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Introduction |
Intro to composition magic cubes and a composition magic square. |
Order 9 Cube |
An order 9 composition magic cube consists of 27 order 3 magic cubes. |
Order 12 Cube |
An order 12 composition magic cube consists of 27 order 4 magic cubes. |
Order 9 Cube-2 |
A new way of constructing composition magic squares and cubes. |
Order 15 Cube |
A cube consisting of 27 order 3 cubes is on my Large Cubes page |
Magic cubes come in as many varieties as magic squares.
However, probably because of the shear volume of numbers to be manipulated when
constructing magic cubes, these variations have been seldom explored.
Some example cube variations that I show on this site are; prime number,
multiply, and inlaid magic cubes.
On this page I show another variety copied from magic squares, the composition magic cube.
To illustrate the concept, I show an order 12 composition magic square, consisting of 16 order 3 magic squares arranged as an order 4 magic square.
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This is the first of 16 order 3 magic squares that make up the order 12 square. |
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This pattern square is index # 112 (of 880) but is the first order 4 associated magic square |
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This magic square is composed of the sums of the 16 order 3 squares within the order 12 magic square. S = 870. The complete order 12 composition magic square. Numbers
used |
My first example of a composition magic cube is an order 9, constructed by assembling 27 order 3 magic cubes, arranged as an order 3 cube.
My second example is an order 12 magic cube. It consists of
27 order 4 magic cubes assembled in an order 3 arrangement.. In each case, the
magic sums of the individual cubes are assembled in the same arrangement, to
form a new order 3 cube with the same constant as the order 9 (or order 12)
composition cube.
Notice that for the order 12 cube, I am using the order 3 as the assembly
pattern and filling the cube with order 4 sub-cubes. In the composition magic
square I used the opposite arrangement!
This order-9 composition magic cube is constructed from 27 order 3 magic cubes
Order 3 index #1 is the pattern for the placement of the 27 sub-cubes, and also for each of the other 26 order 3 cubes.
Plane 1_Top Plane 2 Plane 3_Bottom 2 13 27 16 21 5 24 8 10 22 9 11 3 14 25 17 19 6 18 20 4 23 7 12 1 15 26
The other 26 cubes are constructed to this same pattern, by
using the second group of 27 numbers for the second cube, the third group of 27
numbers for the third cube, etc.
The cubes indicated in the top plane of the pattern order 3 are placed
appropriately in the top 3 planes of the order 9. The order 3 cubes indicated in
the middle and bottom layers are similarly placed in the middle 3 and bottom 3
planes of the order 9 cube.
Plane 1_Top Plane 2_Top-1 29 40 54 326 337 351 704 715 729 43 48 32 340 345 329 718 723 707 49 36 38 346 333 335 724 711 713 30 41 52 327 338 349 705 716 727 45 47 31 342 344 328 720 722 706 50 34 39 347 331 336 725 709 714 569 580 594 218 229 243 272 283 297 583 588 572 232 237 221 286 291 275 589 576 578 238 225 227 292 279 281 570 581 592 219 230 241 273 284 295 585 587 571 234 236 220 288 290 274 590 574 579 239 223 228 293 277 282 461 472 486 515 526 540 83 94 108 475 480 464 529 534 518 97 102 86 481 468 470 535 522 524 103 90 92 462 473 484 516 527 538 84 95 106 477 479 463 531 533 517 99 101 85 482 466 471 536 520 525 104 88 93 Plane 3_Top - 2 Plane 4 Top - 3 51 35 37 348 332 334 726 710 712 407 418 432 542 553 567 110 121 135 44 46 33 341 343 330 719 721 708 427 414 416 562 549 551 130 117 119 28 42 53 325 339 350 703 717 728 423 425 409 558 560 544 126 128 112 591 575 577 240 224 226 294 278 280 56 67 