646:
2516:
812:
2377:
2252:
2582:(but with no boundaries on the enclosing rectangle's width or height) has an important application in combining images into a single larger image. A web page that loads a single larger image often renders faster in the browser than the same page loading multiple small images, due to the overhead involved in requesting each image from the web server. The problem is
178:
27:
2717:
Packing of irregular objects is a problem not lending itself well to closed form solutions; however, the applicability to practical environmental science is quite important. For example, irregularly shaped soil particles pack differently as the sizes and shapes vary, leading to important outcomes for
589:. More commonly, the aim is to pack all the objects into as few containers as possible. In some variants the overlapping (of objects with each other and/or with the boundary of the container) is allowed but should be minimized.
711:
packing of spheres, and is believed to be the optimal of all packings. With 'simple' sphere packings in three dimensions ('simple' being carefully defined) there are nine possible definable packings. The 8-dimensional
1824:
684:
larger than one (in a one-dimensional universe, the circle analogue is just two points). That is, there will always be unused space if people are only packing circles. The most efficient way of packing circles,
2119:
1170:
585:
Usually the packing must be without overlaps between goods and other goods or the container walls. In some variants, the aim is to find the configuration that packs a single container with the maximal
581:, some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly.
1039:
2982:
Haji-Akbari, A.; Engel, M.; Keys, A. S.; Zheng, X.; Petschek, R. G.; Palffy-Muhoray, P.; Glotzer, S. C. (2009). "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra".
1570:
803:
Simulations combining local improvement methods with random packings suggest that the lattice packings for icosahedra, dodecahedra, and octahedra are optimal in the broader class of all packings.
1715:
148:
1362:
1310:
3575:
1907:
2047:
1486:
1219:
1097:
945:
3440:
2169:
1941:
2011:
1981:
1637:
1448:
2372:
1867:
887:
597:
Many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite
1605:
1416:
1389:
1254:
977:
3782:
2401:
141:
3647:
3728:
828:
containers (bins) that are required to pack a given set of item cuboids. The rectangular cuboids to be packed can be rotated by 90 degrees on each axis.
3778:
2417:
1869:. Notice that in an infinite-dimensional Hilbert space this implies that there are infinitely many disjoint open unit balls inside a ball of radius
3480:
3176:
Hudson, T. S.; Harrowell, P. (2011). "Structural searches using isopointal sets as generators: Densest packings for binary hard sphere mixtures".
134:
555:, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.
1720:
3049:
2056:
893:
with no restrictions. It is worth describing in detail here, to give a flavor of the general problem. In this case, a configuration of
1102:
3488:
3153:
519:
3285:
752:
2726:
3915:
2788:
3847:
3721:
3686:
3109:
2743:
990:
96:
1491:
2393:
1642:
3842:
2180:
547:
as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life
2320:- closely related to spreading points in a unit square with the objective of finding the greatest minimal separation,
2290:- closely related to spreading points in a unit circle with the objective of finding the greatest minimal separation,
2327:, between points. To convert between these two formulations of the problem, the square side for unit circles will be
3832:
3786:
3774:
2500:
2469:
2314:
2284:
837:
543:
that involve attempting to pack objects together into containers. The goal is to either pack a single container as
486:
471:
3936:
3822:
3714:
2599:
problems, there are to be no gaps, nor overlaps. Many of the puzzles of this type involve packing rectangles or
2578:: The problem of packing multiple rectangles of varying widths and heights in an enclosing rectangle of minimum
2572:
stowage. For example, it is possible to pack 147 rectangles of size (137,95) in a rectangle of size (1600,1230).
3837:
2849:
Donev, A.; Stillinger, F.; Chaikin, P.; Torquato, S. (2004). "Unusually Dense
Crystal Packings of Ellipsoids".
2130:
704:
476:
376:
3538:. Encyclopedia of Earth. eds Emily Monosson and C. Cleveland. National Council for Science and the Environment
601:. This problem is relevant to a number of scientific disciplines, and has received significant attention. The
1315:
1263:
728:
Cubes can easily be arranged to fill three-dimensional space completely, the most natural packing being the
84:
2280:, and have to pack them in the smallest possible container. Several kinds of containers have been studied:
3796:
3555:
2768:
2645:
1257:
654:
614:
271:
1876:
3827:
2793:
2758:
2695:
2016:
1453:
1178:
1056:
904:
512:
491:
401:
243:
2142:
1912:
3393:
3185:
3068:
3001:
2932:
2868:
1573:
536:
103:
3889:
3751:
2798:
2738:
1986:
1946:
1610:
1421:
626:
559:
344:
266:
108:
3692:
2694:
The study of polyomino tilings largely concerns two classes of problems: to tile a rectangle with
3580:
3322:
3294:
3258:
3209:
3126:
3086:
3058:
3025:
2991:
2964:
2922:
2892:
2858:
2330:
2217:
1839:
1577:
610:
481:
261:
44:
2913:
Torquato, S.; Jiao, Y. (August 2009). "Dense packings of the
Platonic and Archimedean solids".
860:
3894:
3791:
3623:
3604:
3484:
3421:
3201:
3149:
3017:
2956:
2948:
2884:
2778:
2763:
2531:
1580:
it extends to a 1-Lipschitz map that is globally defined; in particular, there exists a point
984:
622:
602:
367:
289:
252:
233:
228:
120:
67:
55:
2465:
and have to pack them into the smallest possible container, where the container type varies:
3701:
3534:
3516:
3504:
3452:
3411:
3401:
3362:
3304:
3250:
3193:
3141:
3118:
3104:
3076:
3009:
2940:
2876:
2831:
2773:
2753:
666:
645:
552:
505:
425:
210:
115:
20:
3318:
3163:
1583:
1394:
1367:
1232:
950:
3314:
3159:
2515:
729:
598:
586:
544:
440:
3653:
Many puzzle books as well as mathematical journals contain articles on packing problems.
3197:
2822:
Lodi, A.; Martello, S.; Monaci, M. (2002). "Two-dimensional packing problems: A survey".
3397:
3189:
3072:
3005:
2936:
2872:
811:
3873:
3852:
3814:
3801:
3766:
3416:
3381:
2676:
2649:
2488:
2453:
2376:
2268:
1834:
733:
698:
686:
658:
630:
618:
606:
574:, possibly of infinite size. Multiple containers may be given depending on the problem.
315:
177:
3667:
3626:
3309:
3280:
2835:
19:
This article is about geometric packing problems. For numerical packing problems, see
3930:
3910:
3857:
3607:
3457:
3366:
3130:
2748:
2610:
on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps:
2480:
890:
846:
717:
571:
320:
299:
196:
16:
Problems which attempt to find the most efficient way to pack objects into containers
3326:
3213:
3090:
2968:
987:. A small computation shows that the distance of each vertex from the barycenter is
3682:
3029:
2896:
2596:
741:
410:
294:
238:
219:
91:
3662:
3353:
Stoyan, Y. G.; Yaskov, G. N. (2010). "Packing identical spheres into a cylinder".
