1018:. Second, after moving, the amazon shoots an arrow from its landing square to another square, using another queenlike move. This arrow may travel in any orthogonal or diagonal direction (even backwards along the same path the amazon just traveled, into or across the starting square if desired). An arrow, like an amazon, cannot cross or enter a square where another arrow has landed or an amazon of either color stands. The square where the arrow lands is marked to show that it can no longer be used. The last player to be able to make a move wins. Draws are impossible.
2761:
eventually each amazon finds itself in a territory blocked off from all other amazons. The amazon can then move about its territory firing arrows until it no longer has any room to move. Since it would be tedious to actually play out all these moves, in practice the game usually ends when all of the amazons are in separate territories. The player with the largest amount of territory will be able to win, as the opponent will have to fill in their own territory more quickly.
1990:
1969:
1941:
2706:
2678:
2671:
2657:
2607:
2600:
2586:
2572:
2565:
2536:
2529:
2522:
2515:
2508:
2494:
2487:
2451:
2444:
2423:
2416:
2380:
2366:
2352:
2345:
2338:
2316:
2309:
2302:
2295:
2288:
2281:
2267:
2231:
2203:
2196:
2189:
2160:
2153:
2146:
2139:
2132:
2125:
2118:
2104:
2097:
2075:
2054:
2040:
2033:
2026:
1962:
1189:
1111:
1090:
116:
95:
2685:
2664:
2593:
2359:
2238:
2210:
2111:
1813:
1600:
1537:
1402:
1366:
1303:
818:
797:
605:
542:
371:
308:
2004:
1997:
1983:
1976:
1955:
1948:
1132:
1125:
1118:
1104:
1097:
1083:
1076:
1069:
137:
130:
123:
109:
102:
88:
81:
74:
2699:
2692:
2650:
2643:
2628:
2621:
2614:
2579:
2550:
2543:
2501:
2472:
2465:
2458:
2437:
2430:
2409:
2394:
2387:
2373:
2331:
2274:
2260:
2253:
2224:
2217:
2182:
2175:
2082:
2068:
2061:
2047:
2019:
1834:
1827:
1820:
1806:
1799:
1792:
1785:
1778:
1771:
1756:
1749:
1742:
1735:
1728:
1721:
1714:
1707:
1700:
1693:
1678:
1671:
1664:
1657:
1650:
1643:
1636:
1629:
1622:
1615:
1593:
1586:
1579:
1572:
1565:
1558:
1551:
1544:
1522:
1515:
1508:
1501:
1494:
1487:
1480:
1473:
1466:
1459:
1444:
1437:
1430:
1423:
1416:
1409:
1395:
1388:
1381:
1359:
1352:
1345:
1338:
1331:
1324:
1317:
1310:
1288:
1281:
1274:
1267:
1260:
1253:
1246:
1239:
1232:
1225:
1210:
1203:
1196:
1182:
1175:
1168:
1161:
1154:
1147:
839:
832:
825:
811:
804:
790:
783:
776:
761:
754:
747:
740:
733:
726:
719:
712:
705:
698:
683:
676:
669:
662:
655:
648:
641:
634:
627:
620:
598:
591:
584:
577:
570:
563:
556:
549:
527:
520:
513:
506:
499:
492:
485:
478:
471:
464:
449:
442:
435:
428:
421:
414:
407:
400:
393:
386:
364:
357:
350:
343:
336:
329:
322:
315:
293:
286:
279:
272:
265:
258:
251:
244:
237:
230:
215:
208:
201:
194:
187:
180:
173:
166:
159:
152:
2764:
Scores are sometimes used for tie-breaking purposes in
Amazons tournaments. When scoring, it is important to note that although the number of moves remaining to a player is usually equal to the number of empty squares in the territories occupied by that player's amazons, it is nonetheless possible to
2760:
The strategy of the game is based on using arrows (as well as one's four amazons) to block the movement of the opponent's amazons and gradually wall off territory, trying to trap the opponents in smaller regions and gain larger areas for oneself. Each move reduces the available playing area, and
2785:
in
December 1992. An approved English translation written by Michael Keller appeared in the magazine World Game Review in January 1994. Other game publications also published the rules, and the game gathered a small but devoted following. The Internet spread the game more widely.
2858:
called GEOGRAPHY-BP3, which is PSPACE-complete, to an
Amazons position. This Amazons position uses only one black queen and one white queen, thus showing that generalized Amazons is PSPACE-complete even if only one queen on each side is
2769:
in which there are fewer moves left than there are empty squares. The simplest such territory is three squares of the same colour, not in a straight line, with the amazon in the middle (for example, a1+b2+c1 with the amazon at b2).