81 353 364 378 650 661 675 584 586 573 233 235 222 287 289 276 76 63 65 373 360 362 670 657 659 568 582 593 217 231 242 271 285 296 72 74 58 369 371 355 666 668 652 483 467 469 537 521 523 105 89 91 596 607 621 164 175 189 299 310 324 476 478 465 530 532 519 98 100 87 616 603 605 184 171 173 319 306 308 460 474 485 514 528 539 82 96 107 612 614 598 180 182 166 315 317 301 Plane 5_Middle Plane 6_Bottom + 3 421 426 410 556 561 545 124 129 113 429 413 415 564 548 550 132 116 118 408 419 430 543 554 565 111 122 133 422 424 411 557 559 546 125 127 114 428 412 417 563 547 552 131 115 120 406 420 431 541 555 566 109 123 134 70 75 59 367 372 356 664 669 653 78 62 64 375 359 361 672 656 658 57 68 79 354 365 376 651 662 673 71 73 60 368 370 357 665 667 654 77 61 66 374 358 363 671 655 660 55 69 80 352 366 377 649 663 674 610 615 599 178 183 167 313 318 302 618 602 604 186 170 172 321 305 307 597 608 619 165 176 187 300 311 322 611 613 600 179 181 168 314 316 303 617 601 606 185 169 174 320 304 309 595 609 620 163 177 188 298 312 323 Plane 7_Bottom + 2 Plane 8_Bottom + 1 623 634 648 191 202 216 245 256 270 637 642 626 205 210 194 259 264 248 643 630 632 211 198 200 265 252 254 624 635 646 192 203 214 246 257 268 639 641 625 207 209 193 261 263 247 644 628 633 212 196 201 266 250 255 434 445 459 488 499 513 137 148 162 448 453 437 502 507 491 151 156 140 454 441 443 508 495 497 157 144 146 435 446 457 489 500 511 138 149 160 450 452 436 504 506 490 153 155 139 455 439 444 509 493 498 158 142 147 2 13 27 380 391 405 677 688 702 16 21 5 394 399 383 691 696 680 22 9 11 400 387 389 697 684 686 3 14 25 381 392 403 678 689 700 18 20 4 396 398 382 693 695 679 23 7 12 401 385 390 698 682 687 Plane 9_Bottom 645 629 631 213 197 199 267 251 253 638 640 627 206 208 195 260 262 249 622 636 647 190 204 215 244 258 269 456 440 442 510 494 496 159 143 145 449 451 438 503 505 492 152 154 141 433 447 458 487 501 512 136 150 161 24 8 10 402 386 388 699 683 685 17 19 6 395 397 384 692 694 681 1 15 26 379 393 404 676 690 701
The numbers in this order 3 magic cube are the magic sums of the 27 cubes in the order 9 composition cube. S = 3285.
Plane 1_Top Plane 2 Plane 3_Bottom 123 1014 2148 1257 1662 366 1905 609 771 1743 690 852 204 1095 1986 1338 1500 447 1419 1581 285 1824 528 933 42 1176 2067
All order 3 cubes are associated, as we expected. The order 9 cube is simple magic with no extra features except that it is associated. Because all the cubes are associated, the middle plane in each of the 3 orthogonal orientations is a magic square.
This order 3 magic cube is the pattern for the placement of
the 27 order 4 cubes in the order 12 magic composition cube shown below. It is
index # 4 of 4.
All the order 3 cubes In the first order 9 and the order 12 cube have exactly the same characteristics.
That is: they all are associated, have 3 magic squares in the 3 center
orthogonal planes, and have all pantriagonals in one direction correct. Also in
all cases there are 2 oblique squares with correct rows, 4 oblique squares with
correct columns, and 3 of the 6 oblique squares have all pandiagonals in one
direction correct. These are the features common to the four basic order 3
cubes.
I_Top II III_Bottom 8 12 22 15 25 2 19 5 18 24 7 11 1 14 27 17 21 4 10 23 9 26 3 13 6 16 20
This is the first of the 27 order 4 cubes.
64 is added to each number in this cube to obtain cube 2, 64 is added to each
number in cube 2 to obtain cube 3, etc. Numbers used are 1 to 1728. 1728 is 12
cubed and also 27 times 64.
I_Top II III IV_Bottom 1 63 62 4 48 18 19 45 32 34 35 29 49 15 14 52 60 6 7 57 21 43 42 24 37 27 26 40 12 54 55 9 56 10 11 53 25 39 38 28 41 23 22 44 8 58 59 5 13 51 50 16 36 30 31 33 20 46 47 17 61 3 2 64
Plane 1 _ Top order 4 cubes represented by the top plane of the order 3
pattern cube are in these top 4 planes.
