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I received this cube via email from
Bogdan Golunski of Germany on November 26, 2003. It is not associated, and must
be classed as a pantriagonal magic cube even though it contains many magic squares.
All 13 of the horizontal planes and all 13 of the planes parallel to the front
of the cube are pandiagonal magic squares. All broken diagonals and 12 of the 13
main diagonals in 1 direction of the vertical planes parallel to the sides of
the cube sum incorrectly. Because of this, only 1 of these 13 planes is a simple magic
square. 3 of the 6 oblique planes are simple magic squares and 3 are pandiagonal magic squares.
Total for the cube, 4 simple magic squares and 29 pandiagonal magic squares.
All pantriagonals in this cube sum correctly. So, because all 3m orthogonal planes in this cube are not magic squares (and the same type), this cube must be classed only as pantriagonal magic! A most unusual cube!
Plane 1 - Top 456 2108 204 915 606 58 1290 1541 1869 823 1157 1857 1403 1044 1768 1454 402 2133 184 854 655 138 1345 1563 1983 764 1618 1914 787 1037 1694 1508 479 2196 209 964 594 31 1256 613 22 1188 1672 1995 835 1060 1817 1434 377 2104 272 898 2041 323 987 664 41 1309 1604 1893 747 1108 1749 1466 355 1823 1515 387 2149 260 882 584 92 1237 1634 1877 684 1163 1907 798 1100 1722 1420 436 2090 277 871 518 155 1314 1679 89 1221 1587 1952 737 1123 1711 1358 484 2165 335 888 637 241 946 563 117 1207 1533 2003 808 1179 1734 1477 422 2057 1409 450 2047 172 995 647 160 1235 1649 1949 703 1081 1790 701 1015 1835 1491 499 2075 292 926 540 75 1278 1586 1974 1336 1655 2028 726 1143 1769 1380 412 2124 229 955 523 7 1009 552 120 1268 1557 1928 780 1077 1806 1366 340 2180 304 Plane 2 - Top-1 1220 1598 1951 730 1127 1708 1360 487 2161 332 894 628 91 943 569 108 1209 1532 2014 807 1172 1738 1474 424 2060 237 447 2049 175 991 644 166 1226 1651 1948 714 1080 1783 1413 1026 1834 1484 503 2072 294 929 536 72 1284 1577 1976 700 1661 2019 728 1142 1780 1379 405 2128 226 957 526 3 1333 554 123 1264 1554 1934 771 1079 1805 1377 339 2173 308 1006 2107 197 919 603 60 1293 1537 1866 829 1148 1859 1402 467 1759 1456 401 2144 183 847 659 135 1347 1566 1979 761 1050 1917 783 1034 1700 1499 481 2195 220 963 587 35 1253 1620 15 1192 1669 1997 838 1056 1814 1440 368 2106 271 909 612 325 986 675 40 1302 1608 1890 749 1111 1745 1463 361 2032 1511 384 2155 251 884 583 103 1236 1627 1881 681 1165 1826 802 1097 1724 1423 432 2087 283 862 520 154 1325 1678 1900 Plane 3 - Top-2 1447 403 2143 194 846 652 139 1344 1568 1982 757 1047 1765 786 1030 1697 1505 472 2197 219 974 586 28 1257 1617 1919 1185 1673 1994 840 1059 1810 1437 374 2097 273 908 623 14 988 674 51 1301 1601 1894 746 1113 1748 1459 358 2038 316 380 2152 257 875 585 102 1247 1626 1874 685 1162 1828 1514 1101 1721 1425 435 2083 280 868 511 156 1324 1689 1899 795 1597 1962 729 1120 1712 1357 489 2164 328 891 634 82 1222 566 114 1200 1534 2013 818 1171 1731 1478 421 2062 240 939 2046 177 994 640 163 1232 1642 1950 713 1091 