1005:
White moves first, and the players alternate moves thereafter. Each move consists of two parts. First, one moves one of one's own amazons one or more empty squares in a straight line (orthogonally or diagonally), exactly as a
961:. The game is played by moving pieces and blocking the opponents from squares, and the last player able to move is the winner. It is a member of the territorial game family, a distant relative of
3085:
997:), which start on the board in the configuration shown at right. A supply of markers (checkers, poker chips, etc.) is also required.
3095:
2837:
Amazons (that is, determining the winner of a game of
Amazons played on a n x n grid, started from an arbitrary configuration) is
3090:
2815:
Usually, in the endgame, the board is partitioned into separate "royal chambers", with queens inside each chamber. We define
1883:
The diagram shows a possible first move by white: d1-d6/g9, i.e. amazon moved from d1 to d6 and fired arrow to g9.
2824:
2819:
to be endgames where each chamber has at most one queen. Determining who wins in a simple
Amazons endgame is
2794:
977:
954:
2755:
A completed
Amazons game. White has just moved f1-e2/f1. White now has 8 moves left, while Black has 31.
2855:
3062:
3047:
3024:
2991:
1989:
1968:
1940:
2705:
2677:
2670:
2656:
2606:
2599:
2585:
2571:
2564:
2535:
2528:
2521:
2514:
2507:
2493:
2486:
2450:
2443:
2422:
2415:
2379:
2365:
2351:
2344:
2337:
2315:
2308:
2301:
2294:
2287:
2280:
2266:
2230:
2202:
2195:
2188:
2159:
2152:
2145:
2138:
2131:
2124:
2117:
2103:
2096:
2074:
2053:
2039:
2032:
2025:
1961:
1188:
1110:
1089:
115:
94:
2684:
2663:
2592:
2358:
2237:
2209:
2110:
1812:
1599:
1536:
1401:
1365:
1302:
817:
796:
604:
541:
370:
307:
2798:
994:
3016:
2943:
2846:
2834:
2933:"Simple Amazons endgames and their connection to Hamilton circuits in cubic subgrid graphs"
2838:
962:
991:
2883:
3079:
2828:
1007:
3028:
2909:
2003:
1996:
1982:
1975:
1954:
1947:
1131:
1124:
1117:
1103:
1096:
1082:
1075:
1068:
136:
129:
122:
108:
101:
87:
80:
73:
2698:
2691:
2649:
2642:
2627:
2620:
2613:
2578:
2549:
2542:
2500:
2471:
2464:
2457:
2436:
2429:
2408:
2393:
2386:
2372:
2330:
2273:
2259:
2252:
2223:
2216:
2181:
2174:
2081:
2067:
2060:
2046:
2018:
1833:
1826:
1819:
1805:
1798:
1791:
1784:
1777:
1770:
1755:
1748:
1741:
1734:
1727:
1720:
1713:
1706:
1699:
1692:
1677:
1670:
1663:
1656:
1649:
1642:
1635:
1628:
1621:
1614:
1592:
1585:
1578:
1571:
1564:
1557:
1550:
1543:
1521:
1514:
1507:
1500:
1493:
1486:
1479:
1472:
1465:
1458:
1443:
1436:
1429:
1422:
1415:
1408:
1394:
1387:
1380:
1358:
1351:
1344:
1337:
1330:
1323:
1316:
1309:
1287:
1280:
1273:
1266:
1259:
1252:
1245:
1238:
1231:
1224:
1209:
1202:
1195:
1181:
1174:
1167:
1160:
1153:
1146:
1014:; it may not cross or enter a square occupied by an amazon of either color or an
838:
831:
824:
810:
803:
789:
782:
775:
760:
753:
746:
739:
732:
725:
718:
711:
704:
697:
682:
675:
668:
661:
654:
647:
640:
633:
626:
619:
597:
590:
583:
576:
569:
562:
555:
548:
526:
519:
512:
505:
498:
491:
484:
477:
470:
463:
448:
441:
434:
427:
420:
413:
406:
399:
392:
385:
363:
356:
349:
342:
335:
328:
321:
314:
292:
285:
278:
271:
264:
257:
250:
243:
236:
229:
214:
207:
200:
193:
186:
179:
172:
165:
158:
151:
981:
973:
3011:
Stefan Reisch (1981). "Hex ist PSPACE-vollständig (Hex is PSPACE-complete)".
