2557:
371:
20:
398:(4)), this results in four new lines (the dashed lines in the diagram) being added, numbered 1 through 4, one for each point that they connect to. The number of new regions introduced by the fifth point can therefore be determined by considering the number of regions added by each of the 4 lines. Set
409:
The number of lines that each new line intersects can be determined by considering the number of points on the "left" of the line and the number of points on the "right" of the line. Since all existing points already have lines between them, the number of points on the left multiplied by the number
204:
1587:
points on the circle, referred to as the exterior vertices, as well as the interior vertices, the intersections of distinct chords in the interior of the circle. The "generic intersection" assumption made above guarantees that each interior vertex is the intersection of no more than two chords.
1595:
is finding the number of interior vertices. As a consequence of the lemma, any two intersecting chords will uniquely determine an interior vertex. These chords are in turn uniquely determined by the four corresponding endpoints of the chords, which are all exterior vertices. Any four exterior
1765:
connecting pairs of adjacent exterior vertices, as well as the chordal line segments (described below) created inside the circle by the collection of chords. Since there are two groups of vertices: exterior and interior, the chordal line segments can be further categorized into three groups:
2087:
1488:
The lemma asserts that the number of regions is maximal if all "inner" intersections of chords are simple (exactly two chords pass through each point of intersection in the interior). This will be the case if the points on the circle are chosen
1059:
2623:
Considering the force-free motion of a particle inside a circle it was shown (see D. Jaud) that for specific reflection angles along the circle boundary the associated area division sequence is given by an arithmetic series.
1835:
Every chord that is cut by another (i.e., chords not in group 1) must contain two group 3 edges, its beginning and ending chordal segments. As chords are uniquely determined by two exterior vertices, there are altogether
622:
1753:
926:
2449:
2716:
Jaud, D. "Integer
Sequences from Circle Divisions by Rational Billiard Trajectories". In "ICGG 2022 - Proceedings of the 20th International Conference on Geometry and Graphics", DOI: 10.1007/978-3-031-13588-0_8
2551:
1198:
748:
2328:
1667:
2223:
1895:
164:
1917:
1832:
edges. Since each edge is defined by two endpoint vertices, only the interior vertices were enumerated, group 2 edges are counted twice while group 3 edges are counted once only.
1770:
Edges directly (not cut by other chords) connecting two exterior vertices. These are chords between adjacent exterior vertices, and form the perimeter of the polygon. There are
1827:
2144:
1563:
1097:
937:
2584:
815:
783:
2657:
2587:
173:
1611:
Therefore, each interior vertex is uniquely determined by a combination of four exterior vertices, where the number of interior vertices is given by
1493:". Under this assumption of "generic intersection", the number of regions can also be determined in a non-inductive way, using the formula for the
488:
1678:
823:
2339:
1604:, so each set of four exterior vertices have exactly one point of intersection formed by their diagonals (chords). Further, by definition
1784:
To find the number of edges in groups 2 and 3, consider each interior vertex, which is connected to exactly four edges. This yields
402:
to count the lines being added. Each new line can cross a number of existing lines, depending on which point it is to (the value of
2460:
223:), the new line passes through a point where two or more old lines (between previously existing points) cross. In the second case (
1108:
633:
2707:
2247:
1617:
1568:
holds. View the diagram (the circle together with all the chords) above as a planar graph. If the general formulas for
2152:
1498:
2633:
1842:
99:
227:), the new line crosses each of the old lines in a different point. It will be useful to know the following fact.
219:
lines can be drawn from the new point to previously existing points. Two cases are possible. In the first case (
2082:{\displaystyle E={\frac {4{n \choose 4}+2\left({n \choose 2}-n\right)}{2}}+n+n=2{n \choose 4}+{n \choose 2}+n.}
378:
In the figure the dark lines are connecting points 1 through 4 dividing the circle into 8 total regions (i.e.,
1523:
vertices (0-dimensional cells). As the graph is connected, the Euler relation for the 2-dimensional sphere S
203:
2600:
2592:
2729:
348:
1903:
The sum of these results divided by two gives the combined number of edges in groups 2 and 3. Adding the
410:
of points on the right is the number of lines that will be crossing the new line. For the line to point
370:
182:
1597:
1494:
1480:
1900:
group 3 edges. This is twice the total number of chords that are not themselves members of group 1.