1782 1406 451 1845 1483 496 2076 291 931 539 68 1281 1583 1967 702 1025 2025 719 1144 1779 1390 404 2121 230 954 528 6 1329 1658 125 1267 1550 1931 777 1070 1807 1376 350 2172 301 1010 551 196 912 607 57 1295 1540 1862 826 1154 1850 1404 466 2118 Plane 4 - Top-3 111 1206 1525 2015 817 1182 1730 1471 425 2059 242 942 562 174 996 643 159 1229 1648 1941 715 1090 1793 1405 444 2050 1494 495 2069 295 928 541 71 1277 1580 1973 693 1027 1844 725 1135 1781 1389 415 2120 223 958 525 8 1332 1654 2022 1269 1553 1927 774 1076 1798 1378 349 2183 300 1003 555 122 911 600 61 1292 1542 1865 822 1151 1856 1395 468 2117 207 394 2145 193 857 651 132 1348 1565 1984 760 1043 1762 1453 1033 1693 1502 478 2188 221 973 597 27 1250 1621 1916 788 1666 1998 837 1061 1813 1433 371 2103 264 910 622 25 1184 676 50 1312 1600 1887 750 1110 1750 1462 354 2035 322 979 2148 254 881 576 104 1246 1637 1873 678 1166 1825 1516 383 1725 1422 437 2086 276 865 517 147 1326 1688 1910 794 1094 1961 740 1119 1705 1361 486 2166 331 887 631 88 1213 1599 Plane 5 - Top-4 1696 1498 475 2194 212 975 596 38 1249 1614 1920 785 1035 1991 841 1058 1815 1436 367 2100 270 901 624 24 1195 1665 52 1311 1611 1886 743 1114 1747 1464 357 2031 319 985 667 250 878 582 95 1248 1636 1884 677 1159 1829 1513 385 2151 1426 434 2088 279 861 514 153 1317 1690 1909 805 1093 1718 739 1130 1704 1354 490 2163 333 890 627 85 1219 1590 1963 1203 1531 2006 819 1181 1741 1470 418 2063 239 944 565 107 993 645 162 1225 1645 1947 706 1092 1792 1416 443 2043 178 506 2068 288 932 538 73 1280 1576 1970 699 1018 1846 1493 1141 1772 1391 414 2131 222 951 529 5 1334 1657 2018 722 1555 1930 770 1073 1804 1369 351 2182 311 1002 548 126 1266 599 54 1296 1539 1867 825 1147 1853 1401 459 2119 206 922 2136 195 856 662 131 1341 1569 1981 762 1046 1758 1450 400 Plane 6 - Top-5 642 164 1228 1641 1944 712 1083 1794 1415 454 2042 171 997 2079 287 925 542 70 1282 1579 1966 696 1024 1837 1495 505 1778 1382 416 2130 233 950 522 9 1331 1659 2021 718 1138 1932 773 1069 1801 1375 342 2184 310 1013 547 119 1270 1552 53 1289 1543 1864 827 1150 1849 1398 465 2110 208 921 610 186 858 661 142 1340 1562 1985 759 1048 1761 1446 397 2142 1501 471 2191 218 966 598 37 1260 1613 1913 789 1032 1698 834 1062 1812 1438 370 2096 267 907 615 26 1194 1676 1990 1313 1610 1897 742 1107 1751 1461 359 2034 315 982 673 43 874 579 101 1239 1638 1883 688 1158 1822 1517 382 2153 253 438 2085 281 864 510 150 1323 1681 1911 804 1104 1717 1419 1129 1715 1353 483 2167 330 892 630 81 1216 1596 1954 741 1528 2012 810 1183 1740 1481 417 2056 243 941 567 110 1199 Plane 7 - Middle 1055 1816 1435 372 2099 263 904 621 17 1196 1675 2001 833 1612 1896 753 1106 1744 1465 356 2036 318 978 670 49 1304 575 98 1245 1629 1885 687 1169 1821 1510 386 2150 255 877 2089 278 866 513 146 1320 1687 1902 806 1103 1728 1418 431 1714 1364 482 2160 334 889 632 84 1212 1593 1960 732 1131 2009 816 1174 1742 1480 428 2055 236 945 564 112 1202 1524 161 1230 1644 1940 709 1089 1785 1417 453 2053 170 990 646 298 924 535 74 1279 1581 1969 692 1021 1843 