[{"data":1,"prerenderedAt":2307},["ShallowReactive",2],{"post-\u002Fblog\u002F2023\u002F2023-12-14-the-bash-game":3},{"id":4,"title":5,"body":6,"description":1993,"draft":2292,"enableComment":2293,"extension":2294,"image":2295,"meta":2296,"navigation":2293,"onday":2298,"path":2299,"seo":2300,"stem":2301,"summary":2302,"tags":2303,"__hash__":2306},"blog\u002Fblog\u002F2023\u002F2023-12-14-the-bash-game.md","The Bash Game",{"type":7,"value":8,"toc":2286},"minimark",[9,14,22,26,29,32,259,370,581,731,962,1027,1031,1034,1237,1562,1947,1951,1984,1987,2243,2246,2282],[10,11,13],"h2",{"id":12},"problem-citation","Problem Citation",[15,16,17,18],"p",{},"A pile of stones, two people take turns to take at least 1 stone and at most 2 stones. Whoever gets the last stone loses. Is there a sure-win strategy of playing first and playing second? That's a famous ",[19,20,21],"strong",{},"Bash Game Problem",[10,23,25],{"id":24},"analysis","Analysis",[15,27,28],{},"There is a pile of n items in total, and two players take turns picking up items from it. It is stipulated that each time at least one item must be taken, and at most m items can be taken. The one who takes the last item wins.",[15,30,31],{},"Let",[15,33,34],{},[35,36,40,115],"span",{"className":37,"translate":39},[38],"katex","no",[35,41,44],{"className":42},[43],"katex-mathml",[45,46,48],"math",{"xmlns":47},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[49,50,51,110],"semantics",{},[52,53,54,58,62,66,69,72,76,79,82,84,87,91,93,96,99,101,103,105,107],"mrow",{},[55,56,57],"mi",{},"n",[59,60,61],"mo",{},"=",[59,63,65],{"stretchy":64},"false","(",[55,67,68],{},"m",[59,70,71],{},"+",[73,74,75],"mn",{},"1",[59,77,78],{"stretchy":64},")",[55,80,81],{},"q",[59,83,71],{},[55,85,86],{},"r",[88,89],"mspace",{"width":90},"1em",[59,92,65],{"stretchy":64},[73,94,95],{},"0",[59,97,98],{},"≤",[55,100,86],{},[59,102,98],{},[55,104,68],{},[59,106,78],{"stretchy":64},[88,108],{"linebreak":109},"newline",[111,112,114],"annotation",{"encoding":113},"application\u002Fx-tex","n=(m+1)q+r \\quad( 0\\leq r \\leq m ) \\\\",[35,116,120,145,170,196,225,244,256],{"className":117,"ariaHidden":119},[118],"katex-html","true",[35,121,124,129,134,138,142],{"className":122},[123],"base",[35,125],{"className":126,"style":128},[127],"strut","height:0.4306em;",[35,130,57],{"className":131},[132,133],"mord","mathnormal",[35,135],{"className":136,"style":137},[88],"margin-right:0.2778em;",[35,139,61],{"className":140},[141],"mrel",[35,143],{"className":144,"style":137},[88],[35,146,148,152,156,159,163,167],{"className":147},[123],[35,149],{"className":150,"style":151},[127],"height:1em;vertical-align:-0.25em;",[35,153,65],{"className":154},[155],"mopen",[35,157,68],{"className":158},[132,133],[35,160],{"className":161,"style":162},[88],"margin-right:0.2222em;",[35,164,71],{"className":165},[166],"mbin",[35,168],{"className":169,"style":162},[88],[35,171,173,176,179,183,187,190,193],{"className":172},[123],[35,174],{"className":175,"style":151},[127],[35,177,75],{"className":178},[132],[35,180,78],{"className":181},[182],"mclose",[35,184,81],{"className":185,"style":186},[132,133],"margin-right:0.03588em;",[35,188],{"className":189,"style":162},[88],[35,191,71],{"className":192},[166],[35,194],{"className":195,"style":162},[88],[35,197,199,202,206,210,213,216,219,222],{"className":198},[123],[35,200],{"className":201,"style":151},[127],[35,203,86],{"className":204,"style":205},[132,133],"margin-right:0.02778em;",[35,207],{"className":208,"style":209},[88],"margin-right:1em;",[35,211,65],{"className":212},[155],[35,214,95],{"className":215},[132],[35,217],{"className":218,"style":137},[88],[35,220,98],{"className":221},[141],[35,223],{"className":224,"style":137},[88],[35,226,228,232,235,238,241],{"className":227},[123],[35,229],{"className":230,"style