1790:
1490:
2671:
2107:
1054:{\displaystyle f(n)=\sum _{k=1}^{n}\left({\frac {1}{6}}k^{3}-k^{2}+{\frac {17}{6}}k-2\right)+1,}
1529:
2687:
1064:
2734:
2690:
2607:= 4) gives the maximum number of regions in the problem of dividing a circle into areas for
2556:
2562:
788:
756:
19:
2711:
2675:
2739:
2723:
2704:
1601:
347:
The lemma establishes an important property for solving the problem. By employing an
1762:
1501:
168:
479:= 3 each cross two lines (there are two points on one side and one on the other).
88:, has a solution by an inductive method. The greatest possible number of regions,
2454:
which yields the same quartic polynomial obtained by using the inductive method
390:= 5 with the dashed lines. When the fifth point is added (i.e., when computing
2228:
Since one of these faces is the exterior of the circle, the number of regions
2695:
617:{\displaystyle f(n)=f(n-1)+\sum _{i=1}^{n-1}{\big (}1+(n-i-1)(i-1){\big )},}
83:
1748:{\displaystyle V=V_{\text{exterior}}+V_{\text{interior}}=n+{n \choose 4}.}
921:{\displaystyle f(n)=f(n-1)+{\frac {1}{6}}n^{3}-n^{2}+{\frac {17}{6}}n-2.}
78:
54:
406:). The new lines will never cross each other, except at the new point.
66:
2444:{\displaystyle r_{G}={\frac {n!}{(n-4)!4!}}+{\frac {n!}{(n-2)!2!}}+1}
1505:
60:
2555:
18:
1511:
A planar graph determines a cell decomposition of the plane with
195:, showing the risk of generalising from only a few observations.
291:. Since the circle has infinitely many points, it has a point
2546:{\displaystyle r_{G}={\frac {1}{24}}n(n^{3}-6n^{2}+23n-18)+1}
1193:{\displaystyle f(n)={\frac {n}{24}}(n^{3}-6n^{2}+23n-18)+1.}
743:{\displaystyle f(n)=f(n-1)+\sum _{i=1}^{n-1}(2-n+ni-i^{2}).}
369:
202:
2661:
382:(4) = 8). This figure illustrates the inductive step from
177:
2619:
Application to mathematical billiards inside the circle
2565:
2463:
2342:
2250:
2155:
2110:
1920:
1845:
1793:
1681:
1620:
1608:
interior vertices are formed by intersecting chords.
1532:
1111:
1067:
940:
826:
791:
759:
636:
491:
102:
2323:{\displaystyle r_{G}={n \choose 4}+{n \choose 2}+1,}
1580:can also be derived, which will solve the problem.
2705:http://www.arbelos.co.uk/Papers/Chords-regions.pdf
2682:. New York: Springer-Verlag, pp. 76–79, 1996.
2578:
2545:
2443:
2322:
2217:
2138:
2081:
1889:
1821:
1747:
1662:{\displaystyle V_{\text{interior}}={n \choose 4},}
1661:
1557:
1192:
1091:
1053:
920:
809:
777:
742:
616:
251:, three points must be on one line: the new point
215:points on the circle and one more point is added,
167:, giving the sequence 1, 2, 4, 8, 16, 31, 57, 99,
158:
2305:
2292:
2280:
2267:
2203:
2190:
2178:
2165:
2064:
2051:
2039:
2026:
1870:
1857:
1813:
1800:
1780:Edges connecting an interior and exterior vertex.