2947:
2983:
958:
3046:
Müller, Martin; Tegos, Theodore (2002), "Experiments in computer
Amazons",
2962:
3070:, MSRI Publications, vol. 42, Cambridge Univ. Press, pp. 261–278
3055:, MSRI Publications, vol. 42, Cambridge Univ. Press, pp. 243–257
2932:
1894:
1022:
27:
2849:
position, which is known to be PSPACE-complete, into an
Amazons position.
2789:
Michael Keller wrote the first known computer version of the game in VAX
3020:
2820:
2790:
986:
2996:
2961:
Furtak, Timothy; Kiyomi, Masashi; Uno, Takeaki; Buro, Michael (2005),
1011:
966:
984:
board. The two players are White and Black; each player has four
3061:
Snatzke, Raymond George (2002), "Exhaustive search in
Amazons",
2807:(The Game of the Amazons) is a trademark of Ediciones de Mente.
2781:
was first published in
Spanish in the Argentine puzzle magazine
2801:, a series of computer-versus-computer competitions.
935:
927:
919:
911:
903:
895:
2911:El Juego de las Amazonas (The Game of the Amazons)
2793:in 1994, and an updated version with graphics in
888:The starting position in The Game of the Amazons
2854:The second way is by reducing a certain kind of
953:often called Amazons for short) is a two-player
2877:
2875:
2873:
2823:. This is proven by reducing it to finding the
2797:in 1995. There are Amazons tournaments at the
972:The Game of the Amazons is played on a 10x10
8:
19:
2845:The first way is by reducing a generalized
2995:
957:invented in 1988 by Walter Zamkauskas of
2963:"Generalized Amazons is PSPACE-complete"
2903:
2901:
2869:
18:
7:
2841:. This can be proved in two ways.
14:
2940:Conference on Computers and Games
2704:
2697:
2690:
2683:
2676:
2669:
2662:
2655:
2648:
2641:
2626:
2619:
2612:
2605:
2598:
2591:
2584:
2577:
2570:
2563:
2548:
2541:
2534:
2527:
2520:
2513:
2506:
2499:
2492:
2485:
2470:
2463:
2456:
2449:
2442:
2435:
2428:
2421:
2414:
2407:
2392:
2385:
2378:
2371:
2364:
2357:
2350:
2343:
2336:
2329:
2314:
2307:
2300:
2293:
2286:
2279:
2272:
2265:
2258:
2251:
2236:
2229:
2222:
2215:
2208:
2201:
2194:
2187:
2180:
2173:
2158:
2151:
2144:
2137:
2130:
2123:
2116:
2109:
2102:
2095:
2080:
2073:
2066:
2059:
2052:
2045:
2038:
2031:
2024:
2017:
2002:
1995:
1988:
1981:
1974:
1967:
1960:
1953:
1946:
1939:
1832:
1825:
1818:
1811:
1804:
1797:
1790:
1783:
1776:
1769:
1754:
1747:
1740:
1733:
1726:
1719:
1712:
1705:
1698:
1691:
1676:
1669:
1662:
1655:
1648:
1641:
1634:
1627:
1620:
1613:
1598:
1591:
1584:
1577:
1570:
1563:
1556:
1549:
1542:
1535:
1520:
1513:
1506:
1499:
1492:
1485:
1478:
1471:
1464:
1457:
1442:
1435:
1428:
1421:
1414:
1407:
1400:
1393:
1386:
1379:
1364:
1357:
1350:
1343:
1336:
1329:
1322:
1315:
1308:
1301:
1286:
1279:
1272:
1265:
1258:
1251:
1244:
1237:
1230:
1223:
1208:
1201:
1194:
1187:
1180:
1173:
1166:
1159:
1152:
1145:
1130:
1123:
1116:
1109:
1102:
1095:
1088:
1081:
1074:
1067:
980:). Some players prefer to use a
837:
830:
823:
816:
809:
802:
795:
788:
781:
774:
759:
752:
745:
738:
731:
724:
717:
710:
703:
696:
681:
674:
667:
660:
653:
646:
639:
632:
625:
618:
603:
596:
589:
582:
575:
568:
561:
554:
547:
540:
525:
518:
511:
504:
497:
490:
483:
476:
469:
462:
447:
440:
433:
426:
419:
412:
405:
398:
391:
384:
369:
362:
355:
348:
341:
334:
327:
320:
313:
306:
291:
284:
277:
270:
263:
256:
249:
242:
235:
228:
213:
206:
199:
192:
185:
178:
171:
164:
157:
150:
135:
128:
121:
114:
107:
100:
93:
86:
79:
72:
3086:Board games introduced in 1988
1:
990:(not to be confused with the
2827:of a cubic subgraph of the
939:Tactics, strategy, position
3112:
2988:Amazons is PSPACE-complete
978:international checkerboard
951:El Juego de las Amazonas;
24:
3096:PSPACE-complete problems
2948:10.1007/3-540-45579-5_17
2811:Computational complexity
2805:El Juego de las Amazonas
2779:El Juego de las Amazonas
3091:Abstract strategy games
3064:More Games of No Chance
3049:More Games of No Chance
2817:simple Amazons endgames
20:The Game of the Amazons
2931:Buro, Michael (2000),
955:abstract strategy game
2856:generalized geography
2767:defective territories
1889:Territory and scoring
16:Two-player board game
2986:(February 2, 2005),
2942:, pp. 250–261,
947:Game of the Amazons
21:
3021:10.1007/bf00288964
2908:Keller, Michael,
2882:Pegg, Ed (1999),
2829:square grid graph
2799:Computer Olympiad
2753:
2752:
1881:
1880:
995:fairy chess piece
943:
942:
886:
885:
3103:
3071:
3069:
3056:
3054:
3033:
3032:
3013:Acta Informatica
3008:
3002:
3000:
2999:
2984:Hearn, Robert A.
2980:
2974:
2972:
2967:
2958:
2952:
2950:
2937:
2928:
2922:
2920:
2919:
2918:
2905:
2896:
2894:
2893:
2892:
2879:
2825:Hamiltonian path
2708:
2701:
2694:
2687:
2680:
2673:
2666:
2659:
2652:
2645:
2630:
2623:
2616:
2609:
2602:
2595:
2588:
2581:
2574:
2567:
2552:
2545:
2538:
2531:
2524:
2517:
2510:
2503:
2496:
2489:
2474:
2467:
2460:
2453:
2446:
2439:
2432:
2425:
2418:
2411:
2396:
2389:
2382:
2375:
2368:
2361:
2354:
2347:
2340:
2333:
2318:
2311:
2304:
2297:
2290:
2283:
2276:
2269:
2262:
2255:
2240:
2233:
2226:
2219:
2212:
2205:
2198:
2191:
2184:
2177:
2162:
2155:
2148:
2141:
2134:
2127:
2120:
2113:
2106:
2099:
2084:
2077:
2070:
2063:
2056:
2049:
2042:
2035:
2028:
2021:
2006:
1999:
1992:
1985:
1978:
1971:
1964:
1957:
1950:
1943:
1895:
1836:
1829:
1822:
1815:
1808:
1801:
1794:
1787:
1780:
1773:
1758:
1751:
1744:
1737:
1730:
1723:
1716:
1709:
1702:
1695:
1680:
1673:
1666:
1659:
1652:
1645:
1638:
1631:
1624:
1617:
1602:
1595:
1588:
1581:
1574:
1567:
1560:
1553:
1546:
1539:
1524:
1517:
1510:
1503:
1496:
1489:
1482:
1475:
1468:
1461:
1446:
1439:
1432:
1425:
1418:
1411:
1404:
1397:
1390:
1383:
1368:
1361:
1354:
1347:
1340:
1333:
1326:
1319:
1312:
1305:
1290:
1283:
1276:
1269:
1262:
1255:
1248:
1241:
1234:
1227:
1212:
1205:
1198:
1191:
1184:
1177:
1170:
1163:
1156:
1149:
1134:
1127:
1120:
1113:
1106:
1099:
1092:
1085:
1078:
1071:
1023:
841:
834:
827:
820:
813:
806:
799:
792:
785:
778:
763:
756:
749:
742:
735:
728:
721:
714:
707:
700:
685:
678:
671:
664:
657:
650:
643:
636:
629:
622:
607:
600:
593:
586:
579:
572:
565:
558:
551:
544:
529:
522:
515:
508:
501:
494:
487:
480:
473:
466:
451:
444:
437:
430:
423:
416:
409:
402:
395:
388:
373:
366:
359:
352:
345:
338:
331:
324:
317:
310:
295:
288:
281:
274:
267:
260:
253:
246:
239:
232:
217:
210:
203:
196:
189:
182:
175:
168:
161:
154:
139:
132:
125:
118:
111:
104:
97:
90:
83:
76:
28:
22:
3111:
3110:
3106:
3105:
3104:
3102:
3101:
3100:
3076:
3075:
3067:
3060:
3052:
3045:
3042:
3040:Further reading
3037:
3036:
3015:(15): 167–191.