":231},[127],"height:0.7719em;vertical-align:-0.136em;",[35,233,86],{"className":234,"style":205},[132,133],[35,236],{"className":237,"style":137},[88],[35,239,98],{"className":240},[141],[35,242],{"className":243,"style":137},[88],[35,245,247,250,253],{"className":246},[123],[35,248],{"className":249,"style":151},[127],[35,251,68],{"className":252},[132,133],[35,254,78],{"className":255},[182],[35,257],{"className":258},[88,109],[15,260,261,316,317,369],{},[35,262,264,288],{"className":263,"translate":39},[38],[35,265,267],{"className":266},[43],[45,268,269],{"xmlns":47},[49,270,271,285],{},[52,272,273,275,277,281,283],{},[88,274],{"width":90},[59,276,65],{"stretchy":64},[278,279,280],"mtext",{},"i",[59,282,78],{"stretchy":64},[88,284],{"width":90},[111,286,287],{"encoding":113},"\\quad (\\text{i})\\quad",[35,289,291],{"className":290,"ariaHidden":119},[118],[35,292,294,297,300,303,310,313],{"className":293},[123],[35,295],{"className":296,"style":151},[127],[35,298],{"className":299,"style":209},[88],[35,301,65],{"className":302},[155],[35,304,307],{"className":305},[132,306],"text",[35,308,280],{"className":309},[132],[35,311,78],{"className":312},[182],[35,314],{"className":315,"style":209},[88],"If ",[35,318,320,338],{"className":319,"translate":39},[38],[35,321,323],{"className":322},[43],[45,324,325],{"xmlns":47},[49,326,327,335],{},[52,328,329,331,333],{},[55,330,86],{},[59,332,61],{},[73,334,95],{},[111,336,337],{"encoding":113},"r=0",[35,339,341,359],{"className":340,"ariaHidden":119},[118],[35,342,344,347,350,353,356],{"className":343},[123],[35,345],{"className":346,"style":128},[127],[35,348,86],{"className":349,"style":205},[132,133],[35,351],{"className":352,"style":137},[88],[35,354,61],{"className":355},[141],[35,357],{"className":358,"style":137},[88],[35,360,362,366],{"className":361},[123],[35,363],{"className":364,"style":365},[127],"height:0.6444em;",[35,367,95],{"className":368},[132],", the second player will win. The strategy is as follows:",[15,371,372,373,404,405,481,482,580],{},"If the first player takes away ",[35,374,376,390],{"className":375,"translate":39},[38],[35,377,379],{"className":378},[43],[45,380,381],{"xmlns":47},[49,382,383,388],{},[52,384,385],{},[55,386,387],{},"k",[111,389,387],{"encoding":113},[35,391,393],{"className":392,"ariaHidden":119},[118],[35,394,396,400],{"className":395},[123],[35,397],{"className":398,"style":399},[127],"height:0.6944em;",[35,401,387],{"className":402,"style":403},[132,133],"margin-right:0.03148em;"," items, the second player takes away ",[35,406,408,431],{"className":407,"translate":39},[38],[35,409,411],{"className":410},[43],[45,412,413],{"xmlns":47},[49,414,415,428],{},[52,416,417,419,421,423,426],{},[55,418,68],{},[59,420,71],{},[73,422,75],{},[59,424,425],{},"−",[55,427,387],{},[111,429,430],{"encoding":113},"m+1-k",[35,432,434,453,472],{"className":433,"ariaHidden":119},[118],[35,435,437,441,444,447,450],{"className":436},[123],[35,438],{"className":439,"style":440},[127],"height:0.6667em;vertical-align:-0.0833em;",[35,442,68],{"className":443},[132,133],[35,445],{"className":446,"style":162},[88],[35,448,71],{"className":449},[166],[35,451],{"className":452,"style":162},[88],[35,454,456,460,463,466,469],{"className":455},[123],[35,457],{"className":458,"style":459},[127],"height:0.7278em;vertical-align:-0.0833em;",[35,461,75],{"className":462},[132],[35,464],{"className":465,"style":162},[88],[35,467,425],{"className":468},[166],[35,470],{"className":471,"style":162},[88],[35,473,475,478],{"className":474},[123],[35,476],{"className":477,"style":399},[127],[35,479,387],{"className":480,"style":403},[132,133]," items. The result is ",[35,483,485,517],{"className":484,"translate":39},[38],[35,486,488],{"className":487},[43],[45,489,490],{"xmlns":47},[49,491,492,514],{},[52,493,494,496,498,500,502,504,506,508,510,512],{},[59,495,65],{"stretchy":64},[55,497,68],{},[59,499,71],{},[73,501,75],{},[59,503,78],{"stretchy":64},[59,505,65],{"stretchy":64},[55,507,81],{},[59,509,425],{},[73,511,75],{},[59,513,78],{"stretchy":64},[111,515,516],{"encoding":113},"(m+1)(q-1)",[35,518,520,541,568],{"className":519,"ariaHidden":119},[118],[35,521,523,526,529,532,535,538],{"className":522},[123],[35,524],{"className":525,"style":151},[127],[35,527,65],{"className":528},[155],[35,530,68],{"className":531},[132,133],[35,533],{"className":534,"style":162},[88],[35,536,71],{"className":537},[166],[35,539],{"className":540,"style":162},[88],[35,542,544,547,550,553,556,559,562,565],{"className":543},[123],[35,545],{"className":546,"style":151},[127],[35,548,75],{"className":549},[132],[35,551,78],{"className":552},[182],[35,554,65],{"className":555},[155],[35,557,81],{"className":558,"style":186},[132,133],[35,560],{"className":561,"style":162},[88],[35,563,425],{"className":564},[166],[35,566],{"className":567,"style":162},[88],[35,569,571,574,577],{"className":570},[123],[35,572],{"className":573,"style":151},[127],[35,575,75],{"className":576},[132],[35,578,78],{"className":579},[182]," items left. By maintaining this method, the second player will win.",[15,582,583,316,636,730],{},[35,584,586,609],{"className":585,"translate":39},[38],[35,587,589],{"className":588},[43],[45,590,591],{"xmlns":47},[49,592,593,606],{},[52,594,595,597,599,602,604],{},[88,596],{"width":90},[59,598,65],{"stretchy":64},[278,600,601],{},"ii",[59,603,78],{"stretchy":64},[88,605],{"width":90},[111,607,608],{"encoding":113},"\\quad (\\text{ii})\\quad",[35,610,612],{"className":611,"ariaHidden":119},[118],[35,613,615,618,621,624,630,633],{"className":614},[123],[35,616],{"className":617,"style":151},[127],[35,619],{"className":620,"style":209},[88],[35,622,65],{"className":623},[155],[35,625,627],{"className":626},[132,306],[35,628,601],{"className":629},[132],[35,631,78],{"className":632},[182],[35,634],{"className":635,"style":209},[88],[35,637,639,659],{"className":638,"translate":39},[38],[35,640,642],{"className":641},[43],[45,643,644],{"xmlns":47},[49,645,646,656],{},[52,647,648,650,654],{},[55,649,86],{},[59,651,653],{"mathvariant":652},"normal","≠",[73,655,95],{},[111,657,658],{"encoding":113},"r≠0",[35,660,662,721],{"className":661,"ariaHidden":119},[118],[35,663,665,669,672,675,718],{"className":664},[123],[35,666],{"className":667,"style":668},[127],"height:0.8889em;vertical-align:-0.1944em;",[35,670,86],{"className":671,"style":205},[132,133],[35,673],{"className":674,"style":137},[88],[35,676,678,711,715],{"className":677},[141],[35,679,681],{"className":680},[141],[35,682,685],{"className":683},[132,684],"vbox",[35,686,689],{"className":687},[688],"thinbox",[35,690,693,696,707],{"className":691},[692],"rlap",[35,694],{"className":695,"style":668},[127],[35,697,700],{"className":698},[699],"inner",[35,701,703],{"className":702},[132],[35,704,706],{"className":705},[141],"",[35,708],{"className":709},[710],"fix",[35,712],{"className":713},[88,714],"nobreak",[35,716,61],{"className":717},[141],[35,719],{"className":720,"style":137},[88],[35,722,724,727],{"className":723},[123],[35,725],{"className":726,"style":365},[127],[35,728,95],{"className":729},[132],", the first player will win. The strategy is as follows:",[15,732,733,734,762,763,791,792,481,864,961],{},"The first player takes away ",[35,735,737,750],{"className":736,"translate":39},[38],[35,738,740],{"className":739},[43],[45,741,742],{"xmlns":47},[49,743,744,748],{},[52,745,746],{},[55,747,86],{},[111,749,86],{"encoding":113},[35,751,753],{"className":752,"ariaHidden":119},[118],[35,754,756,759],{"className":755},[123],[35,757],{"className":758,"style":128},[127],[35,760,86],{"className":761,"style":205},[132,133]," items