1486 507 2078 1388 407 2132 232 961 521 2 1335 1656 2023 721 1134 1775 775 1072 1797 1372 348 2175 312 1012 558 118 1263 1556 1929 1288 1536 1868 824 1152 1852 1394 462 2116 199 923 609 64 849 663 141 1351 1561 1978 763 1045 1763 1449 393 2139 192 474 2187 215 972 589 39 1259 1624 1912 782 1036 1695 1503 Plane 8 - Botom+5 935 534 67 1283 1578 1971 695 1017 1840 1492 498 2080 297 413 2123 234 960 532 1 1328 1660 2020 723 1137 1771 1385 1074 1800 1368 345 2181 303 1014 557 129 1262 1549 1933 772 1535 1861 828 1149 1854 1397 458 2113 205 914 611 63 1299 654 143 1350 1572 1977 756 1049 1760 1451 396 2135 189 855 2190 211 969 595 30 1261 1623 1923 781 1029 1699 1500 476 1809 1439 369 2101 266 900 618 23 1187 1677 2000 844 1054 1898 752 1117 1743 1458 360 2033 320 981 666 46 1310 1603 94 1242 1635 1876 689 1168 1832 1509 379 2154 252 879 578 282 863 515 149 1316 1684 1908 797 1105 1727 1429 430 2082 1363 493 2159 327 893 629 86 1215 1589 1957 738 1122 1716 813 1180 1733 1482 427 2066 235 938 568 109 1204 1527 2005 1227 1646 1943 705 1086 1791 1408 455 2052 181 989 639 165 Plane 9 - Botom+4 754 1116 1754 1457 353 2037 317 983 669 42 1307 1609 1889 1238 1632 1882 680 1170 1831 1520 378 2147 256 876 580 97 867 512 151 1319 1680 1905 803 1096 1729 1428 441 2081 275 492 2170 326 886 633 83 1217 1592 1953 735 1128 1707 1365 1177 1739 1473 429 2065 246 937 561 113 1201 1529 2008 809 1643 1945 708 1082 1788 1414 446 2054 180 1000 638 158 1231 545 66 1276 1582 1968 697 1020 1836 1489 504 2071 299 934 2129 225 962 531 12 1327 1653 2024 720 1139 1774 1381 410 1802 1371 341 2178 309 1005 559 128 1273 1548 1926 776 1071 1860 821 1153 1851 1399 461 2109 202 920 602 65 1298 1546 134 1352 1571 1988 755 1042 1764 1448 398 2138 185 852 660 214 965 592 36 1252 1625 1922 792 1028 1692 1504 473 2192 1432 373 2098 268 903 614 20 1193 1668 2002 843 1065 1808 Plane 10 - Bottom+3 231 953 533 11 1338 1652 2017 724 1136 1776 1384 406 2126 1373 344 2174 306 1011 550 130 1272 1559 1925 769 1075 1799 820 1146 1855 1396 463 2112 198 917 608 56 1300 1545 1871 1343 1573 1987 766 1041 1757 1452 395 2140 188 848 657 140 968 588 33 1258 1616 1924 791 1039 1691 1497 477 2189 216 366 2102 265 905 617 16 1190 1674 1993 845 1064 1819 1431 1118 1753 1468 352 2030 321 980 671 45 1303 1606 1895 745 1628 1879 686 1161 1833 1519 389 2146 249 880 577 99 1241 516 148 1321 1683 1901 800 1102 1720 1430 440 2092 274 860 2169 337 885 626 87 1214 1594 1956 731 1125 1713 1356 494 1736 1479 420 2067 245 948 560 106 1205 1526 2010 812 1173 1942 710 1085 1784 1411 452 2045 182 999 649 157 1224 1647 77 1275 1575 1972 694 1022 1839 1485 501 2077 290 936 544 Plane 11 - Bottom+2 1875 683 1167 1824 1521 388 2157 248 873 581 96 1243 1631 152 1318 1685 1904 796 1099 1726 1421 442 2091 285 859 509 336 896 625 80 1218 1591 1958 734 1121 1710 1362 485 2171 1476 426 2058 247 947 571 105 1198 1530 2007 814 1176 1732 707 1087 1787 1407 449 2051 173 1001 648 168 1223 1640 1946 1286 1574 1965 698 1019 1841 1488 497 2074 296 927 546 76 959 524 13 1337 1663 2016 