1736:
1723:
1650:
1637:
330: + 1 new areas are created by the line
144:
131:
119:
106:
2218:{\displaystyle F={n \choose 4}+{n \choose 2}+2.}
263:where two of the old lines intersect. There are
275:where two of the old lines intersect. For each
471:= 4 each cross zero lines, while the lines to
1982:
1969:
1949:
1936:
1890:{\displaystyle 2\left({n \choose 2}-n\right)}
606:
557:
159:{\displaystyle {n \choose 4}+{n \choose 2}+1}
77:the number of areas created by the edges and
8:
1600:, and all cyclic quadrilaterals are convex
287:crosses the circle in one point other than
49:for first 6 terms of Moser's circle problem
1504:(viewed here as a graph embedded in the 2-
1207:
259:to which the line is drawn, and the point
2570:
2564:
2513:
2497:
2477:
2468:
2462:
2397:
2356:
2347:
2341:
2304:
2291:
2289:
2279:
2266:
2264:
2255:
2249:
2202:
2189:
2187:
2177:
2164:
2162:
2154:
2111:
2109:
2063:
2050:
2048:
2038:
2025:
2023:
1981:
1968:
1966:
1948:
1935:
1933:
1927:
1919:
1869:
1856:
1854:
1844:
1812:
1799:
1797:
1792:
1735:
1722:
1720:
1705:
1692:
1680:
1649:
1636:
1634:
1625:
1619:
1533:
1531:
1160:
1144:
1127:
1110:
1066:
1018:
1009:
996:
982:
971:
960:
939:
899:
890:
877:
863:
825:
790:
758:
728:
688:
677:
635:
605:
604:
556:
555:
543:
532:
490:
181:). Though the first five terms match the
143:
130:
128:
118:
105:
103:
101:
2649:
1911:circular arc edges brings the total to
1777:Edges connecting two interior vertices.
482:So the recurrence can be expressed as
1475:The series alternatively derived from
7:
314:This lemma means that, if there are
2100:into the Euler relation solved for
1576:can both be found, the formula for
440:points on the right, so a total of
295:which will be on none of the lines
2296:
2271:
2194:
2169:
2055:
2030:
1973:
1940:
1861:
1804:
1727:
1641:
1591:Thus the main task in determining
1477:the sum of up to the first 5 terms
351:, one can arrive at a formula for
241:occurs for each of the new lines.
135:
110:
14:
1204:Combinatorics and topology method
271:, and hence finitely many points
1519:edges (1-dimensional cells) and
627:which can be easily reduced to
2640:is the number of straight cuts
2534:
2490:
2420:
2408:
2379:
2367:
1822:{\displaystyle 4{n \choose 4}}
1181:
1137:
1121:
1115:
1077:
1071:
950:
944:
857:
845:
836:
830:
804:
792:
785:natural numbers and the first
772:
760:
734:
700:
667:
655:
646:
640:
601:
589:
586:
568:
522:
510:
501:
495:
463:In this example, the lines to
1:
1515:faces (2-dimensional cells),
1907:edges from group 1, and the
753:Using the sums of the first
322:, then each of them crosses
2691:"Circle Division by Chords"
237:can be chosen so that case
2756:
2139:{\displaystyle \,F=E-V+2,}
1237:
817:squares, this combines to
303:and all of the old points
73:sides in such a way as to
1583:Its vertices include the
1558:{\displaystyle \,V-E+F=2}
1474:
326:at a different point and
65:by means of an inscribed
2678:"How Many Regions." In
2634:Lazy caterer's sequence
1092:{\displaystyle f(0)=1,}
460:lines must be crossed.
429:points on the left and
299:. Then, for this point
2596:
2580:
2547:
2445:
2324:
2219:
2140:
2083:
1891:
1823:
1758:The edges include the
1749:
1663:
1559:
1194:
1093:
1055:
976:
922:
811:
779:
744:
699:
618:
554:
375:
208:
160:
50:
2581:
2579:{\displaystyle r_{G}}
2559:
2548:
2446:
2325:
2237:inside the circle is
2220:
2141:
2084:
1892:
1824:
1750:
1664:
1596:vertices determine a
1560:
1195:
1094:
1056:
956:
923:
812:
810:{\displaystyle (n-1)}
780:
778:{\displaystyle (n-1)}
745:
673:
619:
528:
373:
206:
183:geometric progression
161:
22:
2601:Bernoulli's triangle
2599:The fifth column of
2593:Bernoulli's triangle
2563:
2461:
2340:
2248:
2153:
2108:
1918:
1843:
1791:
1679:
1618:
1598:cyclic quadrilateral
1530:
1495:Euler characteristic
1109:
1065:
938:
824:
789:
757:
634:
489:
100:
2680:The Book of Numbers
2586:(purple) and other
1491:in general position
81:, sometimes called
16:Problem in geometry
2710:2011-09-04 at the
2688:Weisstein, Eric W.