449 511 510 452 705 767 766 708 1345 1407 1406 1348
508 454 455 505 764 710 711 761 1404 1350 1351 1401
504 458 459 501 760 714 715 757 1400 1354 1355 1397
461 499 498 464 717 755 754 720 1357 1395 1394 1360
1473 1535 1534 1476 385 447 446 388 641 703 702 644
1532 1478 1479 1529 444 390 391 441 700 646 647 697
1528 1482 1483 1525 440 394 395 437 696 650 651 693
1485 1523 1522 1488 397 435 434 400 653 691 690 656
577 639 638 580 1409 1471 1470 1412 513 575 574 516
636 582 583 633 1468 1414 1415 1465 572 518 519 569
632 586 587 629 1464 1418 1419 1461 568 522 523 565
589 627 626 592 1421 1459 1458 1424 525 563 562 528
Plane 2 _ Top - 1
496 466 467 493 752 722 723 749 1392 1362 1363 1389
469 491 490 472 725 747 746 728 1365 1387 1386 1368
473 487 486 476 729 743 742 732 1369 1383 1382 1372
484 478 479 481 740 734 735 737 1380 1374 1375 1377
1520 1490 1491 1517 432 402 403 429 688 658 659 685
1493 1515 1514 1496 405 427 426 408 661 683 682 664
1497 1511 1510 1500 409 423 422 412 665 679 678 668
1508 1502 1503 1505 420 414 415 417 676 670 671 673
624 594 595 621 1456 1426 1427 1453 560 530 531 557
597 619 618 600 1429 1451 1450 1432 533 555 554 536
601 615 614 604 1433 1447 1446 1436 537 551 550 540
612 606 607 609 1444 1438 1439 1441 548 542 543 545
Plane 3 _ Top - 2
480 482 483 477 736 738 739 733 1376 1378 1379 1373
485 475 474 488 741 731 730 744 1381 1371 1370 1384
489 471 470 492 745 727 726 748 1385 1367 1366 1388
468 494 495 465 724 750 751 721 1364 1390 1391 1361
1504 1506 1507 1501 416 418 419 413 672 674 675 669
1509 1499 1498 1512 421 411 410 424 677 667 666 680
1513 1495 1494 1516 425 407 406 428 681 663 662 684
1492 1518 1519 1489 404 430 431 401 660 686 687 657
608 610 611 605 1440 1442 1443 1437 544 546 547 541
613 603 602 616 1445 1435 1434 1448 549 539 538 552
617 599 598 620 1449 1431 1430 1452 553 535 534 556
596 622 623 593 1428 1454 1455 1425 532 558 559 529
Plane 4 _ Top - 3
497 463 462 500 753 719 718 756 1393 1359 1358 1396
460 502 503 457 716 758 759 713 1356 1398 1399 1353
456 506 507 453 712 762 763 709 1352 1402 1403 1349
509 451 450 512 765 707 706 768 1405 1347 1346 1408
1521 1487 1486 1524 433 399 398 436 689 655 654 692
1484 1526 1527 1481 396 438 439 393 652 694 695 649
1480 1530 1531 1477 392 442 443 389 648 698 699 645
1533 1475 1474 1536 445 387 386 448 701 643 642 704
625 591 590 628 1457 1423 1422 1460 561 527 526 564
588 630 631 585 1420 1462 1463 1417 524 566 567 521
584 634 635 581 1416 1466 1467 1413 520 570 571 517
637 579 578 640 1469 1411 1410 1472 573 515 514 576
Plane 5 _ Top - 4 order 4 cubes represented by the middle plane of the order 3
pattern cube are in these middle 4 planes.