3227:
2880:
3380:
Teich, E.G.; van Anders, G.; Klotsa, D.; Dshemuchadse, J.; Glotzer, S.C. (2016).
3756:
3678:
2783:
2718:
plant species to adapt root formations and to allow water movement in the soil.
2583:
2462:
2277:
1041:. Moreover, any other point of the space necessarily has a larger distance from
769:
720:
have also been proven to be optimal in their respective real dimensional space.
540:
420:
310:
276:
72:
3520:
3145:
3081:
3044:
2431:
737:
713:
445:
385:
330:
79:
2952:
2568:
has some applications such as loading of boxes on pallets and, specifically,
3657:
3631:
3612:
3406:
3241:
Smalley, I.J. (1963). "Simple regular sphere packings in three dimensions".
2706:
2600:
2541:
2240:
850:
748:
681:
548:
450:
353:
340:
3425:
3205:
3021:
2960:
2888:
740:
can achieve a packing of at least 85%. One of the best packings of regular
2725:
can fit in a given square container has been shown to be complete for the
2863:
2569:
2190:
2049:, the maximum number of disjoint open unit balls inside a ball of radius
677:
3013:
2944:
2251:
3262:
3122:
2722:
2607:
2483:
1983:
is an orthonormal basis, are disjoint and included in a ball of radius
1819:{\displaystyle r_{k}\leq 1+\|a_{0}-a_{j}\|\leq 1+\|x_{0}-x_{j}\|\leq r}
980:
898:
435:
415:
281:
2586:
in general, but there are fast algorithms for solving small instances.
3299:
2803:
2237:
825:
670:
662:
637:
along three positive axis-parallel rays), and unequal-sphere dimers.
551:, storage and transportation issues. Each packing problem has a dual
358:
205:
169:
34:
30:
3507:; Hautus, M.L.J (1971). "Uniformly coloured stained glass windows".
3254:
680:
in other dimensions can never be packed with complete efficiency in
649:
The hexagonal packing of circles on a 2-dimensional
Euclidean plane.
26:
3706:
3585:
3339:
Minkowski, H. Dichteste gitterförmige
Lagerung kongruenter Körper.
2259:
Many variants of 2-dimensional packing problems have been studied.
2135:
People determine the number of spherical objects of given diameter
2114:{\textstyle {\big \lfloor }{\frac {2}{2-(r-1)^{2}}}{\big \rfloor }}
736:
can tile space on its own, but some preliminary results are known.
617:. Many other shapes have received attention, including ellipsoids,
3063:
2996:
2927:
2514:
2375:
2250:
901:
unit balls is available. People place the centers at the vertices
810:
744:
is based on the aforementioned face-centered cubic (FCC) lattice.
644:
25:
1165:{\textstyle r_{k}:=1+{\sqrt {2{\big (}1-{\frac {1}{k}}{\big )}}}}
653:
These problems are mathematically distinct from the ideas in the
2579:
634:
326:
3710:
3550:
Abrahamsen, Mikkel; Miltzow, Tillmann; Nadja, Seiferth (2020),
3107:(March 1995), "Packing tripods", Mathematical entertainments,
2479:
from 1-10, 14-16, 22-25, 33-36, 62-64, 79-81, 98-100, and any
751:
together can fill all of space in an arrangement known as the
665:, possibly of different sizes, on a surface, for instance the
3672:
2705:
A classic puzzle of the second kind is to arrange all twelve
3693:
Best known packings of equal circles in a circle, up to 1100
853:
may be packed inside it has a simple and complete answer in
2556:, allowing for 90° rotation, in a bigger rectangle of size
3675:
A site with tables, graphs, calculators, references, etc.
3441:"Packing 16, 17 or 18 circles in an equilateral triangle"
2297:, between points. Optimal solutions have been proven for
3045:"Dense Crystalline Dimer Packings of Regular Tetrahedra"
2540:: The problem of packing multiple instances of a single
1034:{\textstyle {\sqrt {2{\big (}1-{\frac {1}{k}}{\big )}}}}
1225:
disjoint open unit balls contained in a ball of radius
2059:
1105:
993:
983:
with edge 2; this is easily realized starting from an
3558:
2333:
2145:
2019:
1989:
1949:
1915:
1879:
1842:
1723:
1645:
1613:
1586:
1565:{\displaystyle \|a_{i}-a_{j}\|=2\leq \|x_{i}-x_{j}\|}
1494:
1456:
1424:
1397:
1370:
1318:
1266:
1235:
1181:
1059:
953:
907:
863:
2603:
into a larger rectangle or other square-like shape.
3903:
3882:
3866:
3813:
3765:
3744:
1710:{\displaystyle \|a_{0}-a_{j}\|\leq \|x_{0}-x_{j}\|}
842:The problem of finding the smallest ball such that
3569:
3355:International Transactions in Operational Research
2366:
2216:the spheres arrange to ordered structures, called
2163:
2113:
2041:
2005:
1975:
1935:
1901:
1861:
1818:
1709:
1631:
1599:
1564:
1480:
1442:
1410:
1383:
1356:
1304:
1248:
1213:
1164:
1091:
1033:
971:
939:
881:
3577:-Completeness of Two-Dimensional Packing Problems
3382:"Clusters of Polyhedra in Spherical Confinement"
2709:into rectangles sized 3Ă—20, 4Ă—15, 5Ă—12 or 6Ă—10.
1175:To show that this configuration is optimal, let
3341:Nachr. Akad. Wiss. Göttingen Math. Phys. KI. II
3043:Chen, E. R.; Engel, M.; Glotzer, S. C. (2010).
2721:The problem of deciding whether a given set of
1049:vertices. In terms of inclusions of balls, the
724:Packings of Platonic solids in three dimensions
3658:Links to various MathWorld articles on packing
3509:Proceedings of the London Mathematical Society
37:packed loosely (top) and more densely (bottom)
3722:
2519:The optimal packing of 10 squares in a square
2380:The optimal packing of 15 circles in a square
2255:The optimal packing of 10 circles in a circle
2106:
2062:
1830:disjoint unit open balls in a ball of radius
1155:
1132:
1024:
1001:
513:
142:
8:
1964:
1950:
1807:
1781:
1769:
1743:
1704:
1678:
1672:
1646:
1559:
1533:
1521:
1495:
1351:
1319:
1299:
1267:
2690:(i.e., the box is a multiple of the brick.)
2576:Packing different rectangles in a rectangle
2538:Packing identical rectangles in a rectangle
1909:. For instance, the unit balls centered at
1172:, which is minimal for this configuration.
