20:
50:
checkerboard. The board is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". As in peg solitaire, a move consists of one soldier jumping over an adjacent soldier into an empty cell,
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1033:
1116:
Consider the full starting configuration, where soldiers fill the whole half-plane below the red line. This configuration's score is the sum of the scores of the individual lines. Therefore, if the target square is immediately above the red line, the score is
364:
to the target square. Then we can compute the "score" of a configuration of soldiers by summing the values of the soldiers' squares. For example, a configuration of only two soldiers placed so as to reach the target square on the next jump would have score
838:
612:
1451:
58:
that, regardless of the strategy used, there is no finite sequence of moves that will allow a soldier to advance more than four rows above the horizontal line. His argument uses a carefully chosen weighting of cells (involving the
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vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible.
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Thus, we have shown that when the target square is in the fifth row above the infinite half-plane of soldiers, the starting configuration's score is exactly
155:
2048:
E. Berlekamp, J. Conway and R. Guy, Winning Ways for Your
Mathematical Plays, 2nd ed., Vol. 4, Chap. 23: 803—841, A K Peters, Wellesley, MA, 2004.
63:), and he proved that the total weight can only decrease or remain constant. This argument has been reproduced in a number of popular math books.
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If the target square is in the second row above the red line, every soldier is one space further from the target square, and so the score is
23:
Arrangements of Conway's soldiers to reach rows 1, 2, 3 and 4. The soldiers marked "B" represent an alternative to those marked "A".
858:
1250:{\displaystyle S_{1}=(\varphi +2(\varphi ^{2}+\varphi ^{3}+\varphi ^{4}\ldots ))(1+\varphi +\varphi ^{2}+\varphi ^{3}+\ldots )}
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represents the (small, but positive) contributions of the infinite number of soldiers that remain elsewhere on the plane.
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1028:{\displaystyle \varphi ^{2}+2\varphi ^{3}+2\varphi ^{4}+\ldots =\varphi (\varphi +2\varphi ^{2}+2\varphi ^{3}+\ldots )}
2125:
2115:
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1528:
855:
If this horizontal line of soldiers is immediately below the target square, then the score of the configuration is
852:
Consider now a starting configuration where only one infinite horizontal line is completely filled with soldiers.
833:{\displaystyle \varphi ^{n+2}-\varphi ^{n+1}-\varphi ^{n}=\varphi ^{n}(\varphi ^{2}-\varphi -1)=-2\varphi ^{n+1}}
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When a soldier reaches the target square after some finite number of moves, the ending configuration has score
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73:
series of moves. If diagonal jumps are allowed, the 8th row can be reached, but not the 9th row. In the
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1893:, it is impossible for any soldier to reach a square in the fifth row after a finite number of jumps.
607:{\displaystyle \varphi ^{n-2}-\varphi ^{n-1}-\varphi ^{n}=\varphi ^{n-2}(1-\varphi -\varphi ^{2})=0}
466:
283:
1821:
1446:{\displaystyle S_{1}=(\varphi +2)(1+\varphi +1)=(\varphi +2)^{2}=5+3\varphi \approx 6.85410\ldots }
1738:
419:
316:
1928:
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55:
39:
2051:
R. Honsberger, A problem in checker jumping, in
Mathematical Gems II, Chap. 3: 23—28, MAA, 1976.
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distance from the target square after his jump: In this case the change in score is
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203:{\displaystyle \varphi ={\frac {{\sqrt {5}}-1}{2}}\approx 0.61803\,39887\ldots }
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When a soldier jumps over another soldier, there are three cases to consider:
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78:
1106:{\displaystyle \varphi ^{2}(\varphi +2\varphi ^{2}+2\varphi ^{3}+\ldots )}
47:
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A page describing several variations of the game, with recent references
695:{\displaystyle \varphi ^{n}-\varphi ^{n-1}-\varphi ^{n}=-\varphi ^{n-1}}
2100:
1912:
Bell, George I.; Hirschberg, Daniel S.; Guerrero-Garcia, Pablo (2007).
69:
and Gareth Taylor have shown that the fifth row can be reached via an
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1762:
represents the contribution of the soldier on the target square and
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18:
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So, no jump will ever increase the configuration's total score.
906:{\displaystyle \varphi +2\varphi ^{2}+2\varphi ^{3}+\ldots }
416:
the target square: Let the value of the soldier's square be
81:
version of the game, the highest row that can be reached is
1921:
INTEGERS: Electronic
Journal of Combinatorial Number Theory
1907:
1905:
142:
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At this point, observe another interesting property of
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from the target square: Here the change in score is
110:; Conway's weighting argument demonstrates that row
1328:{\displaystyle \sum _{n=2}^{\infty }\varphi ^{n}=1}
496:; then the total change in score after the jump is
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143:Conway's proof that the fifth row is inaccessible
1953:"Reaching row 5 in Solitaire Army (web version)"
848:Computing the score of the initial configuration
280:Let the target square be labeled with the value
38:or puzzle devised and analyzed by mathematician
1914:"The minimum size required of a solitaire army"
1511:{\displaystyle S_{2}=\varphi S_{1}=3+2\varphi }
463:, and the value of the square he jumps over be
1632:{\displaystyle S_{4}=\varphi S_{3}=1+\varphi }
1573:{\displaystyle S_{3}=\varphi S_{2}=2+\varphi }
8:
1987:Eriksson, Henrik; Lindstrom, Bernt (1995).