897 959 958 900 1537 1599 1598 1540 65 127 126 68
956 902 903 953 1596 1542 1543 1593 124 70 71 121
952 906 907 949 1592 1546 1547 1589 120 74 75 117
909 947 946 912 1549 1587 1586 1552 77 115 114 80
1 63 62 4 833 895 894 836 1665 1727 1726 1668
60 6 7 57 892 838 839 889 1724 1670 1671 1721
56 10 11 53 888 842 843 885 1720 1674 1675 1717
13 51 50 16 845 883 882 848 1677 1715 1714 1680
1601 1663 1662 1604 129 191 190 132 769 831 830 772
1660 1606 1607 1657 188 134 135 185 828 774 775 825
1656 1610 1611 1653 184 138 139 181 824 778 779 821
1613 1651 1650 1616 141 179 178 144 781 819 818 784
Plane 6 _ Top - 5
944 914 915 941 1584 1554 1555 1581 112 82 83 109
917 939 938 920 1557 1579 1578 1560 85 107 106 88
921 935 934 924 1561 1575 1574 1564 89 103 102 92
932 926 927 929 1572 1566 1567 1569 100 94 95 97
48 18 19 45 880 850 851 877 1712 1682 1683 1709
21 43 42 24 853 875 874 856 1685 1707 1706 1688
25 39 38 28 857 871 870 860 1689 1703 1702 1692
36 30 31 33 868 862 863 865 1700 1694 1695 1697
1648 1618 1619 1645 176 146 147 173 816 786 787 813
1621 1643 1642 1624 149 171 170 152 789 811 810 792
1625 1639 1638 1628 153 167 166 156 793 807 806 796
1636 1630 1631 1633 164 158 159 161 804 798 799 801
Plane 7 _ Bottom + 5
928 930 931 925 1568 1570 1571 1565 96 98 99 93
933 923 922 936 1573 1563 1562 1576 101 91 90 104
937 919 918 940 1577 1559 1558 1580 105 87 86 108
916 942 943 913 1556 1582 1583 1553 84 110 111 81
32 34 35 29 864 866 867 861 1696 1698 1699 1693
37 27 26 40 869 859 858 872 1701 1691 1690 1704
41 23 22 44 873 855 854 876 1705 1687 1686 1708
20 46 47 17 852 878 879 849 1684 1710 1711 1681
1632 1634 1635 1629 160 162 163 157 800 802 803 797
1637 1627 1626 1640 165 155 154 168 805 795 794 808
1641 1623 1622 1644 169 151 150 172 809 791 790 812
1620 1646 1647 1617 148 174 175 145 788 814 815 785
Plane 8 _ Bottom + 4
945 911 910 948 1585 1551 1550 1588 113 79 78 116
908 950 951 905 1548 1590 1591 1545 76 118 119 73
904 954 955 901 1544 1594 1595 1541 72 122 123 69
957 899 898 960 1597 1539 1538 1600 125 67 66 128
49 15 14 52 881 847 846 884 1713 1679 1678 1716
12 54 55 9 844 886 887 841 1676 1718 1719 1673
8 58 59 5 840 890 891 837 1672 1722 1723 1669
61 3 2 64 893 835 834 896 1725 1667 1666 1728
1649 1615 1614 1652 177 143 142 180 817 783 782 820
1612 1654 1655 1609 140 182 183 137 780 822 823 777
1608 1658 1659 1605 136 186 187 133 776 826 827 773
1661 1603 1602 1664 189 131 130 192 829 771 770 832
Plane 9 _ Bottom + 3 order 4 cubes represented by the bottom plane of the order 3
pattern cube are in these bottom 4 planes.