first. If the second player takes away ",[35,764,766,779],{"className":765,"translate":39},[38],[35,767,769],{"className":768},[43],[45,770,771],{"xmlns":47},[49,772,773,777],{},[52,774,775],{},[55,776,387],{},[111,778,387],{"encoding":113},[35,780,782],{"className":781,"ariaHidden":119},[118],[35,783,785,788],{"className":784},[123],[35,786],{"className":787,"style":399},[127],[35,789,387],{"className":790,"style":403},[132,133]," items, the first player takes away ",[35,793,795,816],{"className":794,"translate":39},[38],[35,796,798],{"className":797},[43],[45,799,800],{"xmlns":47},[49,801,802,814],{},[52,803,804,806,808,810,812],{},[55,805,68],{},[59,807,71],{},[73,809,75],{},[59,811,425],{},[55,813,387],{},[111,815,430],{"encoding":113},[35,817,819,837,855],{"className":818,"ariaHidden":119},[118],[35,820,822,825,828,831,834],{"className":821},[123],[35,823],{"className":824,"style":440},[127],[35,826,68],{"className":827},[132,133],[35,829],{"className":830,"style":162},[88],[35,832,71],{"className":833},[166],[35,835],{"className":836,"style":162},[88],[35,838,840,843,846,849,852],{"className":839},[123],[35,841],{"className":842,"style":459},[127],[35,844,75],{"className":845},[132],[35,847],{"className":848,"style":162},[88],[35,850,425],{"className":851},[166],[35,853],{"className":854,"style":162},[88],[35,856,858,861],{"className":857},[123],[35,859],{"className":860,"style":399},[127],[35,862,387],{"className":863,"style":403},[132,133],[35,865,867,898],{"className":866,"translate":39},[38],[35,868,870],{"className":869},[43],[45,871,872],{"xmlns":47},[49,873,874,896],{},[52,875,876,878,880,882,884,886,888,890,892,894],{},[59,877,65],{"stretchy":64},[55,879,68],{},[59,881,71],{},[73,883,75],{},[59,885,78],{"stretchy":64},[59,887,65],{"stretchy":64},[55,889,81],{},[59,891,425],{},[73,893,75],{},[59,895,78],{"stretchy":64},[111,897,516],{"encoding":113},[35,899,901,922,949],{"className":900,"ariaHidden":119},[118],[35,902,904,907,910,913,916,919],{"className":903},[123],[35,905],{"className":906,"style":151},[127],[35,908,65],{"className":909},[155],[35,911,68],{"className":912},[132,133],[35,914],{"className":915,"style":162},[88],[35,917,71],{"className":918},[166],[35,920],{"className":921,"style":162},[88],[35,923,925,928,931,934,937,940,943,946],{"className":924},[123],[35,926],{"className":927,"style":151},[127],[35,929,75],{"className":930},[132],[35,932,78],{"className":933},[182],[35,935,65],{"className":936},[155],[35,938,81],{"className":939,"style":186},[132,133],[35,941],{"className":942,"style":162},[88],[35,944,425],{"className":945},[166],[35,947],{"className":948,"style":162},[88],[35,950,952,955,958],{"className":951},[123],[35,953],{"className":954,"style":151},[127],[35,956,75],{"className":957},[132],[35,959,78],{"className":960},[182]," items left. By maintaining this method, the first player will win.",[15,963,964,965,1026],{},"In short, to ensure that the opponent is left with a multiple of ",[35,966,968,990],{"className":967,"translate":39},[38],[35,969,971],{"className":970},[43],[45,972,973],{"xmlns":47},[49,974,975,987],{},[52,976,977,979,981,983,985],{},[59,978,65],{"stretchy":64},[55,980,68],{},[59,982,71],{},[73,984,75],{},[59,986,78],{"stretchy":64},[111,988,989],{"encoding":113},"(m+1)",[35,991,993,1014],{"className":992,"ariaHidden":119},[118],[35,994,996,999,1002,1005,1008,1011],{"className":995},[123],[35,997],{"className":998,"style":151},[127],[35,1000,65],{"className":1001},[155],[35,1003,68],{"className":1004},[132,133],[35,1006],{"className":1007,"style":162},[88],[35,1009,71],{"className":1010},[166],[35,1012],{"className":1013,"style":162},[88],[35,1015,1017,1020,1023],{"className":1016},[123],[35,1018],{"className":1019,"style":151},[127],[35,1021,75],{"className":1022},[132],[35,1024,78],{"className":1025},[182],", one