2611:+ 1 points, where
2597:
2576:
2543:
2441:
2333:which resolves to
2320:
2215:
2136:
2079:
1887:
1819:
1745:
1659:
1555:
1190:
1089:
1051:
918:
807:
775:
740:
614:
376:
209:
156:
51:
2485:
2433:
2392:
2303:
2278:
2201:
2176:
2146:one then obtains
2062:
2037:
2003:
1980:
1947:
1868:
1811:
1734:
1708:
1695:
1648:
1628:
1486:
1485:
1481:Pascal's triangle
1135:
1026:
990:
907:
871:
188:, it deviates at
142:
117:
86:'s circle problem
57:, the problem of
2747:
2701:
2700:
2665:
2664:
2654:
2595:
2585:
2583:
2582:
2577:
2575:
2574:
2552:
2550:
2549:
2544:
2518:
2517:
2502:
2501:
2486:
2478:
2473:
2472:
2450:
2448:
2447:
2442:
2434:
2432:
2406:
2398:
2393:
2391:
2365:
2357:
2352:
2351:
2329:
2327:
2326:
2321:
2310:
2309:
2308:
2295:
2285:
2284:
2283:
2270:
2260:
2259:
2224:
2222:
2221:
2216:
2208:
2207:
2206:
2193:
2183:
2182:
2181:
2168:
2145:
2143:
2142:
2137:
2088:
2086:
2085:
2080:
2069:
2068:
2067:
2054:
2044:
2043:
2042:
2029:
2004:
1999:
1998:
1994:
1987:
1986:
1985:
1972:
1954:
1953:
1952:
1939:
1928:
1896:
1894:
1893:
1888:
1886:
1882:
1875:
1874:
1873:
1860:
1828:
1826:
1825:
1820:
1818:
1817:
1816:
1803:
1754:
1752:
1751:
1746:
1741:
1740:
1739:
1726:
1710:
1709:
1706:
1697:
1696:
1693:
1668:
1666:
1665:
1660:
1655:
1654:
1653:
1640:
1630:
1629:
1626:
1564:
1562:
1561:
1556:
1208:
1199:
1197:
1196:
1191:
1165:
1164:
1149:
1148:
1136:
1128:
1098:
1096:
1095:
1090:
1060:
1058:
1057:
1052:
1041:
1037:
1027:
1019:
1014:
1013:
1001:
1000:
991:
983:
975:
970:
927:
925:
924:
919:
908:
900:
895:
894:
882:
881:
872:
864:
816:
814:
813:
808:
784:
782:
781:
776:
749:
747:
746:
741:
733:
732:
698:
687:
623:
621:
620:
615:
610:
609:
561:
560:
553:
542:
343:Inductive method
255:, the old point
233:. The new point
194:
187:
180:
166:
165:
163:
162:
157:
149:
148:
147:
134:
124:
123:
122:
109:
48:
37:
30:
2755:
2754:
2750:
2749:
2748:
2746:
2745:
2744:
2720:
2719:
2712:Wayback Machine
2686:
2685:
2668:
2656:
2655:
2651:
2647:
2630:
2621:
2591:
2566:
2561:
2560:
2509:
2493:
2464:
2459:
2458:
2407:
2399:
2366:
2358:
2343:
2338:
2337:
2290:
2265:
2251:
2246:
2245:
2236:
2188:
2163:
2151:
2150:
2106:
2105:
2049:
2024:
1967:
1965:
1961:
1934:
1929:
1916:
1915:
1855:
1853:
1849:
1841:
1840:
1798:
1789:
1788:
1721:
1701:
1688:
1677:
1676:
1635:
1621:
1616:
1615:
1528:
1527:
1479:of each row of
1478:
1476:
1220:
1215:
1206:
1156:
1140:
1107:
1106:
1063:
1062:
1005:
992:
981:
977:
936:
935:
886:
873:
822:
821:
787:
786:
755:
754:
724:
632:
631:
487:
486:
349:inductive proof
345:
340:
318:lines crossing
247:. For the case
201:
189:
185:
172:
129:
104:
98:
97:
94:
89:
45:
39:
31:
24:
17:
12:
11:
5:
2753:
2751:
2743:
2742:
2737:
2732:
2722:
2721:
2718:
2717:
2714:
2702:
2683:
2667:
2666:
2648:
2646:
2643:
2642:
2641:
2636:– where
2629:
2626:
2620:
2617:
2573:
2569:
2554:
2553:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2516:
2512:
2508:
2505:
2500:
2496:
2492:
2489:
2484:
2481:
2476:
2471:
2467:
2452:
2451:
2440:
2437:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2405:
2402:
2396:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2364:
2361:
2355:
2350:
2346:
2331:
2330:
2319:
2316:
2313:
2307:
2302:
2299:
2294:
2288:
2282:
2277:
2274:
2269:
2263:
2258:
2254:
2232:
2226:
2225:
2214:
2211:
2205:
2200:
2197:
2192:
2186:
2180:
2175:
2172:
2167:
2161:
2158:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2090:
2089:
2078:
2075:
2072:
2066:
2061:
2058:
2053:
2047:
2041:
2036:
2033:
2028:
2022:
2019:
2016:
2013:
2010:
2007:
2002:
1997:
1993:
1990:
1984:
1979:
1976:
1971:
1964:
1960:
1957:
1951:
1946:
1943:
1938:
1932:
1926:
1923:
1898:
1897:
1885:
1881:
1878:
1872:
1867:
1864:
1859:
1852:
1848:
1830:
1829:
1815:
1810:
1807:
1802:
1796:
1782:
1781:
1778:
1775:
1756:
1755:
1744:
1738:
1733:
1730:
1725:
1719:
1716:
1713:
1704:
1700:
1691:
1687:
1684:
1670:
1669:
1658:
1652:
1647:
1644:
1639:
1633:
1624:
1602:quadrilaterals
1566:
1565:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1484:
1483:
1472:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1449:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1426:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1403:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1380:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1357:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1334:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1311:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1288:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1265:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1242:
1241:
1238:
1236:
1233:
1230:
1227:
1224:
1221:
1216:
1211:
1205:
1202:
1201:
1200:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1163:
1159:
1155:
1152:
1147:
1143:
1139:
1134:
1131:
1126:
1123:
1120:
1117:
1114:
1100:
1099:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1050:
1047:
1044:
1040:
1036:
1033:
1030:
1025:
1022:
1017:
1012:
1008:
1004:
999:
995:
989:
986:
980:
974:
969:
966:
963:
959:
955:
952:
949:
946:
943:
929:
928:
917:
914:
911:
906:
903:
898:
893:
889:
885:
880:
876:
870:
867:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
806:
803:
800:
797:
794:
774:
771:
768:
765:
762:
751:
750:
739:
736:
731:
727:
723:
720:
717:
714:
711:
708:
705:
702:
697:
694:
691:
686:
683:
680:
676:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
625:
624:
613:
608:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
559:
552:
549:
546:
541:
538:
535:
531:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
458:
457:
438:
437:
427:
426:
359:) in terms of
344:
341:
339:
336:
311:will be true.
200:
197:
155:
152:
146:
141:
138:
133:
127:
121:
116:
113:
108:
92:
43:
23:The number of
15:
13:
10:
9:
6:
4:
3:
2:
2752:
2741:
2738:
2736:
2733:
2731:
2730:Combinatorics
2728:
2727:
2725:
2715:
2713:
2709:
2706:
2703:
2698:
2697:
2692:
2689:
2684:
2681:
2677:
2673:
2672:Conway, J. H.
