416:
787:
258:
to obtain some particular solutions of a certain algebraic problem. The problem and
Udayadivākara's solution to the problem are presented below only to illustrate the techniques used by Indian astronomers for solving algebraic equations.
313:
644:
318:
921:
1241:
639:
973:
1197:
851:
426:
To solve the problem, Udayadivākara makes a series of apparently arbitrary assumptions all aimed at reducing the problem to one of solving an indeterminate equation of the form
42:
has not yet been published and is available only in manuscript form. Some of these manuscripts are preserved in the manuscript depositories in
Thiruvananthapuram. According to
1153:
1060:
631:
533:
1978:
1943:
1109:
473:
256:
179:
126:
2193:
1292:
574:
2253:
1025:
999:
1281:
1261:
305:
285:
199:
2278:
2130:
411:{\displaystyle {\begin{aligned}x+y&={\text{a prefect square}},\\x-y&={\text{a prefect square}},\\xy+1&={\text{a prefect square}}.\end{aligned}}}
2556:
1680:
1955:
69:
is an elaborate commentary, it has some historical significance. It has quoted extensively from a now lost work by a little known mathematician
132:
contrary to generally held beliefs. Another important reference to
Jayadeva’s work is the solution of the indeterminate equation of the form
1376:
2110:
2002:
2177:
1897:
2591:
2072:
2596:
2125:
2168:
2135:
2077:
1650:
22:(c. 1073 CE) was an Indian astronomer and mathematician who has authored an influential and elaborate commentary, called
1348:(Chapter titled "Ācārya Jayadeva, the mathematician". Originally published in Ganita, Vol. 5, No. 1 (1954), pp. 1–20.)
2120:
1630:
1369:
2576:
1870:
2273:
2105:
1720:
1474:
862:
782:{\displaystyle {\begin{aligned}x&=12z^{2}+16z+4,\\y&=3z^{2}+4z,\\x+y&=15z^{2}+20z+4.\end{aligned}}}
2453:
1514:
1393:
1202:
2095:
1318:. Vishveshvaranand Institute of Sanskrit and Indological Studies, Panjab University, Hoshiarpur. p. 45
929:
2566:
2561:
2208:
2161:
2115:
2100:
1362:
1158:
798:
2581:
1610:
1114:
46:, historian of the astronomy and mathematics of the Kerala school, Udayadivākara probably hailed from
1620:
1030:
579:
481:
1065:
429:
212:
135:
82:
2586:
2571:
1910:
1776:
1625:
70:
2463:
2248:
2203:
2036:
1918:
1509:
2268:
2154:
2057:
1984:
1797:
538:
2492:
2146:
1914:
1906:
1408:
38:
takes the year 1073 CE as its epoch, probably the commentary was completed about that year.
1996:
1707:
1675:
1004:
978:
8:
2422:
2325:
2258:
2238:
2223:
2062:
2031:
1839:
1825:
1762:
1655:
1600:
1595:
1549:
1499:
2283:
2243:
2198:
1564:
1385:
1266:
1246:
290:
270:
184:
1313:
2448:
2067:
1949:
1769:
1544:
1539:
1413:
1344:
Studies in Indian
Mathematics and Astronomy Selected Articles of Kripa Shankar Shukla
75:
1783:
1665:
1635:
1605:
1748:
1519:
1341:
1990:
1937:
1865:
1811:
1660:
1529:
1569:
1489:
2458:
2412:
2384:
2378:
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2218:
1961:
1881:
1818:
1741:
1727:
1645:
1640:
1574:
1418:
1469:
2550:
2522:
2438:
2360:
2228:
2041:
1973:
1902:
1804:
2396:
1464:
1428:
31:
2401:
2368:
2308:
2263:
2008:
1846:
1790:
1734:
1554:
1534:
1459:
34:. No personal details about Udayadivākara are known. Since the commentary
2504:
2498:
2014:
1967:
1755:
1701:
1484:
1438:
129:
43:
2534:
2443:
2337:
1713:
1670:
1524:
1504:
1479:
2516:
2486:
2417:
2233:
1832:
1494:
1454:
1354:
2528:
2510:
2468:
2303:
2213:
1615:
1579:
1559:
1433:
1681:
Shanti Swarup
Bhatnagar Prize recipients in Mathematical Science
79:
method for solving indeterminate integral equations of the form
1423:
47:
2176:
51:
1342:
Aditya
Kolachana, K. Mahesh and K. Ramasubramanian (2019).