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398:{\displaystyle \varphi ^{1}+\varphi ^{2}}
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1728:{\displaystyle E=\varphi ^{0}+\epsilon }
2101:Plus online magazine describes the game
1901:
270:{\displaystyle \varphi ^{2}=1-\varphi }
1818:; the ending configuration's score is
1971:"Reaching Row Five in Solitaire Army"
1685:{\displaystyle S_{5}=\varphi S_{4}=1}
7:
2096:Interactive version of the game (2)
2091:Interactive version of the game (1)
2062:cut-the-knot.org explains the game
1335:. Applying this identity produces
1304:
917:spaces below the target square is
16:Mathematical puzzle by John Conway
14:
2024:European Journal of Combinatorics
2013:{\displaystyle {\mathbb {Z}}^{d}}
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489:{\displaystyle \varphi ^{n-1}}
306:{\displaystyle \varphi ^{0}=1}
1:
1843:{\displaystyle E=1+\epsilon }
2037:10.1016/0195-6698(95)90054-3
1755:{\displaystyle \varphi ^{0}}
436:{\displaystyle \varphi ^{n}}
333:{\displaystyle \varphi ^{n}}
1876:{\displaystyle S_{5}\geq E}
617:When a soldier remains the
2142:
1989:"Twin jumping checkers in
1775:{\displaystyle \epsilon }
1274:{\displaystyle \varphi }
223:{\displaystyle \varphi }
148:Notation and definitions
1811:{\displaystyle S_{5}=1}
46:, it takes place on an
32:checker-jumping problem
2014:
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1035:. The score of a line
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913:. The score of a line
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42:in 1961. A variant of
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132:{\displaystyle 3n-1}
114:
103:{\displaystyle 3n-2}
85:
2121:Single-player games
2077:"Conway's Soldiers"
210:. (In other words,
139:cannot be reached.
2126:John Horton Conway
2116:Mathematical games
2074:Weisstein, Eric W.
2010:
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362:Manhattan distance
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40:John Horton Conway
25:
1969:; Gareth Taylor.
456:{\displaystyle n}
353:{\displaystyle n}
230:here denotes the
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36:mathematical game
28:Conway's Soldiers
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238:.) Observe that
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34:is a one-person
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2056:External links
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2030:(2): 153–157.
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1601:
1596:
1592:
1584:
1583:
1567:
1564:
1561:
1558:
1553:
1549:
1545:
1542:
1537:
1533:
1525:
1524:
1523:
1505:
1502:
1499:
1496:
1493:
1488:
1484:
1480:
1477:
1472:
1468:
1460:
1459:
1458:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1414:
1406:
1403:
1400:
1394:
1388:
1385:
1382:
1379:
1376:
1367:
1364:
1361:
1355:
1350:
1346:
1338:
1337:
1336:
1322:
1319:
1314:
1310:
1299:
1296:
1293:
1289:
1268:
1241:
1238:
1233:
1229:
1225:
1220:
1216:
1212:
1209:
1206:
1203:
1191:
1186:
1182:
1178:
1173:
1169:
1165:
1160:
1156:
1149:
1146:
1143:
1137:
1132:
1128:
1120:
1119:
1118:
1114:
1113:, and so on.