3010:
3009:
3005:
2982:
2981:
2977:
2965:
2960:
2959:
2955:
2935:
2930:
2929:
2925:
2916:
2914:
2907:
2906:
2899:
2890:
2888:
2881:
2880:
2871:
2866:
2839:PSPACE-complete
2813:
2776:
2758:
2757:
2756:
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1000:
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993:
989:
988:
983:
982:monochromatic
979:
975:
970:
968:
964:
960:
956:
952:
949:(in Spanish,
948:
938:
934:
930:
926:
922:
918:
915:30-60 minutes
914:
910:
906:
902:
898:
894:
882:
880:
877:
874:
871:
868:
865:
862:
859:
856:
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848:
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836:
833:
829:
826:
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819:
815:
812:
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801:
798:
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791:
787:
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769:
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748:
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734:
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684:
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631:
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557:
553:
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536:
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531:
528:
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521:
517:
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489:
486:
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472:
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461:
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453:
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446:
443:
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411:
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397:
394:
390:
387:
383:
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375:
372:
368:
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361:
358:
354:
351:
347:
344:
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305:
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205:
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184:
181:
177:
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167:
163:
160:
156:
153:
149:
146:
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138:
134:
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127:
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117:
113:
110:
106:
103:
99:
96:
92:
89:
85:
82:
78:
75:
71:
68:
67:
64:
62:
59:
56:
53:
50:
47:
44:
41:
38:
35:
32:
30:
29:
23:
3063:
3048:
3012:
3006:
2987:
2978:
2969:
2956:
2939:
2926:
2915:, retrieved
2910:
2889:, retrieved
2884:
2833:
2816:
2814:
2804:
2803:
2795:Visual Basic
2788:
2782:
2778:
2777:
2766:
2763:
2759:
1015:
1004:
985:
971:
950:
946:
944:
912:Playing time
2835:Generalized
2783:El Acertijo
3080:Categories
2917:2014-10-26
2891:2014-10-19
2864:References
974:chessboard
907:20 seconds
904:Setup time
1010:moves in
959:Argentina
928:Age range
2859:allowed.
3029:9125259
2885:Amazons
2821:NP-hard
2791:Fortran
2774:History
987:amazons
976:(or an
896:Players
3027:
992:amazon
936:Skills
920:Chance
3068:(PDF)
3053:(PDF)
3025:S2CID
2992:arXiv
2970:IJCAI
2966:(PDF)
2936:(PDF)
2765:have
1016:arrow
1012:chess
1008:queen
1001:Rules
967:chess
965:and
945:The
923:None
3017:doi
2944:doi
2847:Hex
969:.
3082::
3023:.
2990:,
2968:,
2938:,
2900:^
2872:^
2831:.
2009:10
1936:10
1137:10
1064:10
963:Go
931:4+
142:10
69:10
3072:.
3057:.
3031:.
3019::
3001:.
2994::
2973:.
2951:.
2946::
2921:.
2895:.
2745:j
2742:i
2739:h
2736:g
2733:f
2730:e
2727:d
2724:c
2721:b
2718:a
2711:1
2638:1
2633:2
2560:2
2555:3
2482:3
2477:4
2404:4
2399:5
2326:5
2321:6
2248:6
2243:7
2170:7
2165:8
2092:8
2087:9
2014:9
1927:j
1924:i
1921:h
1918:g
1915:f
1912:e
1909:d
1906:c
1903:b
1900:a
1873:j
1870:i
1867:h
1864:g
1861:f
1858:e
1855:d
1852:c
1849:b
1846:a
1839:1
1766:1
1761:2
1688:2
1683:3
1610:3
1605:4
1532:4
1527:5
1454:5
1449:6
1376:6
1371:7
1298:7
1293:8
1220:8
1215:9
1142:9
1055:j
1052:i
1049:h
1046:g
1043:f
1040:e
1037:d
1034:c
1031:b
1028:a
899:2
878:j
875:i
872:h
869:g
866:f
863:e
860:d
857:c
854:b
851:a
844:1
771:1
766:2
693:2
688:3
615:3
610:4
537:4
532:5
459:5
454:6
381:6
376:7
303:7
298:8
225:8
220:9
147:9
60:j
57:i
54:h
51:g
48:f
45:e
42:d
39:c
36:b
33:a
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