2670:
2669:
2663:
2659:
2653:
2650:
2644:
2639:
2635:
2632:
2631:
2627:
2625:
2618:
2616:
2614:
2610:
2606:
2602:
2594:
2590:sequences in
2589:
2571:
2567:
2558:
2540:
2537:
2531:
2528:
2525:
2522:
2519:
2514:
2510:
2506:
2503:
2498:
2494:
2487:
2482:
2479:
2474:
2469:
2465:
2457:
2456:
2455:
2438:
2435:
2429:
2426:
2423:
2417:
2414:
2411:
2403:
2400:
2394:
2388:
2385:
2382:
2376:
2373:
2370:
2362:
2359:
2353:
2348:
2344:
2336:
2335:
2334:
2317:
2314:
2311:
2300:
2297:
2286:
2275:
2272:
2261:
2256:
2252:
2244:
2243:
2242:
2240:
2235:
2231:
2212:
2209:
2198:
2195:
2184:
2173:
2170:
2159:
2156:
2149:
2148:
2147:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2103:
2099:
2095:
2092:Substituting
2076:
2073:
2070:
2059:
2056:
2045:
2034:
2031:
2020:
2017:
2014:
2011:
2008:
2005:
2000:
1995:
1991:
1988:
1977:
1974:
1962:
1958:
1955:
1944:
1941:
1930:
1924:
1921:
1914:
1913:
1912:
1910:
1906:
1901:
1883:
1879:
1876:
1865:
1862:
1850:
1846:
1839:
1838:
1837:
1833:
1808:
1805:
1794:
1787:
1786:
1785:
1779:
1776:
1773:
1769:
1768:
1767:
1764:
1763:circular arcs
1761:
1742:
1731:
1728:
1717:
1714:
1711:
1702:
1698:
1689:
1685:
1682:
1675:
1674:
1673:
1656:
1645:
1642:
1631:
1622:
1614:
1613:
1612:
1609:
1607:
1603:
1599:
1594:
1589:
1586:
1581:
1579:
1575:
1571:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1526:
1525:
1524:
1522:
1518:
1514:
1509:
1507:
1503:
1500:
1496:
1492:
1482:
1473:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1450:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1427:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1404:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1381:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1358:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1335:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1312:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1289:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1266:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1243:
1239:
1234:
1231:
1228:
1225:
1222:
1219:
1214:
1210:
1209:
1203:
1187:
1184:
1178:
1175:
1172:
1169:
1166:
1161:
1157:
1153:
1150:
1145:
1141:
1132:
1129:
1124:
1118:
1112:
1105:
1104:
1103:
1102:which yields
1086:
1083:
1080:
1074:
1068:
1048:
1045:
1042:
1038:
1034:
1031:
1028:
1023:
1020:
1015:
1010:
1006:
1002:
997:
993:
987:
984:
978:
972:
967:
964:
961:
957:
953:
947:
941:
934:
933:
932:
915:
912:
909:
904:
901:
896:
891:
887:
883:
878:
874:
868:
865:
860:
854:
851:
848:
842:
839:
833:
827:
820:
819:
818:
801:
798:
795:
769:
766:
763:
737:
729:
725:
721:
718:
715:
712:
709:
706:
703:
695:
692:
689:
684:
681:
678:
674:
670:
664:
661:
658:
652:
649:
643:
637:
630:
629:
628:
611:
598:
595:
592:
583:
580:
577:
574:
571:
565:
562:
550:
547:
544:
539:
536:
533:
529:
525:
519:
516:
513:
507:
504:
498:
492:
485:
484:
483:
480:
478:
474:
470:
466:
461:
455:
451:
447:
443:
442:
441:
435:
432:
431:
430:
424:
420:
417:
416:
415:
413:
407:
405:
401:
397:
393:
389:
385:
381:
372:
368:
366:
362:
358:
354:
350:
342:
337:
335:
333:
329:
325:
321:
317:
312:
310:
306:
302:
298:
294:
290:
286:
282:
278:
274:
270:
266:
262:
258:
254:
250:
246:
242:
240:
236:
232:
228:
226:
222:
218:
214:
211:If there are
205:
198:
196:
192:
184:
179:
175:
170:
153:
150:
139:
136:
125:
114:
111:
95:
87:
85:
80:
76:
72:
68:
64:
62:
56:
46:
35:
28:
21:
2694:
2679:
2652:
2637:
2622:
2612:
2608:
2604:
2598:
2453:
2332:
2238:
2233:
2229:
2227:
2101:
2097:
2093:
2091:
1908:
1904:
1902:
1899:
1834:
1831:
1783:
1771:
1759:
1757:
1671:
1610:
1605:
1592:
1590:
1584:
1582:
1577:
1573:
1569:
1567:
1520:
1516:
1512:
1510:
1502:planar graph
1487:
1217:
1212:
1101:
930:
752:
626:
481:
476:
472:
468:
464:
462:
459:
453:
449:
445:
439:
433:
428:
422:
418:
414:, there are
411:
408:
403:
399:
395:
391:
387:
383:
379:
377:
364:
360:
356:
352:
346:
331:
327:
323:
319:
315:
313:
308:
304:
300:
296:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
252:
248:
244:
243:
238:
234:
230:
229:
224:
220:
216:
212:
210:
190:
171:, 256, ... (
90:
82:
74:
70:
58:
52:
41:
33:
26:
2615:≥ 4.
1774:such edges.
283:, the line
267:old points
59:dividing a
2724:Categories
2676:Guy, R. K.
2645:References
394:(5) using
63:into areas
2696:MathWorld
2529:−
2504:−
2415:−
2374:−
2122:−
1989:−
1877:−
1538:−
1499:connected
1176:−
1151:−
1032:−
1003:−
958:∑
931:Finally,
913:−
884:−
852:−
799:−
767:−
722:−
707:−
693:−
675:∑
662:−
596:−
581:−
575:−
548:−
530:∑
517:−
475:= 2 and
79:diagonals
40:regions (
2708:Archived
2628:See also
2241:− 1, or
1707:interior
1694:exterior
1627:interior
467:= 1 and
338:Solution
75:maximise
55:geometry
32:chords (
25:points (
2735:Circles
2662:A000127
2660::
1672:and so
386:= 4 to
307:, case
178:A000127
176::
67:polygon
1506:sphere
452:− 1) (
367:− 1).
61:circle
1497:of a
1061:with
374:Proof
245:Proof
231:Lemma
207:Lemma
199:Lemma
84:Moser
69:with
2740:Area
2674:and
2658:OEIS
2588:OEIS
2096:and
1572:and
1508:S).
1470:256
1447:163
1240:Sum
456:− 1)
279:and
174:OEIS
38:and
1606:all
1467:126
1452:10
1424:99
1401:57
1378:31
1355:16
436:− 1
425:− 1
193:= 6
169:163
53:In
2726::
2693:.
2532:18
2523:23
2483:24
2213:2.
2104:,
1464:84
1461:36
1444:70
1441:56
1438:28
1429:9
1421:35
1418:35
1415:21
1406:8
1398:15
1395:20
1392:15
1383:7
1372:10
1369:10
1360:6
1337:5
1332:8
1314:4
1309:4
1291:3
1286:2
1268:2
1263:1
1245:1
1235:4
1188:1.
1179:18
1170:23
1133:24
1021:17
916:2.
902:17
448:−
421:−
334:.
332:AO
324:AO
320:AO
297:OI
285:OI
96:=
29:),
2699:.
2638:n
2613:n
2609:n
2605:k
2603:(
2572:G
2568:r
2541:1
2538:+
2535:)
2526:n
2520:+
2515:2
2511:n
2507:6
2499:3
2495:n
2491:(
2488:n
2480:1
2475:=
2470:G
2466:r
2439:1
2436:+
2430:!
2427:2
2424:!
2421:)
2418:2
2412:n
2409:(
2404:!
2401:n
2395:+
2389:!
2386:4
2383:!
2380:)
2377:4
2371:n
2368:(
2363:!