1293:
List of astronomers and mathematicians of the Kerala school
1882:
Infinite series expansions for the trigonometric functions
209:
Udayadivākara used his method for solving the equation
2131:
Ramanujan
Institute for Advanced Study in Mathematics
1876:
1269:
1249:
1205:
1161:
1117:
1068:
1033:
1007:
981:
932:
865:
801:
642:
582:
541:
484:
432:
316:
293:
273:
215:
187:
138:
85:
57:
1346:. Hindustan Book Agency/Springer. pp. 133–152.
633:which, together with the earlier equation, yields
1275:
1255:
1235:
1191:
1147:
1103:
1054:
1019:
993:
967:
915:
845:
781:
625:
568:
527:
467:
410:
299:
279:
250:
193:
173:
120:
1315:A History of the Kerala School of Hindu Astronomy
2548:
1111:, Udayadivākara finds the following solutions
2162:
1370:
204:
1062:. Using the method for solving the equation
1311:
2557:Kerala school of astronomy and mathematics
2178:Kerala school of astronomy and mathematics
2169:
2155:
1898:Kerala school of astronomy and mathematics
1377:
1363:
856:which is then transformed to the equation
916:{\displaystyle (30z+20)^{2}=60u^{2}+160.}
2126:Homi Bhabha Centre for Science Education
1337:
1335:
1333:
1236:{\displaystyle (u=8802,\lambda =68180)}
2549:
1384:
478:Udayadivākara begins by assuming that
128:. This shows that the method predates
2519:(Mahishamangalam Sankaran Namputhiri)
2150:
1358:
968:{\displaystyle Nx^{2}+C=\lambda ^{2}}
307:satisfying the following conditions:
1330:
1305:
1651:Subbayya Sivasankaranarayana Pillai
1283:are obtained by back substitution.
1192:{\displaystyle (u=18,\lambda =140)}
846:{\displaystyle 15z^{2}+20z+4=u^{2}}
16:Indian astronomer and mathematician
13:
2111:Institute of Mathematical Sciences
14:
2608:
2254:Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri
2121:Harish-Chandra Research Institute
1631:K. R. Parthasarathy (probabilist)
1148:{\displaystyle (u=2,\lambda =20)}
535:which can be written in the form
2592:Science and technology in Kerala
2597:11th-century Indian astronomers
1055:{\displaystyle \lambda =30z+20}
626:{\displaystyle 3y+4=(3z+2)^{2}}
528:{\displaystyle xy+1=(2y+1)^{2}}
73:. The quotations relate to the
2106:Chennai Mathematical Institute
1475:Melpathur Narayana Bhattathiri
1230:
1206:
1186:
1162:
1142:
1118:
1104:{\displaystyle Nx^{2}+C=y^{2}}
882:
866:
614:
598:
516:
500:
468:{\displaystyle Nx^{2}+C=y^{2}}
251:{\displaystyle Nx^{2}+C=y^{2}}
174:{\displaystyle Nx^{2}+C=y^{2}}
121:{\displaystyle Nx^{2}+1=y^{2}}
1:
1298:
926:This equation is of the form
201:being positive or negative.
