1 - top 659 1538 698 1511 623 1454 1091 242 1130 215 1055 158 353 1196 1313 512 389 1280 1361 476 305 1232 1397 560 71 971 14 947 38 986 1655 827 1598 803 1622 842 1073 185 1028 221 1112 272 641 1481 596 1517 680 1568 1379 530 419 1214 1343 446 371 1250 1427 494 335 1166 1649 764 1604 797 1685 848 65 908 20 941 101 992 650 1547 707 1502 614 1463 1082 251 1139 206 1046 167 356 1193 1316 509 392 1277 1364 473 308 1229 1400 557 74 962 131 926 47 887 1658 818 1715 782 1631 743 1088 188 1133 212 1109 161 656 1484 701 1508 677 1457 1370 539 410 1223 1334 455 362 1259 1418 503 326 1175 1652 761 1601 800 1688 845 68 905 17 944 104 989 Horizontal plane 2 1017 177 633 1576 1125 232 585 1468 1065 280 688 1528 1431 550 375 1162 1323 490 423 1270 1383 442 315 1210 1588 748 88 1005 1696 777 136 897 1636 729 117 957 718 1558 1102 147 610 1491 1150 255 670 1443 1035 195 297 1180 1353 568 405 1240 1305 460 345 1288 1413 520 135 975 1671 730 27 922 1719 838 87 874 1618 778 1012 172 628 1581 1120 237 580 1473 1060 285 693 1533 1438 543 382 1155 1330 483 430 1263 1390 435 322 1203 1593 753 57 1000 1701 808 9 892 1641 856 112 952 711 1551 1059 154 603 1534 1011 262 663 1582 1042 202 292 1185 1348 573 400 1245 1300 465 340 1293 1408 525 142 982 1678 723 34 915 1726 831 94 867 1611 771 Horizontal plane 3 1075 1482 1026 223 679 270 643 186 594 1519 1111 1566 361 1260 1417 504 325 1176 1369 540 409 1224 1333 456 1651 763 1602 799 1686 846 67 907 18 943 102 990 649 241 708 1501 1056 1464 1081 1537 1140 205 624 168 1363 474 307 1230 1399 558 355 1194 1315 510 391 1278 61 972 24 937 37 996 1645 828 1608 793 1621 852 1086 1483 1135 210 678 163 654 187 703 1506 1110 1459 372 1249 1428 493 336 1165 1380 529 420 1213 1344 445 1650 906 1603 798 103 847 66 762 19 942 1687 991 660 252 697 1512 1045 1453 1092 1548 1129 216 613 157 1362 475 306 1231 1398 559 354 1195 1314 511 390 1279 84 817 121 936 1632 877 1668 961 1705 792 48 733 Horizontal plane 4 1142 263 671 1442 1034 203 710 1559 1103 146 602 1499 1304 461 344 1289 1412 521 296 1181 1352 569 404 1241 1718 830 95 866 1619 779 134 974 1679 722 35 923 584 176 1061 284 1121 1529 1016 1472 629 1580 689 233 422 1271 1382 443 314 1211 1430 551 374 1163 1322 491 140 893 1673 728 116 917 1724 749 89 872 1700 773 1019 1550 626 1583 611 230 587 254 1058 287 1043 1526 1301 464 341 1292 1409 524 293 1184 1349 572 401 1244 1727 839 86 875 1610 770 143 983 1670 731 26 914 581 1469 1064 281 692 1532 1013 173 632 1577 1124 236 431 1262 1391 434 323 1202 1439 542 383 1154 1331 482 5 896 1640 857 113 956 1589 752 56 1001 1697 812 Horizontal plane 5 652 1545 1132 213 616 1461 1084 249 700 1509 1048 165 358 1191 1318 507 394 1275 1366 471 310 1227 1402 555 81 964 1605 784 45 988 1653 820 129 940 1629 736 1071 190 591 1522 1107 274 639 1486 1023 226 675 1570 1372 537 412 1221 1336 453 364 1257 1420 501 328 1173 1654 759 22 939 1690 843 70 903 1606 795 106 987 657 1540 1137 208 621 1456 1089 244 705 1504 1053 160 351 1198 1311 514 387 1282 1359 478 303 1234 1395 562 76 969 1708 