2360:n
2354:=
2349:G
2345:r
2318:,
2315:1
2312:+
2306:)
2301:2
2298:n
2293:(
2287:+
2281:)
2276:4
2273:n
2268:(
2262:=
2257:G
2253:r
2239:F
2234:G
2230:r
2210:+
2204:)
2199:2
2196:n
2191:(
2185:+
2179:)
2174:4
2171:n
2166:(
2160:=
2157:F
2134:,
2131:2
2128:+
2125:V
2119:E
2116:=
2113:F
2102:F
2098:E
2094:V
2077:.
2074:n
2071:+
2065:)
2060:2
2057:n
2052:(
2046:+
2040:)
2035:4
2032:n
2027:(
2021:2
2018:=
2015:n
2012:+
2009:n
2006:+
2001:2
1996:)
1992:n
1983:)
1978:2
1975:n
1970:(
1963:(
1959:2
1956:+
1950:)
1945:4
1942:n
1937:(
1931:4
1925:=
1922:E
1909:n
1905:n
1884:)
1880:n
1871:)
1866:2
1863:n
1858:(
1851:(
1847:2
1814:)
1809:4
1806:n
1801:(
1795:4
1772:n
1760:n
1743:.
1737:)
1732:4
1729:n
1724:(
1718:+
1715:n
1712:=
1703:V
1699:+
1690:V
1686:=
1683:V
1657:,
1651:)
1646:4
1643:n
1638:(
1632:=
1623:V
1593:V
1585:n
1578:F
1574:E
1570:V
1553:2
1550:=
1547:F
1544:+
1541:E
1535:V
1521:V
1517:E
1513:F
1489:"
1458:9
1455:1
1435:8
1432:1
1412:7
1409:1
1389:6
1386:1
1375:5
1366:5
1363:1
1352:1
1349:4
1346:6
1343:4
1340:1
1329:—
1326:1
1323:3
1320:3
1317:1
1306:—
1303:—
1300:1
1297:2
1294:1
1283:—
1280:—
1277:—
1274:1
1271:1
1260:—
1257:—
1254:—
1251:—
1248:1
1232:3
1229:2
1226:1
1223:0
1218:n
1213:k
1185:+
1182:)
1173:n
1167:+
1162:2
1158:n
1154:6
1146:3
1142:n
1138:(
1130:n
1125:=
1122:)
1119:n
1116:(
1113:f
1087:,
1084:1
1081:=
1078:)
1075:0
1072:(
1069:f
1049:,
1046:1
1043:+
1039:)
1035:2
1029:k
1024:6
1016:+
1011:2
1007:k
998:3
994:k
988:6
985:1
979:(
973:n
968:1
965:=
962:k
954:=
951:)
948:n
945:(
942:f
910:n
905:6
897:+
892:2
888:n
879:3
875:n
869:6
866:1
861:+
858:)
855:1
849:n
846:(
843:f
840:=
837:)
834:n
831:(
828:f
805:)
802:1
796:n
793:(
773:)
770:1
764:n
761:(
738:.
735:)
730:2
726:i
719:i
716:n
713:+
710:n
704:2
701:(
696:1
690:n
685:1
682:=
679:i
671:+
668:)
665:1
659:n
656:(
653:f
650:=
647:)
644:n
641:(
638:f
612:,
607:)
602:)
599:1
593:i
590:(
587:)
584:1
578:i
572:n
569:(
566:+
563:1
558:(
551:1
545:n
540:1
537:=
534:i
526:+
523:)
520:1
514:n
511:(
508:f
505:=
502:)
499:n
496:(
493:f
477:i
473:i
469:i
465:i
454:i
450:i
446:n
444:(
434:i
423:i
419:n
412:i
404:i
400:i
396:f
392:f
388:n
384:n
380:f
365:n
363:(
361:f
357:n
355:(
353:f
328:k
316:k
309:b
305:O
301:A
293:A
289:O
281:I
277:O
273:I
269:O
265:n
261:I
257:O
253:A
249:a
239:b
235:A
225:b
221:a
217:n
213:n
191:n
186:2
154:1
151:+
145:)
140:2
137:n
132:(
126:+
120:)
115:4
112:n
107:(
93:G
91:r
71:n
47:)
44:G
42:r
36:)
34:c
27:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