1153 1215 1214 1156 257 319 318 260 1089 1151 1150 1092
1212 1158 1159 1209 316 262 263 313 1148 1094 1095 1145
1208 1162 1163 1205 312 266 267 309 1144 1098 1099 1141
1165 1203 1202 1168 269 307 306 272 1101 1139 1138 1104
1025 1087 1086 1028 1281 1343 1342 1284 193 255 254 196
1084 1030 1031 1081 1340 1286 1287 1337 252 198 199 249
1080 1034 1035 1077 1336 1290 1291 1333 248 202 203 245
1037 1075 1074 1040 1293 1331 1330 1296 205 243 242 208
321 383 382 324 961 1023 1022 964 1217 1279 1278 1220
380 326 327 377 1020 966 967 1017 1276 1222 1223 1273
376 330 331 373 1016 970 971 1013 1272 1226 1227 1269
333 371 370 336 973 1011 1010 976 1229 1267 1266 1232
Plane 10 _ Bottom + 2
1200 1170 1171 1197 304 274 275 301 1136 1106 1107 1133
1173 1195 1194 1176 277 299 298 280 1109 1131 1130 1112
1177 1191 1190 1180 281 295 294 284 1113 1127 1126 1116
1188 1182 1183 1185 292 286 287 289 1124 1118 1119 1121
1072 1042 1043 1069 1328 1298 1299 1325 240 210 211 237
1045 1067 1066 1048 1301 1323 1322 1304 213 235 234 216
1049 1063 1062 1052 1305 1319 1318 1308 217 231 230 220
1060 1054 1055 1057 1316 1310 1311 1313 228 222 223 225
368 338 339 365 1008 978 979 1005 1264 1234 1235 1261
341 363 362 344 981 1003 1002 984 1237 1259 1258 1240
345 359 358 348 985 999 998 988 1241 1255 1254 1244
356 350 351 353 996 990 991 993 1252 1246 1247 1249
Plane 11 _ Bottom + 1
1184 1186 1187 1181 288 290 291 285 1120 1122 1123 1117
1189 1179 1178 1192 293 283 282 296 1125 1115 1114 1128
1193 1175 1174 1196 297 279 278 300 1129 1111 1110 1132
1172 1198 1199 1169 276 302 303 273 1108 1134 1135 1105
1056 1058 1059 1053 1312 1314 1315 1309 224 226 227 221
1061 1051 1050 1064 1317 1307 1306 1320 229 219 218 232
1065 1047 1046 1068 1321 1303 1302 1324 233 215 214 236
1044 1070 1071 1041 1300 1326 1327 1297 212 238 239 209
352 354 355 349 992 994 995 989 1248 1250 1251 1245
357 347 346 360 997 987 986 1000 1253 1243 1242 1256
361 343 342 364 1001 983 982 1004 1257 1239 1238 1260
340 366 367 337 980 1006 1007 977 1236 1262 1263 1233
Plane 12 _ Bottom
1201 1167 1166 1204 305 271 270 308 1137 1103 1102 1140
1164 1206 1207 1161 268 310 311 265 1100 1142 1143 1097
1160 1210 1211 1157 264 314 315 261 1096 1146 1147 1093
1213 1155 1154 1216 317 259 258 320 1149 1091 1090 1152
1073 1039 1038 1076 1329 1295 1294 1332 241 207 206 244
1036 1078 1079 1033 1292 1334 1335 1289 204 246 247 201
1032 1082 1083 1029 1288 1338 1339 1285 200 250 251 197
1085 1027 1026 1088 1341 1283 1282 1344 253 195 194 256
369 335 334 372 1009 975 974 1012 1265 1231 1230 1268
332 374 375 329 972 1014 1015 969 1228 1270 1271 1225
328 378 379 325 968 1018 1019 965 1224 1274 1275 1221
381 323 322 384 1021 963 962 1024 1277 1219 1218 1280
This order 3 magic cube is constructed from the magic sums
of the 27 cubes in the order 12 magic cube.
The magic constant for this cube (and the order 12 cube) is 10374.
I_Top II III_Bottom 1922 2946 5506 3714 6274 386 4738 1154 4482 6018 1666 2690 130 3458 6786 4226 5250 898 2434 5762 2178 6530 642 3202 1410 3970 4994
The order 12 cube is simple magic and has no special features except that it is associated. The order 4 cubes and of course the order 3 cubes are also associated.
This order 9 cube consists of 27 order 3 associated magic cubes.
A second method of constructing a composition cube is by forming the sub-cubes by multiplication instead of addition. As expected, the numbers used will be much larger and there will be gaps in the series of numbers used. So the resulting composition will not be a normal cube. A more serious problem results if the numbers in the generating cubes are consecutive. That is, there will be many duplicate numbers.
Using unique prime numbers in the generating cubes ensures that all 729 numbers generated are different. Unfortunately, the two Suzuki cubes I am using as generators have 3 prime numbers in common. They are 683, 839, and 1259.This causes 3 duplicate numbers to appear in the order 9 composition cube. They are 683*839, 683*1259 and 839*1259.
