{"id":111702,"date":"2026-10-01T16:25:07","date_gmt":"2026-10-01T08:25:07","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/111702.html"},"modified":"2026-10-01T16:25:07","modified_gmt":"2026-10-01T08:25:07","slug":"uva-11061-playing-war","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/111702.html","title":{"rendered":"UVa 11061 Playing War"},"content":{"rendered":"

\u9898\u76ee\u63cf\u8ff0<\/h3>\n

\u201cWar\\\\texttt{War}<\/span><\/span>War<\/span><\/span><\/span><\/span><\/span><\/span>\u201d \u662f\u4e00\u6b3e\u7ecf\u5178\u7684\u68cb\u76d8\u6e38\u620f，\u5728\u5df4\u897f\u76f8\u5f53\u6709\u540d\u3002\u6e38\u620f\u4e2d，\u73a9\u5bb6\u4eec\u5171\u540c\u7edf\u6cbb\u6574\u4e2a\u4e16\u754c，\u6bcf\u4f4d\u73a9\u5bb6\u63a7\u5236\u4e00\u4e2a\u6216\u591a\u4e2a\u9886\u5730\u3002\u6bcf\u4f4d\u73a9\u5bb6\u53ef\u80fd\u6709\u4e0d\u540c\u7684\u76ee\u6807，\u4f46\u901a\u5e38\u9886\u5730\u8d8a\u591a\u8d8a\u597d，\u6240\u6709\u4eba\u90fd\u4e3a\u6b64\u800c\u6218\u3002<\/p>\n

\u6bcf\u4e2a\u9886\u5730\u88ab\u5176\u62e5\u6709\u8005\u7684\u82e5\u5e72 \u201c\u519b\u961f\u201d \u5360\u9886\u3002\u6bcf\u56de\u5408，\u73a9\u5bb6\u53ef\u4ee5\u7528\u4e00\u4e2a\u9886\u5730\u4e0a\u7684\u519b\u961f\u653b\u51fb\u76f8\u90bb\u9886\u5730\u5e76\u8bd5\u56fe\u5f81\u670d\u5b83\u3002\u653b\u51fb\u65f6，\u8fdb\u653b\u65b9\u81f3\u5c11\u8981\u5728\u539f\u9886\u5730\u7559\u4e0b 11<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f，\u56e0\u6b64\u82e5\u8fdb\u653b\u65b9\u6709 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f，\u6700\u591a\u53ea\u80fd\u7528 a\u22121a – 1<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e2a\u53bb\u653b\u51fb，\u4e14\u5355\u6b21\u6218\u6597\u4e2d\u53cc\u65b9\u6700\u591a\u53ea\u80fd\u4f7f\u7528 33<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f\u3002\u4e5f\u5c31\u662f\u8bf4，\u65e0\u8bba\u8fdb\u653b\u65b9\u9886\u5730\u6709\u591a\u5c11\u519b\u961f（\u22654\\\\ge 4<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>），\u6bcf\u6b21\u653b\u51fb\u6700\u591a\u4f7f\u7528 33<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span> \u4e2a；\u9632\u5b88\u65b9\u540c\u6837\u6700\u591a\u4f7f\u7528 33<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span> \u4e2a\u3002<\/p>\n

\u6218\u6597\u901a\u8fc7\u6295\u63b7\u666e\u901a 66<\/span><\/span>6<\/span><\/span><\/span><\/span><\/span> \u9762\u9ab0\u5b50\u8fdb\u884c\u3002\u8fdb\u653b\u65b9\u548c\u9632\u5b88\u65b9\u5404\u81ea\u6295\u63b7\u4e0e\u6240\u7528\u519b\u961f\u6570\u76f8\u7b49\u7684\u9ab0\u5b50（\u6700\u591a 33<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span> \u4e2a）\u3002\u7136\u540e\u6309\u4ee5\u4e0b\u89c4\u5219\u51b3\u5b9a\u53cc\u65b9\u635f\u5931\u7684\u519b\u961f\u6570：<\/p>\n

  • \u5c06\u53cc\u65b9\u7684\u9ab0\u5b50\u7ed3\u679c\u5206\u522b\u6309\u975e\u9012\u589e\u987a\u5e8f\u6392\u5217；<\/li>\n
  • \u5c06\u8fdb\u653b\u65b9\u7684\u6700\u9ad8\u9ab0\u5b50\u4e0e\u9632\u5b88\u65b9\u7684\u6700\u9ad8\u9ab0\u5b50\u6bd4\u8f83，\u6b21\u9ad8\u4e0e\u6b21\u9ad8\u6bd4\u8f83，\u4f9d\u6b64\u7c7b\u63a8，\u76f4\u5230\u67d0\u4e00\u65b9\u6ca1\u6709\u66f4\u591a\u9ab0\u5b50\u53ef\u6bd4\u8f83；<\/li>\n
  • \u6bcf\u6b21\u6bd4\u8f83\u4e2d，\u82e5\u8fdb\u653b\u65b9\u9ab0\u5b50\u70b9\u6570\u5927\u4e8e\u9632\u5b88\u65b9，\u5219\u8fdb\u653b\u65b9\u83b7\u80dc（\u9632\u5b88\u65b9\u635f\u5931 11<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f）；\u82e5\u5e73\u5c40\u6216\u5c0f\u4e8e，\u5219\u9632\u5b88\u65b9\u83b7\u80dc（\u8fdb\u653b\u65b9\u635f\u5931 11<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f）\u3002<\/li>\n