1097:
1094:
1089:
1085:
1081:
1078:
1073:
1069:
1065:
1062:
1059:
1051:
1047:
1038:
1019:
1016:
1011:
1007:
1003:
1000:
995:
991:
987:
984:
981:
975:
972:
969:
966:
961:
957:
953:
950:
945:
941:
937:
934:
929:
925:
916:
900:
897:
892:
888:
884:
881:
876:
872:
868:
865:
862:
853:
847:
845:
825:
822:
819:
815:
811:
808:
805:
799:
796:
793:
790:
785:
781:
772:
768:
764:
759:
755:
751:
746:
743:
740:
736:
732:
727:
724:
721:
717:
708:
704:
687:
684:
681:
677:
673:
670:
665:
661:
657:
652:
649:
646:
642:
638:
633:
629:
620:
616:
601:
598:
590:
586:
582:
579:
576:
573:
565:
562:
559:
555:
551:
546:
542:
538:
533:
530:
527:
523:
519:
514:
511:
508:
504:
481:
478:
475:
471:
450:
428:
424:
415:
411:
410:
409:
406:
390:
386:
382:
377:
373:
363:
347:
325:
321:
300:
297:
292:
288:
278:
264:
261:
258:
255:
250:
246:
237:
233:
217:
197:
194:
190:
187:
182:
178:
175:
170:
162:
159:
147:
140:
126:
123:
120:
117:
97:
94:
91:
88:
80:
76:
72:
68:
64:
62:
57:
52:
49:
45:
44:peg solitaire
41:
37:
33:
29:
21:
2080:
2027:
2023:
1982:
1967:Simon Tatham
1961:
1949:Simon Tatham
1943:
1934:math/0612612
1924:
1920:
1888:
1883:. This is a
1784:
1695:
1521:
1456:
1260:
1115:
1036:
914:
854:
851:
843:
706:
618:
413:
407:
279:
236:golden ratio
151:
74:
70:
67:Simon Tatham
65:
61:golden ratio
53:
31:
27:
26:
1522:Similarly:
79:dimensional
2110:Categories
1897:References
232:reciprocal
2082:MathWorld
1868:≥
1838:ϵ
1770:ϵ
1744:φ
1723:ϵ
1711:φ
1664:φ
1627:φ
1605:φ
1568:φ
1546:φ
1506:φ
1481:φ
1441:…
1435:≈
1432:φ
1401:φ
1383:φ
1362:φ
1311:φ
1305:∞
1290:∑
1269:φ
1242:…
1230:φ
1217:φ
1210:φ
1192:…
1183:φ
1170:φ
1157:φ
1144:φ
1098:…
1086:φ
1070:φ
1060:φ
1048:φ
1020:…
1008:φ
992:φ
982:φ
976:φ
970:…
958:φ
942:φ
926:φ
901:…
889:φ
873:φ
863:φ
816:φ
809:−
797:−
794:φ
791:−
782:φ
769:φ
756:φ
752:−
737:φ
733:−
718:φ
685:−
678:φ
674:−
662:φ
658:−
650:−
643:φ
639:−
630:φ
587:φ
583:−
580:φ
577:−
563:−
556:φ
543:φ
539:−
531:−
524:φ
520:−
512:−
505:φ
479:−
472:φ
443:for some
425:φ
387:φ
374:φ
322:φ
289:φ
265:φ
262:−
247:φ
218:φ
198:…
188:≈
176:−
160:φ
124:−
95:−
1735:, where
340:, where
71:infinite
48:infinite
1927:(G07).
1438:6.85410
414:towards
360:is the
234:of the
191:0.61803
152:Define
54:Conway
30:or the
1890:Q.E.D.
56:proved
1974:(PDF)
1929:arXiv
1917:(PDF)
1037:three
195:39887
707:away
619:same
2032:doi
915:two
2112::
2079:.
2028:16
2026:.
2022:.
1951:.
1923:.
1919:.
1904:^
1887:;
405:.
277:.
2085:.
2040:.
2034::
2020:"
2006:d
2000:Z
1976:.
1955:.
1937:.
1931::
1925:7
1871:E
1863:5
1859:S
1835:+
1832:1
1829:=
1826:E
1806:1
1803:=
1798:5
1794:S
1748:0
1720:+
1715:0
1707:=
1704:E
1692:.
1680:1
1677:=
1672:4
1668:S
1661:=
1656:5
1652:S
1639:,
1624:+
1621:1
1618:=
1613:3
1609:S
1602:=
1597:4
1593:S
1580:,
1565:+
1562:2
1559:=
1554:2
1550:S
1543:=
1538:3
1534:S
1518:.
1503:2
1500:+
1497:3
1494:=
1489:1
1485:S
1478:=
1473:2
1469:S
1453:.
1429:3
1426:+
1423:5
1420:=
1415:2
1411:)
1407:2
1404:+
1398:(
1395:=
1392:)
1389:1
1386:+
1380:+
1377:1
1374:(
1371:)
1368:2
1365:+
1359:(
1356:=
1351:1
1347:S
1323:1
1320:=
1315:n
1300:2
1297:=
1294:n
1257:.
1245:)
1239:+
1234:3
1226:+
1221:2
1213:+
1207:+
1204:1
1201:(
1198:)
1195:)
1187:4
1179:+
1174:3
1166:+
1161:2
1153:(
1150:2
1147:+
1141:(
1138:=
1133:1
1129:S
1101:)
1095:+
1090:3
1082:2
1079:+
1074:2
1066:2
1063:+
1057:(
1052:2
1023:)
1017:+
1012:3
1004:2
1001:+
996:2
988:2
985:+
979:(
973:=
967:+
962:4
954:2
951:+
946:3
938:2
935:+
930:2
898:+
893:3
885:2
882:+
877:2
869:2
866:+
840:.
826:1
823:+
820:n
812:2
806:=
803:)
800:1
786:2
778:(
773:n
765:=
760:n
747:1
744:+
741:n
728:2
725:+
722:n
702:.
688:1
682:n
671:=
666:n
653:1
647:n
634:n
614:.
602:0
599:=
596:)
591:2
574:1
571:(
566:2
560:n
552:=
547:n
534:1
528:n
515:2
509:n
482:1
476:n
451:n
429:n
391:2
383:+
378:1
348:n
326:n
301:1
298:=
293:0
259:1
256:=
251:2
183:2
179:1
171:5
163:=
127:1
121:n
118:3
98:2
92:n
89:3
77:-
75:n
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