3729:
3715:
3707:
3702:Circle packing challenge problem in Python
2908:
2906:
520:
506:
192:
160:
149:
135:
40:
3584:
3563:
3562:
3557:
3456:
3415:
3405:
3308:
3298:
3281:"Densest lattice packings of 3-polytopes"
3080:
3062:
2995:
2926:
2862:
2475:: Optimal solutions have been proven for
2358:
2349:
2332:
2144:
2139:that can be packed into a cuboid of size
2105:
2104:
2095:
2067:
2061:
2060:
2058:
2032:
2018:
1996:
1988:
1967:
1957:
1948:
1927:
1916:
1914:
1892:
1878:
1853:
1841:
1801:
1788:
1763:
1750:
1728:
1722:
1698:
1685:
1666:
1653:
1644:
1612:
1591:
1585:
1553:
1540:
1515:
1502:
1493:
1455:
1423:
1402:
1396:
1375:
1369:
1345:
1326:
1317:
1293:
1274:
1265:
1240:
1234:
1205:
1186:
1180:
1154:
1153:
1143:
1131:
1130:
1125:
1110:
1104:
1083:
1064:
1058:
1023:
1022:
1012:
1000:
999:
994:
992:
952:
931:
912:
906:
862:
689:, produces approximately 91% efficiency.
3648:Optimizing Three-Dimensional Bin Packing
3470:
3468:
3274:
3272:
2824:European Journal of Operational Research
757:
3481:The Mathematical Association of America
3136:Gale, David (1998), Gale, David (ed.),
2814:
2383:Optimal solutions have been proven for
384:
366:
339:
251:
218:
195:
168:
43:
2013:centered at the origin. Moreover, for
1357:{\displaystyle \{a_{1},\dots ,a_{k}\}}
1305:{\displaystyle \{x_{1},\dots ,x_{k}\}}
570:, usually a two- or three-dimensional
3140:, Springer-Verlag, pp. 131–136,
3050:Discrete & Computational Geometry
2486:. The wasted space is asymptotically
764:Optimal density of a lattice packing
7:
3570:{\displaystyle \exists \mathbb {R} }
3279:Betke, Ulrich; Henk, Martin (2000).
3178:Journal of Physics: Condensed Matter
2228:People determine the minimum radius
2185:People determine the minimum height
693:Sphere packings in higher dimensions
3663:MathWorld notes on packing squares.
2247:Packing in 2-dimensional containers
1902:{\displaystyle r\geq 1+{\sqrt {2}}}
815:Packing nine L tricubes into a cube
807:Packing in 3-dimensional containers
605:postulated an optimal solution for
3559:
2423:- Optimal solutions are known for
2042:{\displaystyle r<1+{\sqrt {2}}}
1481:{\displaystyle 1\leq i<j\leq k}
1214:{\displaystyle x_{1},\dots ,x_{k}}
1092:{\displaystyle a_{1},\dots ,a_{k}}
940:{\displaystyle a_{1},\dots ,a_{k}}
14:
2622:rectangle can be packed with 1 Ă—
2164:{\displaystyle a\times b\times c}
1099:are included in a ball of radius
889:, and in an infinite-dimensional
3367:10.1111/j.1475-3995.2009.00733.x
1936:{\displaystyle {\sqrt {2}}e_{j}}
857:-dimensional Euclidean space if
824:Determine the minimum number of
753:tetrahedral-octahedral honeycomb
609:hundreds of years before it was
176:
2727:existential theory of the reals
2698:tiles, and to pack one of each
2506:: Good solutions are known for
2407:- good estimates are known for
2175:Identical spheres in a cylinder
820:Different cuboids into a cuboid
3858:Sphere-packing (Hamming) bound
3687:Wolfram Demonstrations Project
3198:10.1088/0953-8984/23/19/194103
3110:The Mathematical Intelligencer
2744:Close-packing of equal spheres
2092:
2079:
966:
954:
45:Covering/packing-problem pairs
1:
3310:10.1016/S0925-7721(00)00007-9
2881:10.1103/PhysRevLett.92.255506
2836:10.1016/s0377-2217(02)00123-6
2648:: A box can be packed with a
2006:{\displaystyle 1+{\sqrt {2}}}
1976:{\displaystyle \{e_{j}\}_{j}}
1632:{\displaystyle 1\leq j\leq k}
1443:{\displaystyle 1\leq j\leq k}
832:Spheres into a Euclidean ball
3458:10.1016/0012-365X(95)90139-C
3386:Proc. Natl. Acad. Sci. U.S.A
2713:Packing of irregular objects
2201:identical spheres of radius
2181:Sphere packing in a cylinder
1826:. This shows that there are
1053:open unit balls centered at
641:Hexagonal packing of circles
2367:{\displaystyle L=2+2/d_{n}}
1862:{\displaystyle r\geq r_{k}}
661:problem deals with packing
3953:
3138:Tracking the Automatic ANT
2789:Slothouber–Graatsma puzzle
2663:if the box has dimensions
2529:
2451:
2266:
2178:
2128:
838:Sphere packing in a sphere
835:
707:structures offer the best
696:
18:
3475:Honsberger, Ross (1976).
3146:10.1007/978-1-4612-2192-0
3082:10.1007/s00454-010-9273-0
2702:-omino into a rectangle.
882:{\displaystyle k\leq n+1}
593:Packing in infinite space
3783:isosceles right triangle
3521:10.1112/plms/s3-23.4.613
2830:(2). Elsevier: 241–252.
2404:isosceles right triangle
2131:Sphere packing in a cube
678:counterparts of a circle
3533:C.Michael Hogan. 2010.
3407:10.1073/pnas.1524875113
2851:Physical Review Letters
97:Maximum independent set
3797:Circle packing theorem
3668:Erich's Packing Center
3571:
3286:Computational Geometry
2769:Kissing number problem
2626:strips if and only if
2606:There are significant
2520:
2418:Packing circles in an
2402:Packing circles in an
2381:
2368:
2256:
2165:
2115:
2043:
2007:
1977:
1937:
1903:
1863:
1820:
1711:
1633:
1601:
1566:
1482:
1444:
1412:
1385:
1358:
1306:
1250:
1215:
1166:
1093:
1035:
973:
941:
883:
816:
655:circle packing theorem
650:
615:Thomas Callister Hales
38:
3627:"de Bruijn's Theorem"
3572:
3439:Melissen, J. (1995).