789 40 885 1660 825 124 933 1624 741 1078 183 706 1515 1114 159 658 1479 1030 207 682 1563 1377 532 417 1216 1341 448 369 1252 1425 496 333 1168 1647 766 15 946 1683 850 63 910 1599 802 99 994 Horizontal plane 6 1014 175 631 1578 1122 235 582 1471 1063 282 690 1531 1440 541 384 1164 1332 481 432 1261 1381 433 324 1201 1722 750 91 870 1699 775 138 894 1675 726 115 919 720 1560 1093 156 601 1489 1152 264 661 1452 1033 193 294 1183 1350 570 402 1243 1302 463 343 1291 1410 523 144 973 1669 732 36 913 1728 829 85 876 1620 769 1015 174 630 1579 1123 234 583 1470 1062 283 691 1530 1429 552 373 1153 1321 492 421 1272 1392 444 313 1212 1591 751 54 1003 1698 810 7 895 1638 859 114 954 577 1549 1068 277 612 1536 1009 253 636 1573 1044 240 295 1182 1351 571 403 1242 1303 462 342 1290 1411 522 133 984 1680 721 25 924 1717 840 96 865 1609 780 Horizontal plane 7 647 1478 1022 227 683 1562 1079 182 590 1523 1115 266 365 1256 1424 500 329 1172 1373 536 413 1217 1337 452 83 911 1706 791 98 878 1667 767 122 935 1682 734 1085 245 704 1505 1052 164 653 1541 1136 209 620 1460 1367 470 302 1226 1403 554 359 1190 1319 515 395 1274 1661 824 128 929 1625 740 77 968 1712 785 41 884 638 1487 1031 218 674 1571 1070 191 599 1514 1106 275 368 1253 1421 497 332 1169 1376 533 416 1220 1340 449 62 902 1607 794 107 995 1646 758 23 938 1691 851 1076 248 593 1520 1049 269 644 1544 1025 224 617 1565 1358 479 311 1235 1394 563 350 1199 1310 506 386 1283 1664 821 125 932 1628 737 80 965 1709 788 44 881 Horizontal plane 8 1149 256 669 1444 1041 196 717 1552 1101 148 609 1492 1299 466 339 1294 1407 526 291 1186 1347 574 399 1246 1720 837 52 873 1612 813 4 981 1672 861 28 921 586 1467 1066 279 694 1527 1018 171 634 1575 1126 231 429 1264 1389 436 321 1204 1437 544 381 1156 1329 484 3 898 1635 862 111 958 1587 754 51 1006 1695 814 1144 261 664 1449 1036 201 712 1557 1096 153 604 1497 1306 459 346 1287 1414 519 298 1179 1354 567 406 1239 1725 832 93 868 1617 772 141 976 1677 724 33 916 579 1474 1095 286 687 1498 1143 178 627 1450 1119 238 424 1269 1384 441 316 1209 1432 549 376 1161 1324 489 10 891 1642 855 118 951 1594 747 58 999 1702 807 Horizontal plane 9 655 1542 1134 211 619 1458 1087 246 702 1507 1051 162 349 1200 1309 505 385 1284 1357 480 312 1236 1393 564 79 967 1710 787 42 882 1663 823 126 931 1626 738 1069 1477 600 1513 684 276 637 181 1032 217 1116 1572 1375 534 415 1219 1339 450 367 1254 1422 498 331 1170 1657 768 132 925 1681 744 73 912 1716 781 97 888 642 247 1027 222 1050 1567 1074 1543 595 1518 618 271 360 1189 1320 516 396 1273 1368 469 301 1225 1404 553 78 966 1711 786 43 883 1662 822 127 930 1627 739 1080 192 589 1524 1105 265 648 1488 1021 228 673 1561 1374 535 414 1218 1338 451 366 1255 1423 499 330 1171 1656 757 13 948 1692 841 72 901 1597 804 108 985 Horizontal plane 10 578 179 635 1574 1118 1535 1010 1475 1067 278 686 239 1436 545 377 1157 1328 485 428 1265 1388 440 320 1205 2 890 59 998 119 959 1586 746 1643 854 1703 815 1148 1556 1097 152 605 197 716 260 665 1448 1037 1493 290 1187 1355 575 398 