The numbers of cube B are multiplied by the number in cube A to form the order 3 cube that goes in that position in the order 9 cube.
Plane 1_Top Plane 2 Plane 3_Bottom 263 2309 2087 1439 1487 1733 2957 863 839 2129 107 2423 1847 1553 1259 683 2999 977 2267 2243 149 1373 1619 1667 1019 797 2843
A. Above layout and multiplier pattern is the Suzuki-prime-1 magic cube.
Plane 1_Top Plane 2 Plane 3_Bottom 2153 929 227 509 1607 1193 647 773 1889 839 947 1523 1787 1103 419 683 1259 1367 317 1433 1559 1013 599 1697 1979 1277 53
B. Above numbers are multiplied by numbers in pattern A. This cube is the Suzuki-prime-2.
The sums of these 27 sub-cubes generated as above, form the cube below. Magic constant for that cube and the order 9 cube which follows, is 15,416,631
Plane 1_Top Plane 2 Plane 3_Bottom 870267 7640481 6905883 4761651 4920483 5734497 9784713 2855667 2776251 7044861 354063 8017707 6111723 5138877 4166031 2260047 9923691 3232893 7501503 7422087 493041 4543257 5357271 5516103 3371871 2637273 9407487
C. This cube is formed from the constants of the 27 order 3 cubes in the order 9 composition magic cube.
Features of the 30 order 3 cubes and the order 9 cube are identical. Each cube is simple and associated, but has no other features. So it follows that each cube has 3 magic squares in the central three orthogonal planes. It also follows that the order 9 cube has 3 x 27 = 81 order 3 magic squares.
Notice that the order 9 and 12 cubes shown previously and their attendant order 3 cubes have all pantriagonals in one direction correct. Also 3 of the 6 oblique squares have all pandiagonals in one direction correct. All four basic order 3 cubes have these features. Presumably, the difference with this construction is that none of the cubes use consecutive numbers.
It is an interesting, but understandable observation, that all 729 numbers in the composition cube are composite (they have at least two factors. The magic constant is also a composite number.
Plane 1_Top 566239 244327 59701 4971277 2145061 524143 4493311 1938823 473749 220657 249061 400549 1937251 2186623 3516607 1750993 1976389 3178501 83371 376879 410017 731953 3308797 3599731 661579 2990671 3253633 4583737 1977841 483283 230371 99403 24289 5216719 2250967 550021 1786231 2016163 3242467 89773 101329 162961 2032897 2294581 3690229 674893 3050857 3319111 33919 153331 166813 768091 3472159 3777457 4880851 2106043 514609 4829179 2083747 509161 320797 138421 33823 1902013 2146849 3452641 1881877 2124121 3416089 125011 141103 226927 718639 3248611 3534253 711031 3214219 3496837 47233 213517 232291 Plane 2_Top-1 133867 422641 313759 1175281 3710563 2754637 1062283 3353809 2489791 469981 290089 110197 4126183 2546827 967471 3729469 2301961 874453 266419 157537 446311 2339017 1383091 3918373 2114131 1250113 3541639 1083661 3421303 2539897 54463 171949 127651 1233307 3893761 2890639 3804523 2348287 892051 191209 118021 44833 4329901 2672569 1015237 2156677 1275271 3612913 108391 64093 181579 2454499 1451377 4111831 1153903 3643069 2704531 1141687 3604501 2675899 75841 239443 177757 4051129 2500501 949873 4008241 2474029 939817 266263 164347 62431 2296471 1357933 3847099 2272159 1343557 3806371 150937 89251 252853 Plane 3_Top-2 170161 203299 496807 1493923 1784857 4361701 1350289 1613251 3942343 179629 331117 359521 1577047 2907031 3156403 1425421 2627533 2852929 520477 335851 13939 4569511 2948593 122377 4130173 2665099 110611 1377463 1645717 4021681 69229 82711 202123 1567681 1872979 4577047 1454107 2680411 2910343 73081 134713 146269 