2794:Strip packing problem
2759:Cutting stock problem
2526:Packing of rectangles
2518:
2501:Packing squares in a
2470:Packing squares in a
2394:Packing circles in a
2379:
2369:
2315:Packing circles in a
2285:Packing circles in a
2254:
2212:. For a small radius
2166:
2116:
2044:
2008:
1978:
1938:
1904:
1864:
1821:
1712:
1634:
1602:
1600:{\displaystyle a_{0}}
1567:
1483:
1445:
1413:
1411:{\displaystyle a_{j}}
1391:in the corresponding
1386:
1384:{\displaystyle x_{j}}
1359:
1307:
1251:
1249:{\displaystyle x_{0}}
1216:
1167:
1094:
1036:
974:
972:{\displaystyle (k-1)}
942:
884:
814:
703:In three dimensions,
648:
537:optimization problems
29:
3779:equilateral triangle
3556:
3477:Mathematical Gems II
3445:Discrete Mathematics
3243:Mathematics Magazine
2420:equilateral triangle
2331:
2224:Polyhedra in spheres
2143:
2057:
2017:
1987:
1947:
1913:
1877:
1840:
1721:
1643:
1611:
1584:
1492:
1454:
1422:
1395:
1368:
1316:
1264:
1260:from the finite set
1233:
1229:centered at a point
1179:
1103:
1057:
991:
951:
905:
861:
797:18/19 = 0.947368...
562:, people are given:
92:Minimum vertex cover
3916:Slothouber–Graatsma
3608:"Klarner's Theorem"
3398:2016PNAS..113E.669T
3190:2011JPCM...23s4103H
3073:2010arXiv1001.0586C
3014:10.1038/nature08641
3006:2009Natur.462..773H
2945:10.1038/nature08239
2937:2009Natur.460..876T
2873:2004PhRvL..92y5506D
2799:Tetrahedron packing
2739:Bin packing problem
2646:de Bruijn's theorem
2218:columnar structures
2125:Spheres in a cuboid
716:and 24-dimensional
560:bin packing problem
482:Nikoli puzzle types
164:Part of a series on
73:Maximum set packing
3673:www.packomania.com
3624:Weisstein, Eric W.
3605:Weisstein, Eric W.
3567:
3123:10.1007/bf03024896
2521:
2448:Packing of squares
2434:are available for
2382:
2364:
2263:Packing of circles
2257:
2243:of a given shape.
2193:with given radius
2161:
2111:
2039:
2003:
1973:
1933:
1899:
1859:
1816:
1707:
1629:
1607:such that for all
1597:
1578:Kirszbraun theorem
1562:
1478:
1440:
1408:
1381:
1354:
1302:
1246:
1221:be the centers of
1211:
1162:
1089:
1031:
969:
937:
879:
817:
789:)/8 = 0.904508...
651:
623:Archimedean solids
487:Puzzle video games
472:Impossible puzzles
368:Puzzle video games
80:Minimum edge cover
39:
3924:
3923:
3883:Other 3-D packing
3867:Other 2-D packing
3792:Apollonian gasket
3105:Stein, Sherman K.
2990:(7274): 773–777.
2921:(7257): 876–879.
2779:Random close pack
2764:Ellipsoid packing
2532:Rectangle packing
2458:People are given
2273:People are given
2102:
2037:
2001:
1921:
1897:
1160:
1151:
1029:
1020:
985:orthonormal basis
801:
800:
782:(5 +
687:hexagonal packing
603:Kepler conjecture
530:
529:
391:
390:
159:
158:
126:
125:
121:Rectangle packing
68:Minimum set cover
56:Covering problems
3944:
3937:Packing problems
3805:
3745:Abstract packing
3738:Packing problems
3731:
3724:
3717:
3708:
3637:
3636:
3618:
3617:
3591:
3589:
3588:
3576:
3574:
3573:
3568:
3566:
3547:
3541:
3531:
3525:
3524:
3501:
3495:
3494:
3472:
3463:
3462:
3460:
3451:(1–3): 333–342.
3436:
3430:
3429:
3419:
3409:
3392:(6): E669–E678.
3377:
3371:
3370:
3350:
3344:
3337:
3331:
3330:
3312:
3302:
3276:
3267:
3266:
3238:
3232:
3231:
3228:"Circle Packing"
3224:
3218:
3217:
3173:
3167:
3166:
3133:
3101:
3095:
3094:
3084:
3066:
3040:
3034:
3033:
2999:
2979:
2973:
2972:
2930:
2910:
2901:
2900:
2866:
2864:cond-mat/0403286
2846:
2840:
2839:
2819:
2774:Knapsack problem
2754:Covering problem
2567:
2555:
2512:
2496:
2478:
2461:
2440:
2429:
2413:
2389:
2373:
2371:
2370:
2365:
2363:
2362:
2353:
2326:
2310:
2303:
2296:
2276:
2236:identical, unit
2235:
2231:
2215:
2211:
2200:
2196:
2188:
2170:
2168:
2167:
2162:
2138:
2120:
2118:
2117:
2112:
2110:
2109:
2103:
2101:
2100:
2099:
2068:
2066:
2065:
2052:
2048:
2046:
2045:
2040:
2038:
2033:
2012:
2010:
2009:
2004:
2002:
1997:
1982:
1980:
1979:
1974:
1972:
1971:
1962:
1961:
1942:
1940:
1939:
1934:
1932:
1931:
1922:
1917:
1908:
1906:
1905:
1900:
1898:
1893:
1872:
1868:
1866:
1865:
1860:
1858:
1857:
1833:
1829:
1825:
1823:
1822:
1817:
1806:
1805:
1793:
1792:
1768:
1767:
1755:
1754:
1733:
1732:
1716:
1714:
1713:
1708:
1703:
1702:
1690:
1689:
1671:
1670:
1658:
1657:
1638:
1636:
1635:
1630:
1606:
1604:
1603:
1598:
1596:
1595:
1571:
1569:
1568:
1563:
1558:
1557:
1545:
1544:
1520:
1519:
1507:
1506:
1487:
1485:
1484:
1479:
1450:. Since for all
1449:
1447:
1446:
1441:
1417:
1415:
1414:
1409:
1407:
1406:
1390:
1388:
1387:
1382:
1380:
1379:
1363:
1361:
1360:
1355:
1350:
1349:
1331:
1330:
1311:
1309:
1308:
1303:
1298:
1297:
1279:
1278:
1255:
1253:
1252:
1247:
1245:
1244:
1228:
1224:
1220:
1218:
1217:
1212:
1210:
1209:
1191:
1190:
1171:
1169:
1168:
1163:
1161:
1159:
1158:
1152:
1144:
1136:
1135:
1126:
1115:
1114:
1098:
1096:
1095:
1090:
1088:
1087:
1069:
1068:
1052:
1048:
1040:
1038:
1037:
1032:
1030:
1028:
1027:
1021:
1013:
1005:
1004:
995:
978:
976:
975:
970:
946:
944:
943:
938:
936:
935:
917:
916:
896:
888:
886:
885:
880:
856:
845:
788:
787:
758:
553:covering problem
533:Packing problems
522:
515:
508:
477:Maze video games
466:
431:Packing problems
426:Optical illusion
404:
193:
189:
180:
161:
151:
144:
137:
116:Polygon covering
85:Maximum matching
61:Packing problems
52:
51:
41:
21:Knapsack problem
3952:
3951:
3947:
3946:
3945:
3943:
3942:
3941:
3927:
3926:
3925:
3920:
3899:
3878:
3862:
3809:
3803:
3802:Tammes problem
3761:
3740:
3735:
3644:
3622:
3621:
3603:
3602:
3599:
3594:
3554:
3553:
3549:
3548:
3544:
3540:. Washington DC
3532:
3528:
3503:
3502:
3498:
3491:
3474:
3473:
3466:
3438:
3437:
3433:
3379:
3378:
3374:
3352:
3351:
3347:
3343:311–355 (1904).