1247 1298 467 338 1286 1406 527 1592 833 1637 860 1616 809 8 977 53 1004 32 953 719 170 662 1451 1127 1490 1151 1466 1094 155 695 194 1433 548 380 1160 1325 488 425 1268 1385 437 317 1208 11 755 50 1007 1694 950 1595 899 1634 863 110 806 1145 1553 1100 149 608 200 713 257 668 1445 1040 1496 299 1178 1346 566 407 1238 1307 458 347 1295 1415 518 1721 980 1676 725 29 776 137 836 92 869 1613 920 Horizontal plane 11 640 1480 1024 225 676 1569 1072 189 592 1521 1113 273 370 1251 1426 495 334 1167 1378 531 418 1215 1342 447 69 909 1713 796 105 880 1665 760 21 928 1684 844 1083 243 699 1510 1047 166 651 1546 1131 214 622 1462 1360 477 304 1233 1396 561 352 1197 1312 513 388 1281 1666 826 130 927 1630 735 82 963 1714 783 39 879 645 1485 1029 220 681 1564 1077 184 597 1516 1108 268 363 1258 1419 502 327 1174 1371 538 411 1222 1335 454 64 904 1600 801 100 993 1648 765 16 945 1689 849 1090 250 598 1503 1054 267 646 1539 1138 219 615 1455 1365 472 309 1228 1401 556 357 1192 1317 508 393 1276 1659 819 123 934 1623 742 75 970 1707 790 46 886 Horizontal plane 12 - Bottom 1146 259 1099 150 1038 199 714 1555 667 1446 606 1495 1308 457 348 1285 1416 517 300 1177 1356 565 408 1237 1590 834 1639 858 1615 811 6 978 55 1002 31 955 588 1476 625 1584 685 1525 1020 180 1057 288 1117 229 426 1267 1386 439 318 1207 1434 547 378 1159 1326 487 12 889 49 1008 120 949 1596 745 1633 864 1704 805 1147 258 1098 151 1039 198 715 1554 666 1447 607 1494 1297 468 337 1296 1405 528 289 1188 1345 576 397 1248 1723 835 1674 727 1614 774 139 979 90 871 30 918 709 1465 672 1441 696 1500 1141 169 1104 145 1128 204 427 1266 1387 438 319 1206 1435 546 379 1158 1327 486 1 900 60 997 109 960 1585 756 1644 853 1693 816
Hendricks Inlaid order-12 magic cube This is an order 12 normal magic cube with 8 order 4 inlaid pantriagonal magic cubes and 48 order 4 pandiagonal magic squares placed within a single layer ‘expansion shell’. The order 12 cube is simple magic and not associated. It consists of the consecutive numbers from 1 to 123 and contains no order 12 magic squares or other extra features (aside from the inlays). The constant sum for the order 12 cube is 10,374 which is the required sum for a normal order 12. The sum for the order 4 cubes and squares is 3,458 (which is 4/12 of 10,374. Obviously these inlaid cubes and squares are not normal because they cannot contain consecutive numbers. This illustration is a feeble attempt to show the parts of the cube.
Variations of this cube just by operations on the 8 inlaid cubes are;
This gives a total of 123,863,040 possible variations involving the inlaid cubes only. Still available are the variations involving the 48 pandiagonal magic squares! Each pandiagonal magic square can have 144
variations due to row, column translocation and each square has 8 aspects.
However, all four squares in a ‘stack’ must have the same transformation
and aspect in order to keep the integrity of the 1-agonals in the order 12
cube. The 12 stacks themselves may be relocated in the cube in any of 12!