can eventually win.",[10,1028,1030],{"id":1029},"extension","Extension",[15,1032,1033],{},"If it is stipulated that the one who takes the last item loses, let",[15,1035,1036],{},[35,1037,1039,1093],{"className":1038,"translate":39},[38],[35,1040,1042],{"className":1041},[43],[45,1043,1044],{"xmlns":47},[49,1045,1046,1090],{},[52,1047,1048,1050,1052,1054,1056,1058,1060,1062,1064,1066,1068,1070,1072,1074,1076,1078,1080,1082,1084,1086,1088],{},[55,1049,57],{},[59,1051,425],{},[73,1053,75],{},[59,1055,61],{},[59,1057,65],{"stretchy":64},[55,1059,68],{},[59,1061,71],{},[73,1063,75],{},[59,1065,78],{"stretchy":64},[55,1067,81],{},[59,1069,71],{},[55,1071,86],{},[88,1073],{"width":90},[59,1075,65],{"stretchy":64},[73,1077,95],{},[59,1079,98],{},[55,1081,86],{},[59,1083,98],{},[55,1085,68],{},[59,1087,78],{"stretchy":64},[88,1089],{"linebreak":109},[111,1091,1092],{"encoding":113},"n-1=(m+1)q+r \\quad( 0≤r≤m ) \\\\",[35,1094,1096,1114,1132,1153,1177,1204,1222,1234],{"className":1095,"ariaHidden":119},[118],[35,1097,1099,1102,1105,1108,1111],{"className":1098},[123],[35,1100],{"className":1101,"style":440},[127],[35,1103,57],{"className":1104},[132,133],[35,1106],{"className":1107,"style":162},[88],[35,1109,425],{"className":1110},[166],[35,1112],{"className":1113,"style":162},[88],[35,1115,1117,1120,1123,1126,1129],{"className":1116},[123],[35,1118],{"className":1119,"style":365},[127],[35,1121,75],{"className":1122},[132],[35,1124],{"className":1125,"style":137},[88],[35,1127,61],{"className":1128},[141],[35,1130],{"className":1131,"style":137},[88],[35,1133,1135,1138,1141,1144,1147,1150],{"className":1134},[123],[35,1136],{"className":1137,"style":151},[127],[35,1139,65],{"className":1140},[155],[35,1142,68],{"className":1143},[132,133],[35,1145],{"className":1146,"style":162},[88],[35,1148,71],{"className":1149},[166],[35,1151],{"className":1152,"style":162},[88],[35,1154,1156,1159,1162,1165,1168,1171,1174],{"className":1155},[123],[35,1157],{"className":1158,"style":151},[127],[35,1160,75],{"className":1161},[132],[35,1163,78],{"className":1164},[182],[35,1166,81],{"className":1167,"style":186},[132,133],[35,1169],{"className":1170,"style":162},[88],[35,1172,71],{"className":1173},[166],[35,1175],{"className":1176,"style":162},[88],[35,1178,1180,1183,1186,1189,1192,1195,1198,1201],{"className":1179},[123],[35,1181],{"className":1182,"style":151},[127],[35,1184,86],{"className":1185,"style":205},[132,133],[35,1187],{"className":1188,"style":209},[88],[35,1190,65],{"className":1191},[155],[35,1193,95],{"className":1194},[132],[35,1196],{"className":1197,"style":137},[88],[35,1199,98],{"className":1200},[141],[35,1202],{"className":1203,"style":137},[88],[35,1205,1207,1210,1213,1216,1219],{"className":1206},[123],[35,1208],{"className":1209,"style":231},[127],[35,1211,86],{"className":1212,"style":205},[132,133],[35,1214],{"className":1215,"style":137},[88],[35,1217,98],{"className":1218},[141],[35,1220],{"className":1221,"style":137},[88],[35,1223,1225,1228,1231],{"className":1224},[123],[35,1226],{"className":1227,"style":151},[127],[35,1229,68],{"className":1230},[132,133],[35,1232,78],{"className":1233},[182],[35,1235],{"className":1236},[88,109],[15,1238,1239,316,1290,1340,1341,404,1369,481,1441,1561],{},[35,1240,1242,1263],{"className":1241,"translate":39},[38],[35,1243,1245],{"className":1244},[43],[45,1246,1247],{"xmlns":47},[49,1248,1249,1261],{},[52,1250,1251,1253,1255,1257,1259],{},[88,1252],{"width":90},[59,1254,65],{"stretchy":64},[278,1256,280],{},[59,1258,78],{"stretchy":64},[88,1260],{"width":90},[111,1262,287],{"encoding":113},[35,1264,1266],{"className":1265,"ariaHidden":119},[118],[35,1267,1269,1272,1275,1278,1284,1287],{"className":1268},[123],[35,1270],{"className":1271,"style":151},[127],[35,1273],{"className":1274,"style