3338:
3334:
3278:
3277:
3270:
3255:10.2307/2688954
3240:
3239:
3235:
3226:
3225:
3221:
3175:
3174:
3170:
3156:
3135:
3134:. Reprinted in
3103:
3102:
3098:
3042:
3041:
3037:
2981:
2980:
2976:
2912:
2911:
2904:
2848:
2847:
2843:
2821:
2820:
2816:
2812:
2735:
2715:
2677:natural numbers
2593:
2557:
2545:
2534:
2528:
2507:
2487:
2476:
2459:
2456:
2450:
2435:
2424:
2408:
2384:
2354:
2329:
2328:
2325:
2321:
2305:
2298:
2295:
2291:
2274:
2271:
2265:
2249:
2233:
2232:that will pack
2229:
2226:
2213:
2202:
2198:
2197:that will pack
2194:
2186:
2183:
2177:
2141:
2140:
2136:
2133:
2127:
2091:
2072:
2055:
2054:
2050:
2015:
2014:
1985:
1984:
1963:
1953:
1945:
1944:
1923:
1911:
1910:
1875:
1874:
1873:if and only if
1870:
1849:
1838:
1837:
1831:
1827:
1797:
1784:
1759:
1746:
1724:
1719:
1718:
1717:, so that also
1694:
1681:
1662:
1649:
1641:
1640:
1609:
1608:
1587:
1582:
1581:
1549:
1536:
1511:
1498:
1490:
1489:
1452:
1451:
1420:
1419:
1398:
1393:
1392:
1371:
1366:
1365:
1341:
1322:
1314:
1313:
1289:
1270:
1262:
1261:
1256:. Consider the
1236:
1231:
1230:
1226:
1222:
1201:
1182:
1177:
1176:
1106:
1101:
1100:
1079:
1060:
1055:
1054:
1050:
1046:
989:
988:
949:
948:
927:
908:
903:
902:
894:
859:
858:
854:
843:
840:
834:
822:
809:
785:
783:
747:Tetrahedra and
730:cubic honeycomb
726:
701:
695:
643:
607:packing spheres
599:Euclidean space
595:
587:packing density
535:are a class of
526:
497:
496:
467:
464:
457:
456:
455:
441:Problem solving
405:
400:
393:
392:
325:
272:Disentanglement
190:
187:
155:
24:
17:
12:
11:
5:
3950:
3948:
3940:
3939:
3929:
3928:
3922:
3921:
3919:
3918:
3913:
3907:
3905:
3901:
3900:
3898:
3897:
3892:
3886:
3884:
3880:
3879:
3877:
3876:
3874:Square packing
3870:
3868:
3864:
3863:
3861:
3860:
3855:
3853:Kissing number
3850:
3845:
3840:
3835:
3830:
3825:
3819:
3817:
3815:Sphere packing
3811:
3810:
3808:
3807:
3799:
3794:
3789:
3771:
3769:
3767:Circle packing
3763:
3762:
3760:
3759:
3754:
3748:
3746:
3742:
3741:
3736:
3734:
3733:
3726:
3719:
3711:
3705:
3704:
3696:
3695:
3690:
3676:
3670:
3665:
3660:
3651:
3650:
3643:
3642:External links
3640:
3639:
3638:
3619:
3598:
3595:
3593:
3592:
3565:
3561:
3552:Framework for
3542:
3536:Abiotic factor
3526:
3515:(4): 613–628.
3496:
3489:
3483:. p. 67.
3464:
3431:
3372:
3345:
3332:
3293:(3): 157–186.
3268:
3249:(5): 295–299.
3233:
3219:
3184:(19): 194103.
3168:
3154:
3096:
3057:(2): 253–280.
3035:
2974:
2902:
2857:(25): 255506.
2841:
2813:
2811:
2808:
2807:
2806:
2801:
2796:
2791:
2786:
2781:
2776:
2771:
2766:
2761:
2756:
2751:
2746:
2741:
2734:
2731:
2714:
2711:
2692:
2691:
2650:harmonic brick
2643:
2592:
2591:Related fields
2589:
2588:
2587:
2573:
2530:Main article:
2527:
2524:
2523:
2522:
2498:
2454:Square packing
2452:Main article:
2449:
2446:
2443:
2442:
2415:
2399:
2391:
2361:
2357:
2352:
2348:
2345:
2342:
2339:
2336:
2323:
2312:
2293:
2269:Circle packing
2267:Main article:
2264:
2261:
2248:
2245:
2225:
2222:
2179:Main article:
2176:
2173:
2160:
2157:
2154:
2151:
2148:
2126:
2123:
2108:
2098:
2094:
2090:
2087:
2084:
2081:
2078:
2075:
2071:
2064:
2036:
2031:
2028:
2025:
2022:
2000:
1995:
1992:
1970:
1966:
1960:
1956:
1952:
1930:
1926:
1920:
1896:
1891:
1888:
1885:
1882:
1856:
1852:
1848:
1845:
1835:if and only if
1815:
1812:
1809:
1804:
1800:
1796:
1791:
1787:
1783:
1780:
1777:
1774:
1771:
1766:
1762:
1758:
1753:
1749:
1745:
1742:
1739:
1736:
1731:
1727:
1706:
1701:
1697:
1693:
1688:
1684:
1680:
1677:
1674:
1669:
1665:
1661:
1656:
1652:
1648:
1628:
1625:
1622:
1619:
1616:
1594:
1590:
1572:this map is 1-
1561:
1556:
1552:
1548:
1543:
1539:
1535:
1532:
1529:
1526:
1523:
1518:
1514:
1510:
1505:
1501:
1497:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1439:
1436:
1433:
1430:
1427:
1405:
1401:
1378:
1374:
1353:
1348:
1344:
1340:
1337:
1334:
1329:
1325:
1321:
1301:
1296:
1292:
1288:
1285:
1282:
1277:
1273:
1269:
1243:
1239:
1208:
1204:
1200:
1197:
1194:
1189:
1185:
1157:
1150:
1147:
1142:
1139:
1134:
1129:
1124:
1121:
1118:
1113:
1109:
1086:
1082:
1078:
1075:
1072:
1067:
1063:
1026:
1019:
1016:
1011:
1008:
1003:
998:
968:
965:
962:
959:
956:
934:
930:
926:
923:
920:
915:
911:
878:
875:
872:
869:
866:
836:Main article:
833:
830:
821:
818:
808:
805:
799:
798:
795:
791:
790:
780:
776:
775:
772:
766:
765:
762:
734:Platonic solid
725:
722:
699:Sphere packing