717 1140 1773 1386 409 2122 228 346 2177 302 1008 556 121 1274 1558 1936 768 1068 1803 1370 1145 1848 1400 460 2114 201 913 605 62 1291 1547 1870 831 1564 1989 765 1052 1756 1445 399 2137 190 851 653 137 1349 591 29 1255 1622 1915 793 1038 1702 1496 470 2193 213 970 2095 269 902 619 19 1186 1671 1999 836 1066 1818 1442 365 1755 1467 363 2029 314 984 668 47 1306 1602 1892 751 1109 Plane 12 - Bottom+1 2179 305 1004 553 127 1265 1560 1935 779 1067 1796 1374 343 1847 1393 464 2111 203 916 601 59 1297 1538 1872 830 1156 1980 767 1051 1767 1444 392 2141 187 853 656 133 1346 1570 32 1251 1619 1921 784 1040 1701 1507 469 2186 217 967 593 262 906 616 21 1189 1667 1996 842 1057 1820 1441 376 2094 1469 362 2040 313 977 672 44 1308 1605 1888 748 1115 1746 679 1164 1830 1512 390 2156 259 872 574 100 1240 1633 1878 1322 1682 1906 799 1095 1723 1427 433 2093 284 870 508 145 895 636 79 1211 1595 1955 736 1124 1706 1359 491 2162 338 423 2064 238 949 570 116 1197 1523 2011 811 1178 1735 1472 1084 1789 1410 445 2048 179 992 650 167 1234 1639 1939 711 1585 1964 691 1023 1838 1490 500 2070 293 933 537 78 1285 530 4 1339 1662 2027 716 1133 1777 1383 411 2125 224 956 Plane 13 - Bottom 1686 1903 801 1098 1719 1424 439 2084 286 869 519 144 1315 635 90 1210 1588 1959 733 1126 1709 1355 488 2168 329 897 2061 244 940 572 115 1208 1522 2004 815 1175 1737 1475 419 1786 1412 448 2044 176 998 641 169 1233 1650 1938 704 1088 1975 690 1016 1842 1487 502 2073 289 930 543 69 1287 1584 10 1330 1664 2026 727 1132 1770 1387 408 2127 227 952 527 307 1007 549 124 1271 1551 1937 778 1078 1795 1367 347 2176 1392 457 2115 200 918 604 55 1294 1544 1863 832 1155 1858 758 1053 1766 1455 391 2134 191 850 658 136 1342 1567 1986 1254 1615 1918 790 1031 1703 1506 480 2185 210 971 590 34 899 620 18 1191 1670 1992 839 1063 1811 1443 375 2105 261 364 2039 324 976 665 48 1305 1607 1891 744 1112 1752 1460 1160 1827 1518 381 2158 258 883 573 93 1244 1630 1880 682
This cube is not associated. It is perfect and so contains 9m pandiagonal
magic squares. That is 3 * 13 = 39 orthogonal order 13 pandiagonal magic
squares, 6 oblique, and 6m-6 broken oblique order 13 pandiagonal magic
squares.
The fact that there are 6m-6 broken oblique pandiagonal magic squares in
a perfect cube was first mentioned by Rosser and Walker in 1938 [1]. It was
mentioned again in Liao's paper [2].
This cube contains 507
1-agonals (rows, columns and pillars) [3]
1014 pan-2-agonals (pan-diagonals)
676 pan-3-agonals (pan-triagonals)
The discrepancy between the above agonals (number lines) is due to the fact that
the same line of numbers appear in several different magic squares.
This cube appeared in a technical report by
F. Liao, T.
Katayama and K. Takaba of Kyoto University in 1999. [2]
Following is a listing of this cube.
Then I show an example broken oblique plane.
[1] B. Rosser and R. J. Walker, Magic
Squares: Published papers and Supplement, 1939, a bound volume at Cornell
University, catalogued as QA 165 R82+pt.1-4. All papers are very technical.
There are NO diagrams.