ways. Multiplying these two numbers together gives the total variations for this one cube. Remember, the cube itself can also be shown in any of 48 aspects. AND we haven't even started to look at the variations possible in the 'shell'. For instance, all lines of 4 numbers that border an inlaid square or cube plane also sum to 3458. The location of these lines can be rearranged, a long as care is taken to keep complete stacks together to preserve the integrity of the lines at right angles. John Hendricks devotes about 20 pages in his book to this cube. He shows the solution set he used, as well as algebraic and numerical listings, and construction figures. [1][2] Following is the listing for the 12 horizontal
planes of the cube. I will indicate the inlaid horizontal magic squares
and planes of the order 4 cubes with blue text. Horizontal plane 1 – Top Top 4 pandiagonal magic squares 942 1230 355 222 1651 643 1075 67 1518 1363 510 798 966 474 1400 401 1183 619 1110 174 1700 101 1483 763 751 1267 317 1340 534 1122 607 1543 41 1616 258 978 882 1328 546 1255 329 703 1026 1628 246 1555 29 847 859 389 1195 462 1412 1014 715 113 1471 186 1688 870 775 487 1386 1495 90 1098 666 1674 199 378 1207 919 811 1242 343 234 1639 1062 630 55 1530 1351 522 955 727 306 1280 569 1303 1146 583 6 1580 269 1603 1002 990 1435 437 1172 414 595 1134 1711 161 1448 138 739 835 1160 426 1423 449 1038 691 1460 126 1723 149 894 906 557 1315 294 1292 679 1050 281 1591 18 1568 823 930 499 1374 1507 78 655 1087 1662 211 366 1219 786 Horizontal plane 2 Top plane of the top 4 pantriagonal magic cubes 940 1245 352 1371 490 640 1089 1534 63 1660 201 789 760 563 1308 422 1165 904 681 157 1718 12 1571 1113 1125 289 1274 456 1439 693 892 143 1464 266 1585 748 610 1296 311 1417 434 1042 831 1442 121 1607 288 975 963 1310 565 1163 420 819 1054 1716 155 1573 14 622 772 1209 364 1359 526 1101 664 1498 75 1648 237 921 813 508 1353 382 1215 1060 633 219 1642 93 1504 952 735 1394 457 1271 336 879 706 1632 263 1465 98 1138 1150 1188 395 1333 542 718 867 1550 37 1691 180 723 585 397 1190 540 1331 1017 856 35 1548 182 1693 1000 988 479 1416 314 1249 844 1029 241 1610 120 1487 597 933 496 1389 346 1227 657 1072 207 1678 57 1516 796 Horizontal plane 3 2nd plane of the top 4 pantriagonal magic cubes 790 483 1378 357 1240 1090 639 196 1665 70 1527 939 970 1429 446 1284 299 826 1047 1595 276 1454 133 615 603 1175 432 1298 553 1035 838 1561 2 1728 167 982 1120 410 1153 575 1320 688 897 24 1583 145 1706 753 765 444 1427 301 1286 909 676 278 1597 131 1452 1108 958 519 1366 369 1204 627 1066 232 1653 82 1491 807 915 1222 375 1348 513 670 1095 1509 88 1635 226 778 993 552 1343 385 1178 849 1024 170 1681 47 1560 592 580 326 1261 467 1404 1012 861 108 1475 253 1622 1005 1143 1259 324 1406 469 711 874 1477 110 1620 251 730 742 1321 530 1200 407 886 699 1703 192 1538 25 1131 795 1234 339 1384 501 1071 658 1521 52 1671 214 934 Horizontal plane 4 3rd plane of the top 4 pantriagonal magic cubes 947 358 1239 484 1377 647 1082 69 1528 195 1666 782 1119 312 1295 433 1418 687 898 122 1441 287 1608 754 766 566 1309 419 1164 910 675 156 1715 13 1574 1107 969 1307 564 1166 421 825 1048 1717 158 1572 11 616 604 1273 290 1440 455 1036 837 1463 144 1586 265 981 779 370 1203 520 1365 1094 671 81 1492 231 1654 914 806 1347 514 1221 376 1067 626 1636 225 1510 87 959 1144 1189 398 1332 539 712 873 1547 36 1694 181 729 741 1415 480 1250 313 885 700 1609 242 1488 119 1132 994 458 1393 335 1272 850 1023 