697:Main article:
694:
691:
659:circle packing
657:. The related
642:
639:
594:
591:
583:
582:
575:
528:
527:
525:
524:
517:
510:
502:
499:
498:
495:
494:
489:
484:
479:
474:
468:
463:
462:
459:
458:
454:
453:
448:
443:
438:
433:
428:
423:
418:
413:
407:
406:
399:
398:
395:
394:
389:
388:
382:
381:
380:
379:
371:
370:
364:
363:
362:
361:
356:
348:
347:
337:
336:
335:
334:
323:
318:
313:
305:
304:
303:
302:
297:
292:
287:
279:
274:
269:
264:
256:
255:
249:
248:
247:
246:
244:Self-reference
241:
236:
231:
223:
222:
216:
215:
214:
213:
208:
200:
199:
191:
186:
185:
182:
181:
173:
172:
166:
165:
157:
156:
154:
153:
146:
139:
131:
128:
127:
124:
123:
118:
112:
111:
106:
100:
99:
94:
88:
87:
82:
76:
75:
70:
64:
63:
58:
48:
47:
15:
13:
10:
9:
6:
4:
3:
2:
3949:
3938:
3935:
3934:
3932:
3917:
3914:
3912:
3909:
3908:
3906:
3902:
3896:
3893:
3891:
3888:
3887:
3885:
3881:
3875:
3872:
3871:
3869:
3865:
3859:
3856:
3854:
3851:
3849:
3848:Close-packing
3846:
3844:
3843:In a cylinder
3841:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3820:
3818:
3816:
3812:
3806:
3800:
3798:
3795:
3793:
3790:
3788:
3784:
3780:
3776:
3773:
3772:
3770:
3768:
3764:
3758:
3755:
3753:
3750:
3749:
3747:
3743:
3739:
3732:
3727:
3725:
3720:
3718:
3713:
3712:
3709:
3703:
3700:
3699:
3698:
3694:
3691:
3688:
3684:
3680:
3679:"Box Packing"
3677:
3674:
3671:
3669:
3666:
3664:
3661:
3659:
3656:
3655:
3654:
3649:
3646:
3645:
3641:
3634:
3633:
3628:
3625:
3620:
3615:
3614:
3609:
3606:
3601:
3600:
3596:
3587:
3582:
3578:
3546:
3543:
3539:
3537:
3530:
3527:
3522:
3518:
3514:
3510:
3506:
3505:Klarner, D.A.
3500:
3497:
3492:
3490:0-88385-302-7
3486:
3482:
3478:
3471:
3469:
3465:
3459:
3454:
3450:
3446:
3442:
3435:
3432:
3427:
3423:
3418:
3413:
3408:
3403:
3399:
3395:
3391:
3387:
3383:
3376:
3373:
3368:
3364:
3360:
3356:
3349:
3346:
3342:
3336:
3333:
3328:
3324:
3320:
3316:
3311:
3306:
3301:
3296:
3292:
3288:
3287:
3282:
3275:
3273:
3269:
3264:
3260:
3256:
3252:
3248:
3244:
3237:
3234:
3229:
3223:
3220:
3215:
3211:
3207:
3203:
3199:
3195:
3191:
3187:
3183:
3179:
3172:
3169:
3165:
3161:
3157:
3155:0-387-98272-8
3151:
3147:
3143:
3139:
3132:
3128:
3124:
3120:
3116:
3112:
3111:
3106:
3100:
3097:
3092:
3088:
3083:
3078:
3074:
3070:
3065:
3060:
3056:
3052:
3051:
3046:
3039:
3036:
3031:
3027:
3023:
3019:
3015:
3011:
3007:
3003:
2998:
2993:
2989:
2985:
2978:
2975:
2970:
2966:
2962:
2958:
2954:
2950:
2946:
2942:
2938:
2934:
2929:
2924:
2920:
2916:
2909:
2907:
2903:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2870:
2865:
2860:
2856:
2852:
2845:
2842:
2837:
2833:
2829:
2825:
2818:
2815:
2809:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2780:
2777:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2749:Conway puzzle
2747:
2745:
2742:
2740:
2737:
2736:
2732:
2730:
2728:
2724:
2719:
2712:
2710:
2708:
2703:
2701:
2697:
2689:
2685:
2681:
2678:
2674:
2670:
2666:
2662:
2658:
2654:
2651:
2647:
2644:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2612:
2611:
2609:
2604:
2602:
2598:
2595:In tiling or
2590:
2585:
2581:
2577:
2574:
2571:
2565:
2561:
2553:
2549:
2543:
2539:
2536:
2535:
2533:
2525:
2517:
2510:
2505:
2504:
2499:
2494:
2490:
2485:
2482:
2474:
2473:
2468:
2467:
2466:
2464:
2455:
2447:
2445:
2438:
2433:
2427:
2422:
2421:
2416:
2411:
2406:
2405:
2400:
2398:
2397:
2392:
2387:
2378:
2359:
2355:
2350:
2346:
2343:
2340:
2337:
2334:
2319:
2318:
2313:
2308:
2301:
2289:
2288:
2283:
2282:
2281:
2279:
2270:
2262:
2260:
2253:
2246:
2244:
2242:
2239:
2223:
2221:
2219:
2209:
2205:
2192:
2182:
2174:
2172:
2158:
2155:
2152:
2149:
2146:
2132:
2124:
2122:
2096:
2088:
2085:
2082:
2076:
2073:
2069:
2034:
2029:
2026:
2023:
2020:
1998:
1993:
1990:
1968:
1958:
1954:
1928:
1924:
1918:
1894:
1889:
1886:
1883:
1880:
1854:
1850:
1846:
1843:
1836:
1813:
1810:
1802:
1798:
1794:
1789:
1785:
1778:
1775:
1772:
1764:
1760:
1756:
1751:
1747:
1740:
1737:
1734:
1729:
1725:
1699:
1695:
1691:
1686:
1682:
1675:
1667:
1663:
1659:
1654:
1650:
1626:
1623:
1620:
1617:
1614:
1592:
1588:
1579:
1575:
1554:
1550:
1546:
1541:
1537:
1530:
1527:
1524:
1516:
1512:
1508:
1503:
1499:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1437:
1434:
1431:
1428:
1425:
1403:
1399:
1376:
1372:
1346:
1342:
1338:
1335:
1332:
1327:
1323:
1294:
1290:
1286:
1283:
1280:
1275:
1271:
1259:
1241:
1237:
1206:
1202:
1198:
1195:
1192:
1187:
1183:
1173:
1148:
1145:
1140:
1137:
1127:
1122:
1119:
1116:
1111:
1107:
1084:
1080:
1076:
1073:
1070:
1065:
1061:
1044:
1017:
1014:
1009:
1006:
996:
986:
982:
963:
960:
957:
947:of a regular
932:
928:
924:
921:
918:
913:
909:
900:
892:
891:Hilbert space
876:
873:
870:
867:
864:
852:
848:
839:
831:
829:
827:
819:
813:
806:
804:
796:
793:
792:
781:
779:dodecahedron
778:
777:
773:
771:
768:
767:
763:
760:
759:
756:
754:
750:
745:
743:
739:
735:
731:
723:
721:
719:
718:Leech lattice
715:
710:
706:
700:
692:
690:
688:
683:
679:
674:
672:
668:
664:
660:
656:
647:
640:
638:
636:
632:
628:
624:
620:
616:
612:
608:
604:
600:
592:
590:
588:
580:
576:
573:
572:convex region
569:
565:
564:
563:
561:
556:
554:
550:
546:
542:
538:
534:
523:
518:
516:
511:
509:
504:
503:
501:
500:
493:
492:Puzzle topics
490:
488:
485:
483:
480:
478:
475:
473:
470:
469:
461:
460:
452:
449:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
422:
419:
417:
414:
412:
409:
408:
403:
397:
396:
387:
383:
378:
375:
374:
373:
372:
369:
365:
360:
357:
355:
352:
351:
350:
349:
346:
342:
338:
332:
328:
324:
322:
319:
317:
314:
312:
309:
308:
307:
306:
301:
298:
296:
293:
291:
288:
286:
284:
280:
278:
275:
273:
270:
268:
265:
263:
260:
259:
258:
257:
254:
250:
245:
242:
240:
237:
235:
232:
230:
227:
226:
225:
224:
221:
217:
212:
209:
207:
204:
203:
202:
201:
198:
194:
184:
183:
179:
175:
174:
171:
167:
163:
162:
152:
147:
145:
140:
138:
133:
132:
130:
129:
122:
119:
117:
114:
113:
110:
107:
105:
102:
101:
98:
95:
93:
90:
89:
86:
83:
81:
78:
77:
74:
71:
69:
66:
65:
62:
59:
57:
54:
53:
50:
49:
46:
42:
36:
32:
28:
22:
3785: /
3781: /
3777: /
3737:
3697:
3683:Ed Pegg, Jr.
3652:
3630:
3611:
3551:
3545:
3535:
3529:
3512:
3508:
3499:
3476:
3448:
3444:
3434:
3389:
3385:
3375:
3358:
3354:
3348:
3340:
3335:
3300:math/9909172
3290:
3284:
3246:
3242:
3236:
3222:
3181:
3177:
3171:
3137:
3117:(2): 37–39,
3114:
3108:
3099:
3054:
3048:
3038:
2987:
2983:
2977:
2918:
2914:
2854:
2850:
2844:
2827:
2823:
2817:
2720:
2716:
2704:
2699:
2693:
2687:
2683:
2679:
2672:
2668:
2664:
2660:
2656:
2652:
2639:
2635:
2631:
2627:
2623:
2619:
2615:
2605:
2597:tessellation
2594:
2575:
2563:
2559:
2551:
2547:
2537:
2508:
2502:
2492:
2471:
2463:unit squares
2457:
2444:
2436:
2425:
2419:
2409:
2403:
2395:
2385:
2316:
2306:
2299:
2286:
2278:unit circles
2272:
2258:
2227:
2207:
2203:
2184:
2134:
1174:
1042:
979:dimensional
841:
823:
802:
774:0.836357...
746:
727:
708:
705:close-packed
702:
675:
652:
596:
584:
578:
567:
557:
532:
531:
430:
411:Brain teaser
282:
267:Construction
104:Bin covering
60:
3890:Tetrahedron
3833:In a sphere
3804:(on sphere)
3775:In a circle
2784:Set packing
2707:pentominoes
2601:polyominoes
2584:NP-complete
2432:conjectures
1576:and by the
1045:one of the
794:octahedron
770:icosahedron
742:dodecahedra
732:. No other
633:(unions of
613:correct by
541:mathematics
386:Metapuzzles
262:Combination
109:Bin packing
3823:Apollonian
3597:References
3586:2004.07558
2129:See also:
851:unit balls
738:Tetrahedra
714:E8 lattice
682:dimensions
627:tetrahedra
625:including
446:Puzzlehunt
331:Logic maze
253:Mechanical
239:Logic grid
229:Dissection
3895:Ellipsoid
3838:In a cube
3632:MathWorld
3613:MathWorld
3560:∃
3361:: 51–70.
3131:124703268
3064:1001.0586
2997:1012.5138
2953:0028-0836
2928:0908.4107
2696:congruent
2675:for some
2542:rectangle
2396:rectangle
2241:polyhedra
2156:×
2150:×
2086:−
2077:−
1884:≥
1847:≥
1811:≤
1808:‖
1795:−
1782:‖
1773:≤
1770:‖
1757:−
1744:‖
1735:≤
1705:‖
1692:−
1679:‖
1676:≤
1673:‖
1660:−
1647:‖
1624:≤
1618:≤
1574:Lipschitz
1560:‖
1547:−
1534:‖
1531:≤
1522:‖
1509:−
1496:‖
1473:≤
1461:≤
1435:≤
1429:≤
1418:for each
1336:…
1284:…
1196:…
1141:−
1074:…
1010:−
961:−
922:…
897:pairwise
868:≤
749:octahedra
577:A set of
568:container
549:packaging
451:Syllogism
354:Crossword
234:Induction
211:Situation
3931:Category
3426:26811458
3327:12118403
3214:25505460
3206:21525553
3091:18523116
3022:20010683
2969:52819935
2961:19675649
2889:15245027
2733:See also
2723:polygons
2638:divides
2630:divides
2608:theorems
2570:woodpulp
2544:of size
2412:< 300
2191:cylinder
2107:⌋
2063:⌊
1943:, where
1639:one has
1043:at least
847:disjoint
619:Platonic
285:problems
197:Guessing
3904:Puzzles
3689:, 2007.