":209},[88],[35,1276,65],{"className":1277},[155],[35,1279,1281],{"className":1280},[132,306],[35,1282,280],{"className":1283},[132],[35,1285,78],{"className":1286},[182],[35,1288],{"className":1289,"style":209},[88],[35,1291,1293,1310],{"className":1292,"translate":39},[38],[35,1294,1296],{"className":1295},[43],[45,1297,1298],{"xmlns":47},[49,1299,1300,1308],{},[52,1301,1302,1304,1306],{},[55,1303,86],{},[59,1305,61],{},[73,1307,95],{},[111,1309,337],{"encoding":113},[35,1311,1313,1331],{"className":1312,"ariaHidden":119},[118],[35,1314,1316,1319,1322,1325,1328],{"className":1315},[123],[35,1317],{"className":1318,"style":128},[127],[35,1320,86],{"className":1321,"style":205},[132,133],[35,1323],{"className":1324,"style":137},[88],[35,1326,61],{"className":1327},[141],[35,1329],{"className":1330,"style":137},[88],[35,1332,1334,1337],{"className":1333},[123],[35,1335],{"className":1336,"style":365},[127],[35,1338,95],{"className":1339},[132],", the second player will win. The strategy is as follows: If the first player takes away ",[35,1342,1344,1357],{"className":1343,"translate":39},[38],[35,1345,1347],{"className":1346},[43],[45,1348,1349],{"xmlns":47},[49,1350,1351,1355],{},[52,1352,1353],{},[55,1354,387],{},[111,1356,387],{"encoding":113},[35,1358,1360],{"className":1359,"ariaHidden":119},[118],[35,1361,1363,1366],{"className":1362},[123],[35,1364],{"className":1365,"style":399},[127],[35,1367,387],{"className":1368,"style":403},[132,133],[35,1370,1372,1393],{"className":1371,"translate":39},[38],[35,1373,1375],{"className":1374},[43],[45,1376,1377],{"xmlns":47},[49,1378,1379,1391],{},[52,1380,1381,1383,1385,1387,1389],{},[55,1382,68],{},[59,1384,71],{},[73,1386,75],{},[59,1388,425],{},[55,1390,387],{},[111,1392,430],{"encoding":113},[35,1394,1396,1414,1432],{"className":1395,"ariaHidden":119},[118],[35,1397,1399,1402,1405,1408,1411],{"className":1398},[123],[35,1400],{"className":1401,"style":440},[127],[35,1403,68],{"className":1404},[132,133],[35,1406],{"className":1407,"style":162},[88],[35,1409,71],{"className":1410},[166],[35,1412],{"className":1413,"style":162},[88],[35,1415,1417,1420,1423,1426,1429],{"className":1416},[123],[35,1418],{"className":1419,"style":459},[127],[35,1421,75],{"className":1422},[132],[35,1424],{"className":1425,"style":162},[88],[35,1427,425],{"className":1428},[166],[35,1430],{"className":1431,"style":162},[88],[35,1433,1435,1438],{"className":1434},[123],[35,1436],{"className":1437,"style":399},[127],[35,1439,387],{"className":1440,"style":403},[132,133],[35,1442,1444,1480],{"className":1443,"translate":39},[38],[35,1445,1447],{"className":1446},[43],[45,1448,1449],{"xmlns":47},[49,1450,1451,1477],{},[52,1452,1453,1455,1457,1459,1461,1463,1465,1467,1469,1471,1473,1475],{},[59,1454,65],{"stretchy":64},[55,1456,68],{},[59,1458,71],{},[73,1460,75],{},[59,1462,78],{"stretchy":64},[59,1464,65],{"stretchy":64},[55,1466,81],{},[59,1468,425],{},[73,1470,75],{},[59,1472,78],{"stretchy":64},[59,1474,71],{},[73,1476,75],{},[111,1478,1479],{"encoding":113},"(m+1)(q-1)+1",[35,1481,1483,1504,1531,1552],{"className":1482,"ariaHidden":119},[118],[35,1484,1486,1489,1492,1495,1498,1501],{"className":1485},[123],[35,1487],{"className":1488,"style":151},[127],[35,1490,65],{"className":1491},[155],[35,1493,68],{"className":1494},[132,133],[35,1496],{"className":1497,"style":162},[88],[35,1499,71],{"className":1500},[166],[35,1502],{"className":1503,"style":162},[88],[35,1505,1507,1510,1513,1516,1519,1522,1525,1528],{"className":1506},[123],[35,1508],{"className":1509,"style":151},[127],[35,1511,75],{"className":1512},[132],[35,1514,78],{"className":1515},[182],[35,1517,65],{"className":1518},[155],[35,1520,81],{"className":1521,"style":186},[132,133],[35,1523],{"className":1524,"style":162},[88],[35,1526,425],{"className":1527},[166],[35,1529],{"className":1530,"style":162},[88],[35,1532,1534,1537,1540,1543,1546,1549],{"className":1533},[123],[35,1535],{"className":1536,"style":151},[127],[35,1538,75],{"className":1539},[132],[35,1541,78],{"className":1542},[182],[35,1544],{"className":1545,"style":162},[88],[35,1547,71],{"className":1548},[166],[35,1550],{"className":1551,"style":162},[88],[35,1553,1555,1558],{"className":1554},[123],[35,1556],{"className":1557,"style":365},[127],[35,1559,75],{"className":1560},[132]," item