2383:Laghuvivṛti (commentary on
2096:Indian Statistical Institute
7:
2505:Mahodayapuram (Kodungallur)
2294:Śaṅkara Varma of Kaṭattanāṭ
2116:Indian Institute of Science
2101:Bhaskaracharya Pratishthana
1979:A. A. Krishnaswamy Ayyangar
1944:Shankar Balakrishna Dikshit
1871:Hindu–Arabic numeral system
1286:
421:
58:Historical significance of
10:
2613:
2525:(Nilakantha Somayaji, etc)
2495:(Govinda Bhattathiri, etc)
2454:Madhava's correction terms
1611:Subrahmanyan Chandrasekhar
262:
205:A problem and its solution
2513:(Madhava of Sangamagrama)
2489:(Azhvanchery Thamprakkal)
2477:
2431:
2317:
2279:Śaṅkara of Mahiṣamaṅgalam
2184:
2086:
2050:
2024:
1928:
1890:
1856:
1693:
1621:Veeravalli S. Varadarajan
1588:
1447:
1401:
1392:
1243:from which the values of
792:Now, Udayadivākara puts
65:Apart from the fact that
2531:(Putumana Somayaji, etc)
2269:Parameśvara of Vațaśśeri
2037:Henry Thomas Colebrooke
1626:S. R. Srinivasa Varadhan
569:{\displaystyle x-y=3y+4}
2249:Madhava of Sangamagrama
1510:Mādhava of Saṅgamagrāma
576:. He next assumes that
267:Find positive integers
2577:History of mathematics
2478:Places associated with
2393:Mahābhāskarīya Bhāshya
1277:
1257:
1237:
1193:
1149:
1105:
1056:
1021:
995:
969:
917:
847:
783:
627:
570:
529:
469:
412:
301:
281:
252:
195:
175:
122:
2480:members of the school
2390:Madhyamanayanaprakara
2365:Jyotiśśāstra-saṃgraha
2357:Jātaka-sāara-saṃgraha
2354:Grahacāraṇinibandhana
2324:Aryabhatiya-bhashya (
2214:Dāmodara of Vațaśreņi
2204:Azhvāñceri Taṃprākkaḷ
1721:Brāhmasphuṭasiddhānta
1278:
1258:
1238:
1194:
1150:
1106:
1057:
1022:
1020:{\displaystyle C=160}
996:
970:
918:
848:
784:
628:
571:
530:
470:
413:
302:
282:
253:
196:
176:
123:
2529:Sivapuram (Thrissur)
2464:Madhava's sine table
1997:T. A. Saraswati Amma
1708:Bakhshali manuscript
1676:Kannan Soundararajan
1312:K. V. Sarma (1972).
1267:
1247:
1203:
1159:
1115:
1066:
1031:
1005:
994:{\displaystyle N=60}
979:
930:
863:
799:
640:
580:
539:
482:
430:
314:
291:
271:
213:
185:
136:
83:
2326:Nīlakaņțha Sōmayāji
2259:Nīlakaņțha Sōmayāji
2224:Govinda Bhattathiri
2073:Islamic mathematics
2032:Walter Eugene Clark
1826:Vasishtha Siddhanta
1798:Siddhānta Shiromani
1777:Paitamaha Siddhanta
1656:Tilak Raj Prabhakar
1601:Satyendra Nath Bose
1596:Srinivasa Ramanujan
1550:Nilakantha Somayaji
2567:Astronomy in India
2562:Indian mathematics
2244:Kochukrishnan Asan
2187:and mathematicians
1565:Gangesha Upadhyaya
1386:Indian mathematics
1273:
1253:
1233:
1189:
1145:
1101:
1052:
1017:
991:
965:
913:
843:
779:
777:
623:
566:
525:
465:
408:
406:
297:
277:
248:
191:
171:
118:
2582:History of Kerala
2544:
2543:
2507:(Sankaranarayana)
2449:Kaṭapayādi system
2444:Drigganita system
2331:Candrachayaganita
2284:Putumana Somayāji
2144:
2143:
1950:Sudhakara Dvivedi
1770:Paulisa Siddhanta
1689:
1688:
1545:Jagannatha Samrat
1540:Achyuta Pisharati
1276:{\displaystyle y}
1256:{\displaystyle x}
399:
368:
340:
300:{\displaystyle y}
280:{\displaystyle x}
194:{\displaystyle C}
2604:
2501:(Sankara Varman)
2406:Sidhhantadarpana
2171:
2164:
2157:
2148:
2147:
1784:Romaka Siddhanta
1666:Akshay Venkatesh
1636:M. S. Narasimhan
1606:P.C. Mahalanobis
1399:
1398:
1379:
1372:
1365:
1356:
1355:
1349:
1347:
1339:
1328:
1327:
1325:
1323:
1309:
1282:
1280:
1279:
1274:
1262:
1260:
1259:
1254:
1242:
1240:
1239:
1234:
1198:
1196:
1195:
1190:
1154:
1152:
1151:
1146:
1110:
1108:
1107:
1102:
1100:
1099:
1081:
1080:
1061:
1059:
1058:
1053:
1026:
1024:
1023:
1018:
1000:
998:
997:
992:
974:
972:
971:
966:
964:
963:
945:
944:
922:
920:
919:
914:
906:
905:
890:
889:
852:
850:
849:
844:
842:
841:
814:
813:
788:
786:
785:
780:
778:
759:
758:
714:
713:
669:
668:
632:
630:
629:
624:
622:
621:
575:
573:
572:
567:
534:
532:
531:
526:
524:
523:
474:
472:
471:
466:
464:
463:
445:
444:
417:
415:
414:
409:
407:
400:
398:a prefect square
397:
369:
367:a prefect square
366:
341:
339:a prefect square
338:
306:
304:
303:
298:
286:
284:
283:
278:
257:
255:
254:
249:
247:
246:
228:
227:
200:
198:
197:
192:
180:
178:
177:
172:
170:
169:
151:
150:
127:
125:
124:
119:
117:
116:
98:
97:
28:Laghu-bhāskarīya
2612:
2611:
2607:
2606:
2605:
2603:
2602:
2601:
2587:Medieval Kerala
2572:Hindu astronomy
2547:
2546:
2545:
2540:
2479:
2473:
2469:Parahita system
2432:Concepts/Topics
2427:
2313:
2289:Śankara Vāriyar
2274:Śaṅkaranārāyaṇa
2199:Acyuta Piṣāraṭi
2194:(Complete list)
2186:
2180:
2175:
2145:
2140:
2088:
2082:
2046:
2020:
1991:C. T. Rajagopal
1938:Bapudeva Sastri
1930:
1924:
1886:
1866:Brahmi numerals
1858:
1852:
1812:Surya Siddhanta
1685:
1661:Manjul Bhargava
1584:
1443:
1388:
1383:
1353:
1352:
1340:
1331:
1321:
1319:
1310:
1306:
1301:
1289:
1268:
1265:
1264:
1248:
1245:
1244:
1204:
1201:
1200:
1160:
1157:
1156:
1116:
1113:
1112:
1095:
1091:
1076:
1072:
1067:
1064:
1063:
1032:
1029:
1028:
1006:
1003:
1002:
980:
977:
976:
959:
955:
940:
936:
931:
928:
927:
901:
897:
885:
881:
864:
861:
860:
837:
833:
809:
805:
800:
797:
796:
776:
775:
754:
750:
740:
728:
727:
709:
705:
695:
689:
688:
664:
660:
650:
643:
641:
638:
637:
617:
613:
581:
578:
577:
540:
537:
536:
519:
515:
483:
480:
479:
459:
455:
440:
436:
431:
428:
427:
424:
405:
404:
396:
389:
374:
373:
365:
358:
346:
345:
337:
330:
317:
315:
312:
311:
292:
289:
288:
272:
269:
268:
265:
242:
238:
223:
219:
214:
211:
210:
207:
186:
183:
182:
165:
161:
146:
142:
137:
134:
133:
112:
108:
93:
89:
84:
81:
80:
63:
17:
12:
11:
5:
2610:
2600:
2599:
2594:
2589:
2584:
2579:
2574:
2569:
2564:
2559:
2542:
2541:
2539:
2538:
2532:
2526:
2520:
2514:
2508:
2502:
2496:
2490:
2483:
2481:
2475:
2474:
2472:
2471:
2466:
2461:
2459:Madhava series
2456:
2451:
2446:
2441:
2435:
2433:
2429:
2428:
2426:
2425:
2420:
2415:
2413:Tantrasamgraha
2410:
2407:
2404:
2399:
2394:
2391:
2388:
2385:Tantrasamgraha
2381:
2379:Kriyakramakari
2376:
2374:Karaṇapaddhati
2371:
2366:
2363:
2358:
2355:
2352:
2349:
2346:
2345:Gaṇitayuktayah
2343:
2340:
2335:
2332:
2329:
2321:
2319:
2315:
2314:
2312:
2311:
2306:
2301:
2296:
2291:
2286:
2281:
2276:
2271:
2266:
2261:
2256:
2251:
2246:
2241:
2236:
2231:
2226:
2221:
2216:
2211:
2206:
2201:
2196:
2190:
2188:
2182:
2181:
2174:
2173:
2166:
2159:
2151:
2142:
2141:
2139:
2138:
2133:
2128:
2123:
2118:
2113:
2108:
2103:
2098:
2092:
2090:
2084:
2083:
2081:
2080:
2075:
2070:
2065:
2060:
2054:
2052:
2048:
2047:
2045:
2044:
2039:
2034:
2028:
2026:
2022:
2021:
2019:
2018:
2012:
2006:
2000:
1994:
1988:
1982:
1976:
1971:
1965:
1962:P. C. Sengupta
1959:
1953:
1947:
1941:
1934:
1932:
1926:
1925:
1923:
1922:
1900:
1894:
1892:
1888:
1887:
1885:
1884:
1879:
1873:
1868:
1862:
1860:
1854:
1853:
1851:
1850:
1843:
1836:
1829:
1822:
1819:Tantrasamgraha
1815:
1808:
1801:
1794:
1787:
1780:
1773:
1766:
1759:
1752:
1745:
1742:Karanapaddhati
1738:
1731:
1728:Ganita Kaumudi
1724:
1717:
1710:
1705:
1697:
1695:
1691:
1690:
1687:
1686:
1684:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1648:
1646:Harish-Chandra
1643:
1641:C. S. Seshadri
1638:
1633:
1628:
1623:
1618:
1613:
1608:
1603:
1598:
1592:
1590:
1586:
1585:
1583:
1582:
1577:
1575:Sankara Variar
1572:
1567:
1562:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1477:
1472:
1467:
1462:
1457:
1451:
1449:
1445:
1444:
1442:
1441:
1436:
1431:
1426:
1421:
1416:
1411:
1405:
1403:
1396:
1394:Mathematicians
1390:
1389:
1382:
1381:
1374:
1367:
1359:
1351:
1350:
1329:
1303:
1302:
1300:
1297:
1296:
1295:
1288:
1285:
1272:
1252:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1098:
1094:
1090:
1087:
1084:
1079:
1075:
1071:
1051:
1048:
1045:
1042:
1039:
1036:
1016:
1013:
1010:
990:
987:
984:
962:
958:
954:
951:
948:
943:
939:
935:
924:
923:
912:
909:
904:
900:
896:
893:
888:
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2523:Thrikkandiyur
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2500:
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2264:Nīlakaṇṭha II
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2229:Govindasvāmin
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2051:Other regions
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2042:David Pingree
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2016:
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1956:M. Rangacarya
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1929:Historians of
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334:
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274:
260:
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78:
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72:
68:
61:
55:
53:
49:
45:
41:
37:
33:
29:
25:
21:
20:Udayadivākara
2409:Spuṭanirṇaya
2402:Sadratnamālā
2369:Kanakkusāraṃ
2309:Vidyamadhava
2298:
2089:institutions
2009:K. S. Shukla
1981:(1892– 1953)
1847:Yavanajataka
1845:
1838:
1831:
1824:
1817:
1810:
1805:Śulba Sūtras
1803:
1796:
1791:Sadratnamala
1789:
1782:
1775:
1768:
1761:
1754:
1747:
1740:
1735:Kanakkusaram
1733:
1726:
1719:
1712:
1700:
1570:Varāhamihira
1535:Parameshvara
1490:Govindasvāmi
1460:Āryabhaṭa II
1343:
1320:. Retrieved
1314:
1307:
925:
855:
791:
477:
425:
266:
208:
74:
66:
64:
59:
39:
35:
27:
23:
19:
18:
2537:(Haridatta)
2499:Kadathanadu
2239:Jyeṣṭhadeva
2185:Astronomers
2025:Translators