[2] F. Liao,
T. Katayama and K. Takaba, On the Construction of Pandiagonal Magic Cubes,
Kyoto Univ. Technical Report # 99021, 1999
[3] H. D. Heinz and J. R. Hendricks,
Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0, page
165.
Plane 1 - Top 1 184 367 550 733 916 1099 1282 1465 1648 1831 2014 2197 1746 1929 2112 98 281 464 647 830 1013 1027 1197 1380 1563 1294 1477 1660 1843 2026 2040 26 196 379 562 745 928 1111 842 856 1039 1222 1392 1575 1758 1941 2124 110 293 476 659 221 391 574 757 940 1123 1306 1489 1672 1855 1869 2052 38 1953 2136 122 305 488 671 685 868 1051 1234 1417 1587 1770 1501 1684 1698 1881 2064 50 233 416 586 769 952 1135 1318 880 1063 1246 1429 1612 1782 1965 2148 134 317 500 514 697 428 611 781 964 1147 1330 1513 1527 1710 1893 2076 62 245 2160 146 329 343 526 709 892 1075 1258 1441 1624 1807 1977 1539 1722 1905 2088 74 257 440 623 806 976 1159 1342 1356 1087 1270 1453 1636 1819 2002 2172 158 172 355 538 721 904 635 818 1001 1171 1185 1368 1551 1734 1917 2100 86 269 452 Plane 2 931 1114 1297 1480 1663 1846 2016 2030 16 199 382 565 748 479 662 845 846 1029 1212 1395 1578 1761 1944 2127 113 296 2042 28 211 394 577 760 943 1126 1309 1492 1675 1858 1872 1590 1773 1956 2139 125 308 491 674 688 871 1041 1224 1407 1138 1321 1504 1687 1701 1884 2067 40 223 406 589 772 955 517 700 883 1066 1236 1419 1602 1785 1968 2151 137 320 503 65 235 418 601 784 967 1150 1333 1516 1530 1713 1896 2079 1797 1980 2163 149 332 346 529 712 895 1078 1261 1431 1614 1345 1359 1542 1725 1908 2091 77 260 430 613 796 979 1162 724 907 1090 1273 1456 1626 1809 1992 2175 161 175 358 541 272 455 625 808 991 1174 1188 1371 1554 1737 1920 2103 89 2004 2187 4 187 370 553 736 919 1102 1285 1468 1651 1821 1383 1566 1749 1932 2115 101 284 467 650 820 1003 1017 1200 Plane 3 1848 1862 2045 31 214 397 580 763 946 1129 1312 1495 1665 1227 1410 1593 1776 1959 2142 128 311 494 664 678 861 1044 775 958 1141 1324 1507 1690 1691 1874 2057 43 226 409 592 323 506 520 690 873 1056 1239 1422 1605 1788 1971 2154 140 1886 2069 55 238 421 604 787 970 1153 1336 1519 1533 1716 1434 1617 1800 1983 2166 152 335 349 532 715 885 1068 1251 982 1165 1348 1362 1545 1728 1911 2081 67 250 433 616 799 361 544 727 910 1080 1263 1446 1629 1812 1995 2178 164 178 2106 79 262 445 628 811 994 1177 1191 1374 1557 1740 1923 1641 1824 2007 2190 7 190 373 556 739 922 1105 1275 1458 1020 1203 1386 1569 1752 1935 2118 104 274 457 640 823 1006 568 751 934 1117 1300 1470 1653 1836 2019 2033 19 202 385 116 299 469 652 835 849 1032 1215 1398 1581 1764 1947 2130 Plane 4 412 595 778 961 1144 1314 1497 1680 1694 1877 2060 46 229 2157 143 313 496 510 693 876 1059 1242 1425 1608 1791 1974 1523 1706 1889 2072 58 241 424 607 790 973 1156 1339 1509 1071 1254 1437 1620 1803 1986 2169 155 338 339 522 705 888 619 802 985 1168 1351 1365 1535 1718 1901 2084 70 253 436 167 181 364 534 717 900 1083 1266 1449 1632 1815 1998 2181 1730 1913 2096 82 265 448 631 814 997 1180 1194 1377 1560 1278 1461 1644 1827 2010 2193 10 193 