left. By maintaining this method, the first player will take the last item.",[15,1563,1564,316,1615,1698,1699,762,1727,791,1755,481,1827,1946],{},[35,1565,1567,1588],{"className":1566,"translate":39},[38],[35,1568,1570],{"className":1569},[43],[45,1571,1572],{"xmlns":47},[49,1573,1574,1586],{},[52,1575,1576,1578,1580,1582,1584],{},[88,1577],{"width":90},[59,1579,65],{"stretchy":64},[278,1581,601],{},[59,1583,78],{"stretchy":64},[88,1585],{"width":90},[111,1587,608],{"encoding":113},[35,1589,1591],{"className":1590,"ariaHidden":119},[118],[35,1592,1594,1597,1600,1603,1609,1612],{"className":1593},[123],[35,1595],{"className":1596,"style":151},[127],[35,1598],{"className":1599,"style":209},[88],[35,1601,65],{"className":1602},[155],[35,1604,1606],{"className":1605},[132,306],[35,1607,601],{"className":1608},[132],[35,1610,78],{"className":1611},[182],[35,1613],{"className":1614,"style":209},[88],[35,1616,1618,1635],{"className":1617,"translate":39},[38],[35,1619,1621],{"className":1620},[43],[45,1622,1623],{"xmlns":47},[49,1624,1625,1633],{},[52,1626,1627,1629,1631],{},[55,1628,86],{},[59,1630,653],{"mathvariant":652},[73,1632,95],{},[111,1634,658],{"encoding":113},[35,1636,1638,1689],{"className":1637,"ariaHidden":119},[118],[35,1639,1641,1644,1647,1650,1686],{"className":1640},[123],[35,1642],{"className":1643,"style":668},[127],[35,1645,86],{"className":1646,"style":205},[132,133],[35,1648],{"className":1649,"style":137},[88],[35,1651,1653,1680,1683],{"className":1652},[141],[35,1654,1656],{"className":1655},[141],[35,1657,1659],{"className":1658},[132,684],[35,1660,1662],{"className":1661},[688],[35,1663,1665,1668,1677],{"className":1664},[692],[35,1666],{"className":1667,"style":668},[127],[35,1669,1671],{"className":1670},[699],[35,1672,1674],{"className":1673},[132],[35,1675,706],{"className":1676},[141],[35,1678],{"className":1679},[710],[35,1681],{"className":1682},[88,714],[35,1684,61],{"className":1685},[141],[35,1687],{"className":1688,"style":137},[88],[35,1690,1692,1695],{"className":1691},[123],[35,1693],{"className":1694,"style":365},[127],[35,1696,95],{"className":1697},[132],", the first player will win. The strategy is as follows: The first player takes away ",[35,1700,1702,1715],{"className":1701,"translate":39},[38],[35,1703,1705],{"className":1704},[43],[45,1706,1707],{"xmlns":47},[49,1708,1709,1713],{},[52,1710,1711],{},[55,1712,86],{},[111,1714,86],{"encoding":113},[35,1716,1718],{"className":1717,"ariaHidden":119},[118],[35,1719,1721,1724],{"className":1720},[123],[35,1722],{"className":1723,"style":128},[127],[35,1725,86],{"className":1726,"style":205},[132,133],[35,1728,1730,1743],{"className":1729,"translate":39},[38],[35,1731,1733],{"className":1732},[43],[45,1734,1735],{"xmlns":47},[49,1736,1737,1741],{},[52,1738,1739],{},[55,1740,387],{},[111,1742,387],{"encoding":113},[35,1744,1746],{"className":1745,"ariaHidden":119},[118],[35,1747,1749,1752],{"className":1748},[123],[35,1750],{"className":1751,"style":399},[127],[35,1753,387],{"className":1754,"style":403},[132,133],[35,1756,1758,1779],{"className":1757,"translate":39},[38],[35,1759,1761],{"className":1760},[43],[45,1762,1763],{"xmlns":47},[49,1764,1765,1777],{},[52,1766,1767,1769,1771,1773,1775],{},[55,1768,68],{},[59,1770,71],{},[73,1772,75],{},[59,1774,425],{},[55,1776,387],{},[111,1778,430