1654909 3050557 3312241 4213291 2718733 112837 211753 136639 5671 4795117 3094171 128419 1466749 1752391 4282363 1451221 1733839 4237027 96403 115177 281461 1548361 2854153 3098989 1531969 2823937 3066181 101767 187591 203683 4486393 2894959 120151 4438897 2864311 118879 294871 190273 7897 Plane 4_Top-3 3098167 1336831 326653 3201511 1381423 337549 3731149 1609957 393391 1207321 1362733 2191597 1247593 1408189 2264701 1453987 1641151 2639359 456163 2062087 2243401 471379 2130871 2318233 549361 2483389 2701747 3976591 1715863 419269 3343609 1442737 352531 2710627 1169611 285793 1549633 1749109 2812981 1302967 1470691 2365219 1056301 1192273 1917457 585499 2646751 2879473 492301 2225449 2421127 399103 1804147 1962781 2956069 1275517 311671 3485707 1504051 367513 3589051 1548643 378409 1151947 1300231 2091079 1358341 1533193 2465737 1398613 1578649 2538841 435241 1967509 2140507 513223 2320027 2524021 528439 2388811 2598853 Plane 5_Middle 732451 2312473 1716727 756883 2389609 1773991 882097 2784931 2067469 2571493 1587217 602941 2657269 1640161 623053 3096871 1911499 726127 1457707 861961 2441983 1506331 890713 2523439 1755529 1038067 2940901 940123 2968129 2203471 790477 2495671 1852729 640831 2023213 1501987 3300589 2037241 773893 2775211 1712959 650707 2249833 1388677 527521 1871011 1106353 3134359 1573189 930247 2635441 1275367 754141 2136523 698857 2206411 1637989 824071 2601733 1931467 848503 2678869 1988731 2453551 1514419 575287 2893153 1785757 678361 2978929 1838701 698473 1390849 822427 2329981 1640047 969781 2747443 1688671 998533 2828899 Plane 6_Bottom+3 931033 1112347 2718271 962089 1149451 2808943 1121251 1339609 3273637 982837 1811701 1967113 1015621 1872133 2032729 1183639 2181847 2369011 2847781 1837603 76267 2942773 1898899 78811 3429607 2213041 91849 1195009 1427731 3488983 1004791 1200469 2933617 814573 973207 2378251 1261501 2325373 2524849 1060699 1955227 2122951 859897 1585081 1721053 3655213 2358619 97891 3073387 1983181 82309 2491561 1607743 66727 888331 1061329 2593597 1047493 1251487 3058291 1078549 1288591 3148963 937759 1728607 1876891 1105777 2038321 2213173 1138561 2098753 2278789 2717167 1753321 72769 3204001 2067463 85807 3298993 2128759 88351 Plane 7_Bottom+2 6366421 2747053 671239 1858039 801727 195901 1806367 779431 190453 2480923 2800279 4503511 724057 817261 1314349 703921 794533 1277797 937369 4237381 4609963 273571 1236679 1345417 265963 1202287 1308001 1470499 634507 155041 6456847 2786071 680773 2103481 907633 221779 573037 646801 1040209 2516161 2840053 4567477 819703 925219 1487971 216511 978739 1064797 950683 4297567 4675441 309709 1400041 1523143 2193907 946651 231313 1715941 740413 180919 6120979 2641147 645361 854941 964993 1551937 668683 754759 1213831 2385277 2692321 4329889 323023 1460227 1588621 252649 1142101 1242523 901231 4074019 4432237 Plane 8_Bottom+1 1505113 4751899 3527701 439267 1386841 1029559 427051 1348273 1000927 5284159 3261571 1238983 1542181 951889 361597 1499293 925417 351541 2995441 1771243 5018029 874219 516937 1464511 849907 502561 1423783 347647 1097581 814819 1526491 4819393 3577807 497293 1570039 1165561 1220521 753349 286177 5359213 3307897 1256581 1745899 1077631 409363 691879 409117 1159051 3037987 1796401 5089303 989701 585223 1657969 518671 1637533 1215667 405673 1280779 950821 1447087 4568701 3391699 1820953 1123957 426961 1424239 879091 333943 5080441 3135829 1191217 1032247 610381 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October 14, 2009
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