376 559 729 912 1095 826 1009 1023 1206 1389 1572 1755 1925 2108 94 277 460 643 205 388 571 754 924 1107 1290 1473 1656 1839 2022 2036 22 1950 2120 106 289 472 655 838 852 1035 1218 1401 1584 1767 1485 1668 1851 1865 2048 34 217 400 583 766 949 1119 1302 864 1047 1230 1413 1596 1779 1962 2145 118 301 484 667 681 Plane 5 1329 1512 1526 1709 1892 2075 61 244 427 610 793 963 1146 708 891 1074 1257 1440 1623 1806 1989 2159 145 328 342 525 256 439 622 805 988 1158 1341 1355 1538 1721 1904 2087 73 2001 2184 157 171 354 537 720 903 1086 1269 1452 1635 1818 1367 1550 1733 1916 2099 85 268 451 634 817 1000 1183 1184 915 1098 1281 1464 1647 1830 2013 2196 13 183 366 549 732 463 646 829 1012 1026 1209 1379 1562 1745 1928 2111 97 280 2039 25 208 378 561 744 927 1110 1293 1476 1659 1842 2025 1574 1757 1940 2123 109 292 475 658 841 855 1038 1221 1404 1122 1305 1488 1671 1854 1868 2051 37 220 403 573 756 939 670 684 867 1050 1233 1416 1599 1769 1952 2135 121 304 487 49 232 415 598 768 951 1134 1317 1500 1683 1697 1880 2063 1794 1964 2147 133 316 499 513 696 879 1062 1245 1428 1611 Plane 6 2090 76 259 442 612 795 978 1161 1344 1358 1541 1724 1907 1638 1808 1991 2174 160 174 357 540 723 906 1089 1272 1455 1173 1187 1370 1553 1736 1919 2102 88 271 454 637 807 990 552 735 918 1101 1284 1467 1650 1833 2003 2186 3 186 369 100 283 466 649 832 1002 1016 1199 1382 1565 1748 1931 2114 1845 2028 2029 15 198 381 564 747 930 1113 1296 1479 1662 1211 1394 1577 1760 1943 2126 112 295 478 661 844 858 1028 759 942 1125 1308 1491 1674 1857 1871 2054 27 210 393 576 307 490 673 687 870 1053 1223 1406 1589 1772 1955 2138 124 1883 2066 52 222 405 588 771 954 1137 1320 1503 1686 1700 1418 1601 1784 1967 2150 136 319 502 516 699 882 1065 1248 966 1149 1332 1515 1529 1712 1895 2078 64 247 417 600 783 345 528 711 894 1077 1260 1443 1613 1796 1979 2162 148 331 Plane 7 810 993 1176 1190 1373 1556 1739 1922 2105 91 261 444 627 189 372 555 738 921 1104 1287 1457 1640 1823 2006 2189 6 1934 2117 103 286 456 639 822 1005 1019 1202 1385 1568 1751 1482 1652 1835 2018 2032 18 201 384 567 750 933 1116 1299 848 1031 1214 1397 1580 1763 1946 2129 115 298 481 651 834 396 579 762 945 1128 1311 1494 1677 1847 1861 2044 30 213 2141 127 310 493 676 677 860 1043 1226 1409 1592 1775 1958 1689 1703 1873 2056 42 225 408 591 774 957 1140 1323 1506 1055 1238 1421 1604 1787 1970 2153 139 322 505 519 702 872 603 786 969 1152 1335 1518 1532 1715 1898 2068 54 237 420 151 334 348 531 714 897 1067 1250 1433 1616 1799 1982 2165 1727 1910 2093 66 249 432 615 798 981 1164 1347 1361 1544 1262 1445 1628 1811 1994 2177 163 177 360 543 726 909 1092 Plane 8 1571 1754 1937 2107 93 276 459 642 825 1008 1022 1205 1388 1106 1289 1472 1655 1838 2021 2035 21 204 387 570 753 936 654 837 851 1034 1217 1400 1583 1766 1949 2132 105 288 471 33 216 399 582 765 948 1131 1301 1484 1667 1850 1864 2047 1778 1961 2144 130 300 483 666 680 863 1046 1229 1412 1595 1326 1496 1679 1693 1876 2059 45 228 411 594 777 960 1143 692 875 1058 1241 1424 