],{"encoding":113},[35,1780,1782,1800,1818],{"className":1781,"ariaHidden":119},[118],[35,1783,1785,1788,1791,1794,1797],{"className":1784},[123],[35,1786],{"className":1787,"style":440},[127],[35,1789,68],{"className":1790},[132,133],[35,1792],{"className":1793,"style":162},[88],[35,1795,71],{"className":1796},[166],[35,1798],{"className":1799,"style":162},[88],[35,1801,1803,1806,1809,1812,1815],{"className":1802},[123],[35,1804],{"className":1805,"style":459},[127],[35,1807,75],{"className":1808},[132],[35,1810],{"className":1811,"style":162},[88],[35,1813,425],{"className":1814},[166],[35,1816],{"className":1817,"style":162},[88],[35,1819,1821,1824],{"className":1820},[123],[35,1822],{"className":1823,"style":399},[127],[35,1825,387],{"className":1826,"style":403},[132,133],[35,1828,1830,1865],{"className":1829,"translate":39},[38],[35,1831,1833],{"className":1832},[43],[45,1834,1835],{"xmlns":47},[49,1836,1837,1863],{},[52,1838,1839,1841,1843,1845,1847,1849,1851,1853,1855,1857,1859,1861],{},[59,1840,65],{"stretchy":64},[55,1842,68],{},[59,1844,71],{},[73,1846,75],{},[59,1848,78],{"stretchy":64},[59,1850,65],{"stretchy":64},[55,1852,81],{},[59,1854,425],{},[73,1856,75],{},[59,1858,78],{"stretchy":64},[59,1860,71],{},[73,1862,75],{},[111,1864,1479],{"encoding":113},[35,1866,1868,1889,1916,1937],{"className":1867,"ariaHidden":119},[118],[35,1869,1871,1874,1877,1880,1883,1886],{"className":1870},[123],[35,1872],{"className":1873,"style":151},[127],[35,1875,65],{"className":1876},[155],[35,1878,68],{"className":1879},[132,133],[35,1881],{"className":1882,"style":162},[88],[35,1884,71],{"className":1885},[166],[35,1887],{"className":1888,"style":162},[88],[35,1890,1892,1895,1898,1901,1904,1907,1910,1913],{"className":1891},[123],[35,1893],{"className":1894,"style":151},[127],[35,1896,75],{"className":1897},[132],[35,1899,78],{"className":1900},[182],[35,1902,65],{"className":1903},[155],[35,1905,81],{"className":1906,"style":186},[132,133],[35,1908],{"className":1909,"style":162},[88],[35,1911,425],{"className":1912},[166],[35,1914],{"className":1915,"style":162},[88],[35,1917,1919,1922,1925,1928,1931,1934],{"className":1918},[123],[35,1920],{"className":1921,"style":151},[127],[35,1923,75],{"className":1924},[132],[35,1926,78],{"className":1927},[182],[35,1929],{"className":1930,"style":162},[88],[35,1932,71],{"className":1933},[166],[35,1935],{"className":1936,"style":162},[88],[35,1938,1940,1943],{"className":1939},[123],[35,1941],{"className":1942,"style":365},[127],[35,1944,75],{"className":1945},[132]," item left. By maintaining this method, the second player will take the last item.",[10,1948,1950],{"id":1949},"mini-game","Mini Game",[15,1952,1953,1954,1983],{},"Two people take turns counting, each time reporting at least one, at most ten. Whoever reaches ",[35,1955,1957,1971],{"className":1956,"translate":39},[38],[35,1958,1960],{"className":1959},[43],[45,1961,1962],{"xmlns":47},[49,1963,1964,1969],{},[52,1965,1966],{},[73,1967,1968],{},"100",[111,1970,1968],{"encoding":113},[35,1972,1974],{"className":1973,"ariaHidden":119},[118],[35,1975,1977,1980],{"className":1976},[123],[35,1978],{"className":1979,"style":365},[127],[35,1981,1968],{"className":1982},[132]," first wins.",[15,1985,1986],{},"Attached code:",[1988,1989,1994],"pre",{"className":1990,"code":1991,"language":1992,"meta":1993,"style":1993},"language-cpp shiki shiki-themes github-dark","#include\u003Ccstdio>\n#include\u003Cstring.h>\n#include\u003Cqueue>\n#include\u003Calgorithm>\n#include\u003Ciostream>\nusing namespace std;\nint main()\n{\n    int T;\n    scanf(\"%d\",&T);\n    while(T--){\n        int n,m;\n        scanf(\"%d%d\",&n,&m);\n        if(n%(m+1)) printf(\"first\\n\");\n        else printf(\"second\\n\");\n    }\n    return 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