3417:4760782
3394:Bibcode
3319:1765181
3263:2688954
3186:Bibcode
3164:1661863
3069:Bibcode
3030:4412674
3002:Bibcode
2933:Bibcode
2897:7982407
2869:Bibcode
2673:a b c r
2484:integer
2439:< 28
2428:< 13
1364:taking
981:simplex
899:tangent
784:√
709:lattice
663:circles
631:tripods
579:objects
545:densely
436:Paradox
416:Dilemma
329: (
316:Sliding
290:Folding
170:Puzzles
35:circles
31:Spheres
3911:Conway
3828:Finite
3787:square
3685:, the
3487:
3424:
3414:
3325:
3317:
3261:
3212:
3204:
3162:
3152:
3129:
3089:
3028:
3020:
2984:Nature
2967:
2959:
2951:
2915:Nature
2895:
2887:
2804:Tetris
2503:circle
2481:square
2472:square
2430:, and
2317:square
2304:, and
2287:circle
2238:volume
2206:(<
826:cuboid
761:Solid
671:sphere
611:proven
402:Topics
359:Sudoku
345:Number
300:Tiling
206:Riddle
3581:arXiv
3511:. 3.
3323:S2CID
3295:arXiv
3259:JSTOR
3210:S2CID
3127:S2CID
3087:S2CID
3059:arXiv
3026:S2CID
2992:arXiv
2965:S2CID
2923:arXiv
2893:S2CID
2859:arXiv
2810:Notes
2669:a b q
2661:a b c
2189:of a
1312:into
849:open
669:or a
667:plane
635:cubes
558:In a
465:Lists
377:Mazes
321:Chess
295:Stick
220:Logic
188:Types
3485:ISBN
3422:PMID
3202:PMID
3150:ISBN
3018:PMID
2957:PMID
2949:ISSN
2885:PMID
2580:area
2511:≤ 35
2388:≤ 30
2309:= 19
2302:≤ 13
2024:<
1467:<
676:The
621:and
421:Joke
343:and
341:Word
327:Maze
311:Tour
277:Lock
3757:Set
3752:Bin
3681:by
3517:doi
3453:doi
3449:145
3412:PMC
3402:doi
3390:113
3363:doi
3305:doi
3251:doi
3194:doi
3142:doi
3119:doi
3077:doi
3010:doi
2988:462
2941:doi
2919:460
2877:doi
2832:doi
2828:141
2665:a p
2657:a b
2634:or
2614:An
2053:is
1258:map
539:in
33:or
3933::
3629:.
3610:.
3579:,
3513:23
3479:.
3467:^
3447:.
3443:.
3420:.
3410:.
3400:.
3388:.
3384:.
3359:17
3357:.
3321:.
3315:MR
3313:.
3303:.
3291:16
3289:.
3283:.
3271:^
3257:.
3247:36
3245:.
3208:.
3200:.
3192:.
3182:23
3180:.
3160:MR
3158:,
3148:,
3125:,
3115:17
3113:,
3085:.
3075:.
3067:.
3055:44
3053:.
3047:.
3024:.
3016:.
3008:.
3000:.
2986:.
2963:.
2955:.
2947:.
2939:.
2931:.
2917:.
2905:^
2891:.
2883:.
2875:.
2867:.
2855:92
2853:.
2826:.
2729:.
2686:,
2682:,
2671:Ă—
2667:Ă—
2659:Ă—
2655:Ă—
2618:Ă—
2374:.
2220:.
2171:.
2121:.
1488:,
1117::=
755:.
673:.
629:,
566:A
283:Go
3730:e
3723:t
3716:v
3635:.
3616:.
3590:.
3583::
3564:R
3523:.
3519::
3493:.
3461:.
3455::
3428:.
3404::
3396::
3369:.
3365::
3329:.
3307::
3297::
3265:.
3253::
3230:.
3216:.
3196::
3188::
3144::
3121::
3093:.
3079::
3071::
3061::
3032:.
3012::
3004::
2994::
2971:.
2943::
2935::
2925::
2899:.
2879::
2871::
2861::
2838:.
2834::
2700:n
2688:r
2684:q
2680:p
2653:a
2642:.
2640:b
2636:n
2632:a
2628:n
2624:n
2620:b
2616:a
2566:)
2564:W
2562:,
2560:L
2558:(
2554:)
2552:w
2550:,
2548:l
2546:(
2513:.
2509:n
2497:.
2495:)
2493:a
2491:(
2489:O
2477:n
2460:n
2441:.
2437:n
2426:n
2414:.
2410:n
2390:.
2386:n
2360:n
2356:d
2351:/
2347:2
2344:+
2341:2
2338:=
2335:L
2324:n
2322:d
2311:.
2307:n
2300:n
2294:n
2292:d
2275:n
2234:n
2230:R
2214:R
2210:)
2208:R
2204:r
2199:n
2195:R
2187:h
2159:c
2153:b
2147:a
2137:d
2097:2
2093:)
2089:1
2083:r
2080:(
2074:2
2070:2
2051:r
2035:2
2030:+
2027:1
2021:r
1999:2
1994:+
1991:1
1969:j
1965:}
1959:j
1955:e
1951:{
1929:j
1925:e
1919:2
1895:2
1890:+
1887:1
1881:r
1871:r
1855:k
1851:r
1844:r
1832:r
1828:k
1814:r
1803:j
1799:x
1790:0
1786:x
1779:+
1776:1
1765:j
1761:a
1752:0
1748:a
1741:+
1738:1
1730:k
1726:r
1700:j
1696:x
1687:0
1683:x
1668:j
1664:a
1655:0
1651:a
1627:k
1621:j
1615:1
1593:0
1589:a
1555:j
1551:x
1542:i
1538:x
1528:2
1525:=
1517:j
1513:a
1504:i
1500:a
1476:k
1470:j
1464:i
1458:1
1438:k
1432:j
1426:1
1404:j
1400:a
1377:j
1373:x
1352:}
1347:k
1343:a
1339:,
1333:,
1328:1
1324:a
1320:{
1300:}
1295:k
1291:x
1287:,
1281:,
1276:1
1272:x
1268:{
1242:0
1238:x
1227:r
1223:k
1207:k
1203:x
1199:,
1193:,
1188:1
1184:x
1156:)
1149:k
1146:1
1138:1
1133:(
1128:2
1123:+
1120:1
1112:k
1108:r
1085:k
1081:a
1077:,
1071:,
1066:1
1062:a
1051:k
1047:k
1025:)
1018:k
1015:1
1007:1
1002:(
997:2
967:)
964:1
958:k
955:(
933:k
929:a
925:,
919:,
914:1
910:a
895:k
877:1
874:+
871:n
865:k
855:n
844:k
786:5
521:e
514:t
507:v
333:)
150:e
143:t
136:v
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