264 1631 97 1466 591 579 396 1187 541 1334 1011 862 38 1549 179 1692 1006 926 1383 502 1233 340 650 1079 1672 213 1522 51 803 Horizontal plane 5 4th plane of the top 4 pantriagonal magic cubes 781 1372 489 1246 351 1081 648 1659 202 1533 64 948 609 1154 409 1319 576 1041 832 1584 23 1705 146 976 964 1428 443 1285 302 820 1053 1598 277 1451 132 621 759 445 1430 300 1283 903 682 275 1596 134 1453 1114 1126 431 1176 554 1297 694 891 1 1562 168 1727 747 949 1360 525 1210 363 636 1057 1647 238 1497 76 816 924 381 1216 507 1354 661 1104 94 1503 220 1641 769 586 323 1260 470 1405 1018 855 109 1478 252 1619 999 987 529 1322 408 1199 843 1030 191 1704 26 1537 598 736 1344 551 1177 386 880 705 1682 169 1559 48 1137 1149 1262 325 1403 468 717 868 1476 107 1621 254 724 804 345 1228 495 1390 1080 649 58 1515 208 1677 925 Horizontal plane 6 2nd set of the 4 magic squares 785 497 1376 1505 80 1088 656 1664 209 368 1217 929 761 475 1397 404 1182 1112 620 175 1697 104 1482 965 980 1266 320 1337 535 605 1121 1542 44 1613 259 752 845 1325 547 1254 332 1028 704 1625 247 1554 32 881 872 392 1194 463 1409 713 1013 116 1470 187 1685 860 956 1244 341 236 1637 629 1061 53 1532 1349 524 812 920 488 1385 1496 89 665 1097 1673 200 377 1208 776 1004 307 1277 572 1302 581 1145 7 1577 272 1602 728 737 1434 440 1169 415 1136 596 1710 164 1445 139 989 896 1157 427 1422 452 689 1037 1457 127 1722 152 836 821 560 1314 295 1289 1052 680 284 1590 19 1565 905 797 1229 356 221 1652 1076 644 68 1517 1364 509 941 Horizontal plane 7 3rd set of the 4 magic squares 788 1220 365 212 1661 1085 653 77 1508 1373 500 932 824 1253 331 1326 548 1049 677 1553 31 1626 248 908 893 464 1410 391 1193 692 1040 188 1686 115 1469 833 740 403 1181 476 1398 1133 593 103 1481 176 1698 992 1001 1338 536 1265 319 584 1148 1614 260 1541 43 725 953 521 1352 1529 56 632 1064 1640 233 344 1241 809 917 1205 380 197 1676 668 1100 92 1493 1388 485 773 869 1421 451 1158 428 716 1016 1721 151 1458 128 857 848 296 1290 559 1313 1025 701 20 1566 283 1589 884 977 571 1301 308 1278 608 1124 271 1601 8 1578 749 764 1170 416 1433 439 1109 617 1446 140 1709 163 968 800 512 1361 1520 65 1073 641 1649 224 353 1232 944 Horizontal plane 8 1st plane of the bottom 4 magic cubes 792 1391 494 1225 348 1092 637 1680 205 1514 59 937 746 1161 418 1312 567 890 695 1575 16 1714 153 1127 1115 1419 436 1294 309 683 902 1605 286 1444 123 758 624 454 1437 291 1276 1056 817 268 1587 141 1462 961 973 424 1167 561 1306 829 1044 10 1569 159 1720 612 960 1355 506 1213 384 625 1068 1644 217 1502 95 805 913 362 1211 528 1357 672 1093 73 1500 239 1646 780 721 316 1251 477 1414 865 720 118 1485 243 1612 1152 1140 538 1329 399 1192 708 877 184 1695 33 1546 733 599 1335 544 1186 393 1031 842 1689 178 1552 39 986 998 1269 334 1396 459 854 1019 1467 100 1630 261 587 793 350 1247 492 1369 1069 660 61 1536 203 1658 936 Horizontal plane 9 2nd plane of the bottom 4 magic cubes 938 337 1236 503 1382 638 1091 50 1523 216 1669 791 984 303 1288 442 1425 840 1033 129 1450 280 1599 601 613 573 1318 412 1155 1045 828 147 1708 22 1581 972 1106 1300 555 1173 430 674 911 1726 165 1563 4 767 755 1282 297 1431 448 899 686 1456 135 1593 274 1118 770 373 1224 515 1346 1103 662 86 1511 228 1633 923 815 1368 517 1202 371 1058 635 1655 230 1489 84 950 1007 1198 405 1323 532 863 1010 1540 27 1701 190 578 590 1408 471 1257 322 1022 851 1618 249 1479 112 995 1129 465 1402 328 1263 697 888 255 1624 106 1473 744 732 387 1180 550 1341 876 709 45 