2017:(1919–2005)
2015:K. V. Sarma
2011:(1918–2007)
2005:(1918–1992)
1999:(1918–2000)
1993:(1903–1978)
1987:(1901–1954)
1985:A. N. Singh
1970:(1888–1958)
1968:B. B. Datta
1964:(1876–1962)
1958:(1861–1916)
1952:(1855–1910)
1946:(1853–1898)
1940:(1821–1900)
1931:mathematics
1875:Symbol for
1859:innovations
1756:Lokavibhaga
1702:Aryabhatiya
1500:Jyeṣṭhadeva
1485:Brahmagupta
1470:Bhāskara II
1455:Āryabhaṭa I
1439:Yajnavalkya
130:Bhaskara II
44:K. V. Sarma
2551:Categories
2535:Tirunavaya
2423:Yuktibhāṣā
2397:Pañcabodha
2338:Drigganita
2334:Dasādhyāyi
2304:Vararuci I
2209:Citrabhānu
1974:T. Hayashi
1857:Pioneering
1840:Yuktibhāṣā
1763:Pātīgaṇita
1714:Bijaganita
1671:Ravi Vakil
1525:Munishvara
1505:Kamalakara
1480:Brahmadeva
1465:Bhāskara I
1414:Baudhayana
1322:28 January
1299:References
32:Bhāskara I
2517:Peruvanam
2493:Alathiyur
2487:Athavanad
2418:Venvaroha
2342:Dṛkkaraṇa
2318:Treatises
2234:Haridatta
2219:Devācārya
2003:S. N. Sen
1911:New Delhi
1833:Veṇvāroha
1694:Treatises
1495:Halayudha
1448:Classical
1419:Katyayana
1409:Apastamba
1222:λ
1178:λ
1134:λ
1035:λ
957:λ
546:−
353:−
76:cakravala
2511:Kudallur
2351:Golasara
2348:Golavada
1919:Varanasi
1877:zero (0)
1749:Līlāvatī
1616:C.R. Rao
1580:Virasena
1560:Sridhara
1530:Narayana
1515:Mahāvīra
1287:See also
422:Solution
71:Jayadeva
2058:Babylon
1891:Centres
1555:Śrīpati
1434:Pingala
1402:Ancient
263:Problem
67:Sundari
60:Sundari
40:Sundari
36:Sundari
24:Sundari
2087:Modern
2078:Europe
2068:Greece
1915:Ujjain
1907:Jaipur
1589:Modern
1429:Pāṇini
1424:Manava
48:Kerala
2063:China
1228:68180
975:with
52:India
26:, on
2136:TIFR
1324:2023
1263:and
1216:8802
1199:and
1027:and
911:160.
287:and
1184:140
1015:160
1001:,
30:of
2553::
1917:,
1913:,
1909:,
1332:^
1172:18
1155:,
1140:20
1050:20
1041:30
989:60
895:60
879:20
870:30
819:20
803:15
773:4.
764:20
748:15
674:16
658:12
475:.
181:,
54:.
50:,
2387:)
2328:)
2170:e
2163:t
2156:v
1921:)
1905:(
1378:e
1371:t
1364:v
1326:.
1271:y
1251:x
1231:)
1225:=
1219:,
1213:=
1210:u
1207:(
1187:)
1181:=
1175:,
1169:=
1166:u
1163:(
1143:)
1137:=
1131:,
1128:2
1125:=
1122:u
1119:(
1097:2
1093:y
1089:=
1086:C
1083:+
1078:2
1074:x
1070:N
1047:+
1044:z
1038:=
1012:=
1009:C
986:=
983:N
961:2
953:=
950:C
947:+
942:2
938:x
934:N
908:+
903:2
899:u
892:=
887:2
883:)
876:+
873:z
867:(
839:2
835:u
831:=
828:4
825:+
822:z
816:+
811:2
807:z
770:+
767:z
761:+
756:2
752:z
745:=
738:y
735:+
732:x
725:,
722:z
719:4
716:+
711:2
707:z
703:3
700:=
693:y
686:,
683:4
680:+
677:z
671:+
666:2
662:z
655:=
648:x
619:2
615:)
611:2
608:+
605:z
602:3
599:(
596:=
593:4
590:+
587:y
584:3
564:4
561:+
558:y
555:3
552:=
549:y
543:x
521:2
517:)
513:1
510:+
507:y
504:2
501:(
498:=
495:1
492:+
489:y
486:x
461:2
457:y
453:=
450:C
447:+
442:2
438:x
434:N
402:.
394:=
387:1
384:+
381:y
378:x
371:,
363:=
356:y
350:x
343:,
335:=
328:y
325:+
322:x
295:y
275:x
244:2
240:y
236:=
233:C
230:+
225:2
221:x
217:N
189:C
167:2
163:y
159:=
156:C
153:+
148:2
144:x
140:N
114:2
110:y
106:=
103:1
100:+
95:2
91:x
87:N
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