1607 1790 1973 2156 142 325 495 509 240 423 606 789 972 1155 1338 1521 1522 1705 1888 2071 57 1985 2168 154 337 351 521 704 887 1070 1253 1436 1619 1802 1364 1547 1717 1900 2083 69 252 435 618 801 984 1167 1350 899 1082 1265 1448 1631 1814 1997 2180 166 180 363 546 716 447 630 813 996 1179 1193 1376 1559 1742 1912 2095 81 264 2192 9 192 375 558 741 911 1094 1277 1460 1643 1826 2009 Plane 9 291 474 657 840 854 1037 1220 1403 1586 1756 1939 2122 108 1867 2050 36 219 402 585 755 938 1121 1304 1487 1670 1853 1415 1598 1781 1951 2134 120 303 486 669 683 866 1049 1232 950 1133 1316 1499 1682 1696 1879 2062 48 231 414 597 780 498 512 695 878 1061 1244 1427 1610 1793 1976 2146 132 315 2074 60 243 426 609 792 975 1145 1328 1511 1525 1708 1891 1622 1805 1988 2171 144 327 341 524 707 890 1073 1256 1439 1170 1340 1354 1537 1720 1903 2086 72 255 438 621 804 987 536 719 902 1085 1268 1451 1634 1817 2000 2183 169 170 353 84 267 450 633 816 999 1182 1196 1366 1549 1732 1915 2098 1829 2012 2195 12 195 365 548 731 914 1097 1280 1463 1646 1208 1391 1561 1744 1927 2110 96 279 462 645 828 1011 1025 743 926 1109 1292 1475 1658 1841 2024 2038 24 207 390 560 Plane 10 1052 1235 1405 1588 1771 1954 2137 123 306 489 672 686 869 587 770 953 1136 1319 1502 1685 1699 1882 2065 51 234 404 135 318 501 515 698 881 1064 1247 1430 1600 1783 1966 2149 1711 1894 2077 63 246 429 599 782 965 1148 1331 1514 1528 1259 1442 1625 1795 1978 2161 147 330 344 527 710 893 1076 794 977 1160 1343 1357 1540 1723 1906 2089 75 258 441 624 173 356 539 722 905 1088 1271 1454 1637 1820 1990 2173 159 1918 2101 87 270 453 636 819 989 1172 1186 1369 1552 1735 1466 1649 1832 2015 2185 2 185 368 551 734 917 1100 1283 1014 1015 1198 1381 1564 1747 1930 2113 99 282 465 648 831 380 563 746 929 1112 1295 1478 1661 1844 2027 2041 14 197 2125 111 294 477 660 843 857 1040 1210 1393 1576 1759 1942 1673 1856 1870 2053 39 209 392 575 758 941 1124 1307 1490 Plane 11 1969 2152 138 321 504 518 701 884 1054 1237 1420 1603 1786 1517 1531 1714 1897 2080 53 236 419 602 785 968 1151 1334 896 1079 1249 1432 1615 1798 1981 2164 150 333 347 530 713 431 614 797 980 1163 1346 1360 1543 1726 1909 2092 78 248 2176 162 176 359 542 725 908 1091 1274 1444 1627 1810 1993 1555 1738 1921 2104 90 273 443 626 809 992 1175 1189 1372 1103 1286 1469 1639 1822 2005 2188 5 188 371 554 737 920 638 821 1004 1018 1201 1384 1567 1750 1933 2116 102 285 468 17 200 383 566 749 932 1115 1298 1481 1664 1834 2017 2031 1762 1945 2128 114 297 480 663 833 847 1030 1213 1396 1579 1310 1493 1676 1859 1860 2043 29 212 395 578 761 944 1127 689 859 1042 1225 1408 1591 1774 1957 2140 126 309 492 675 224 407 590 773 956 1139 1322 1505 1688 1702 1885 2055 41 Plane 12 - Bottom plus 1 533 703 886 1069 1252 1435 1618 1801 1984 2167 153 336 350 68 251 434 617 800 983 1166 1349 1363 1546 1729 1899 2082 1813 1996 2179 165 179 362 545 728 898 1081 1264 1447 1630 1192 1375 1558 1741 1924 2094 80 263 446 629 812 995 1178 740 923 1093 1276 1459 1642 1825 2008 2191 8 191 