1558 172 1683 1141 935 1380 481 1238 359 659 1070 1667 194 1525 72 794 Horizontal plane 10 3rd plane of the bottom 4 magic cubes 783 504 1381 338 1235 1083 646 215 1670 49 1524 946 1105 1438 453 1275 292 673 912 1588 267 1461 142 768 756 1168 423 1305 562 900 685 1570 9 1719 160 1117 983 417 1162 568 1311 839 1034 15 1576 154 1713 602 614 435 1420 310 1293 1046 827 285 1606 124 1443 971 951 516 1345 374 1223 634 1059 227 1634 85 1512 814 922 1201 372 1367 518 663 1102 1490 83 1656 229 771 1130 543 1336 394 1185 698 887 177 1690 40 1551 743 731 333 1270 460 1395 875 710 99 1468 262 1629 1142 1008 1252 315 1413 478 864 1009 1486 117 1611 244 577 589 1330 537 1191 400 1021 852 1696 183 1545 34 996 802 1237 360 1379 482 1078 651 1526 71 1668 193 927 Horizontal plane 11 4th plane of the bottom 4 magic cubes 945 1226 347 1392 493 645 1084 1513 60 1679 206 784 623 556 1299 429 1174 1055 818 166 1725 3 1564 962 974 298 1281 447 1432 830 1043 136 1455 273 1594 611 745 1287 304 1426 441 889 696 1449 130 1600 279 1128 1116 1317 574 1156 411 684 901 1707 148 1582 21 757 777 1214 383 1356 505 1096 669 1501 96 1643 218 916 808 527 1358 361 1212 1065 628 240 1645 74 1499 957 600 1401 466 1264 327 1032 841 1623 256 1474 105 985 997 1179 388 1342 549 853 1020 1557 46 1684 171 588 722 406 1197 531 1324 866 719 28 1539 189 1702 1151 1139 472 1407 321 1258 707 878 250 1617 111 1480 734 928 491 1370 349 1248 652 1077 204 1657 62 1535 801 Horizontal plane 12 4th set of the 4 magic squares 943 511 1362 1519 66 642 1074 1650 223 354 1231 799 907 1256 330 1327 545 678 1051 1556 30 1627 245 822 834 461 1411 390 1196 1039 690 185 1687 114 1472 895 991 402 1184 473 1399 594 1135 102 1484 173 1699 738 726 1339 533 1268 318 1147 582 1615 257 1544 42 1003 774 1206 379 198 1675 1099 667 91 1494 1387 486 918 810 523 1350 1531 54 1063 631 1638 235 342 1243 954 858 1424 450 1159 425 1015 714 1724 150 1459 125 871 883 293 1291 558 1316 702 1027 17 1567 282 1592 846 750 570 1304 305 1279 1123 606 270 1604 5 1579 979 967 1171 413 1436 438 618 1111 1447 137 1712 162 762 931 1218 367 210 1663 654 1086 79 1506 1375 498 787This is one of the 8 inlaid order 4 pantriagonal magic cubes. It is in the top left back position of the order 12 cube above, so it appears in planes 2, 3, 4, and 5. It is pantriagonal (as are the other 7) so all 16 pantriagonals in each of the four directions sum correctly to 3458. It is also ‘complete’, because Every pantriagonal contains m/2 complement pairs, spaced m/2 apart. Top II III IV 563 1308 422 1165 1429 446 1284 299 312 1295 433 1418 1154 409 1319 576 289 1274 456 1439 1175 432 1298 553 566 1309 419 1164 1428 443 1285 302 1296 311 1417 434 410 1153 575 1320 1307 564 1166 421 445 1430 300 1283 1310 565 1163 420 444 1427 301 1286 312 1295 433 1418 431 1176 554 1297This is another of the eight inlaid cubes. This one is located in the bottom right front of the order 12 cube. therefore, the top plane is located in the in the lower right quadrant of plane 8 of the listing for the mother cube. Top II III IV 118 1485 243 1612 1540 27 1701 190 177 1690 40 1551 1623 256 1474 105 184 1695 33 1546 1618 249 1479 112 99 1468 262 1629 1557 46 1684 171 1689 178 1552 39 255 1624 106 1473 1486 117 1611 244 28 1539 189 1702 1467 100 1630 261 45 1558 172 1683 1696 183 1545 34 250 1617 111 1480
[1] John R. Hendricks, Inlaid
Magic Squares and Cubes, self-published, 1999, 0-9684700-1-7, 188+
pages.
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