374 557 275 458 641 824 1007 1021 1204 1387 1570 1753 1936 2119 92 2020 2034 20 203 386 569 752 935 1118 1288 1471 1654 1837 1399 1582 1765 1948 2131 117 287 470 653 836 850 1033 1216 947 1130 1313 1483 1666 1849 1863 2046 32 215 398 581 764 482 665 679 862 1045 1228 1411 1594 1777 1960 2143 129 312 2058 44 227 410 593 776 959 1142 1325 1508 1678 1692 1875 1606 1789 1972 2155 141 324 507 508 691 874 1057 1240 1423 1154 1337 1520 1534 1704 1887 2070 56 239 422 605 788 971 Plane 13 - Bottom 1450 1633 1816 1999 2182 168 182 352 535 718 901 1084 1267 998 1181 1195 1378 1548 1731 1914 2097 83 266 449 632 815 377 547 730 913 1096 1279 1462 1645 1828 2011 2194 11 194 2109 95 278 461 644 827 1010 1024 1207 1390 1573 1743 1926 1657 1840 2023 2037 23 206 389 572 742 925 1108 1291 1474 1036 1219 1402 1585 1768 1938 2121 107 290 473 656 839 853 584 767 937 1120 1303 1486 1669 1852 1866 2049 35 218 401 119 302 485 668 682 865 1048 1231 1414 1597 1780 1963 2133 1695 1878 2061 47 230 413 596 779 962 1132 1315 1498 1681 1243 1426 1609 1792 1975 2158 131 314 497 511 694 877 1060 791 974 1157 1327 1510 1524 1707 1890 2073 59 242 425 608 326 340 523 706 889 1072 1255 1438 1621 1804 1987 2170 156 1902 2085 71 254 437 620 803 986 1169 1352 1353 1536 1719
This is one of the 6m-6 broken oblique planes from the above cube. All are pandiagonal magic squares. this is the case for all perfect magic cubes! [1][2]
[1] B. Rosser and R. J. Walker, Magic Squares: Published papers and Supplement, 1939, a bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4. All papers are very technical. There are NO diagrams. The bound book contains:
[2] F. Liao, T. Katayama and K. Takaba, On the Construction of Pandiagonal Magic Cubes, Kyoto Univ. Technical Report # 99021, 1999
This is a simple magic cube and is not associated. Only
rows, columns pillars and the 4 main triagonals sum correctly to 19215.
This cube contains no magic squares. I received it from Abinhav Soni as an email
attachment on Nov. 29, 2003
I will show the top horizontal plane only.
1100 1141 1224 1258 1292 1326 1066 592 730 826 572 2033 1779 1875 1214 1248 1282 1372 1063 1097 1131 1735 1824 1913 686 775 864 610 593 676 367 401 484 525 559 813 909 655 1773 1869 2007 381 1050 1084 1167 1201 1235 1318 1352 1956 2045 426 858 947 357 1818 1500 1534 1575 1658 1692 1383 1466 992 395 1905 2001 1747 464 903 1271 1305 1339 1030 1120 1154 1188 1792 509 941 687 489 1950 2039 1679 1419 1453 1487 1521 1604 1645 527 1995 1741 1830 547 986 781 1964 2445 2541 1944 1690 64 160 2472 2513 2596 2630 2664 2698 2438 1392 109 198 2058 2490 2579 1982 2586 2620 2654 2744 2435 2469 2503 2528 2624 2027 1430 154 292 1753 250 333 24 58 141 182 216 241 330 1798 2573 2662 1729 1475 2079 2113 2196 2230 2264 2347 2381 2707 1767 1562 286 32 1836 2618 471 505 546 629 663 354 437 77 1881 2656 2402 1861 1607 324 2300 2334 2368 2059 2149 2183 2217 1899 1652 26 115 1919 2701 2496 650 390 424 458 492 575 616
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