    \u73a9\u5bb6\u53ef\u4ee5\u5728\u540c\u4e00\u56de\u5408\u5185\u5bf9\u4e00\u4e2a\u654c\u65b9\u9886\u5730\u53d1\u8d77\u8fde\u7eed\u591a\u6b21\u653b\u51fb，\u6bcf\u6b21\u4f7f\u7528\u5269\u4f59\u7684\u519b\u961f\u5e76\u9075\u5b88\u6bcf\u6b21\u6700\u591a 33<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span> \u4e2a\u7684\u9650\u5236\u3002\u901a\u5e38\u73a9\u5bb6\u4f1a\u5c3d\u91cf\u591a\u7528\u519b\u961f，\u4f46\u4e5f\u6709\u4e00\u4e9b \u201c\u795e\u7ecf\u8d28\u201d \u7684\u73a9\u5bb6，\u4e00\u65e6\u5f00\u59cb\u653b\u51fb\u67d0\u4e2a\u654c\u65b9\u9886\u5730，\u5c31\u4f1a\u4e00\u76f4\u653b\u51fb\u4e0b\u53bb，\u76f4\u5230\u5f81\u670d\u8be5\u9886\u5730（\u9632\u5b88\u65b9\u519b\u961f\u4e3a 00<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>） \u6216 \u65e0\u6cd5\u7ee7\u7eed\u653b\u51fb（\u8fdb\u653b\u65b9\u53ea\u5269 11<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f） \u4e3a\u6b62\u3002<\/p>\n

    Elbdson\\\\texttt{Elbdson}<\/span><\/span>Elbdson<\/span><\/span><\/span><\/span><\/span><\/span> \u5c31\u662f\u8fd9\u6837\u7684\u795e\u7ecf\u8d28\u73a9\u5bb6，\u540c\u65f6\u4ed6\u4e5f\u662f\u4e2a\u4f18\u79c0\u7684\u7a0b\u5e8f\u5458\u3002\u56e0\u6b64，\u5f53\u4ed6\u9080\u8bf7\u670b\u53cb\u73a9 War\\\\texttt{War}<\/span><\/span>War<\/span><\/span><\/span><\/span><\/span><\/span> \u65f6，\u7b2c\u4e00\u4ef6\u4e8b\u5c31\u662f\u5199\u7a0b\u5e8f\u8ba1\u7b97\u81ea\u5df1\u8fdb\u653b\u65f6\u7684\u4f18\u52bf\u3002\u6362\u53e5\u8bdd\u8bf4，Elbdson\\\\texttt{Elbdson}<\/span><\/span>Elbdson<\/span><\/span><\/span><\/span><\/span><\/span> \u60f3\u77e5\u9053：\u4e3a\u4e86\u653b\u51fb\u4e00\u4e2a\u62e5\u6709 XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f\u7684\u654c\u65b9\u9886\u5730，\u5e76\u4e14\u6709\u8d85\u8fc7 50%50\\\\%<\/span><\/span>50%<\/span><\/span><\/span><\/span><\/span> \u7684\u6982\u7387\u5f81\u670d\u5b83，\u4ed6\u81f3\u5c11\u9700\u8981\u5728\u8be5\u9886\u5730\u62e5\u6709\u591a\u5c11\u4e2a\u519b\u961f\u3002\u73b0\u5728\u8bf7\u4f60\u5b8c\u6210\u540c\u6837\u7684\u8ba1\u7b97\u3002<\/p>\n

    \u8f93\u5165\u683c\u5f0f<\/h3>\n

    \u8f93\u5165\u5305\u542b\u591a\u4e2a\u6d4b\u8bd5\u96c6，\u6bcf\u4e2a\u6d4b\u8bd5\u96c6\u4e3a\u4e00\u884c\u4e00\u4e2a\u6574\u6570 XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span>（1\u2264X\u226410001 \\\\le X \\\\le 1000<\/span><\/span>1<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>X<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>1000<\/span><\/span><\/span><\/span><\/span>），\u8868\u793a\u654c\u65b9\u9886\u5730\u7684\u519b\u961f\u6570\u3002\u8f93\u5165\u4ee5 X=0X = 0<\/span><\/span>X<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u7ed3\u675f，\u8be5\u884c\u4e0d\u5904\u7406\u3002<\/p>\n

    \u8f93\u51fa\u683c\u5f0f<\/h3>\n

    \u5bf9\u4e8e\u6bcf\u4e2a\u6d4b\u8bd5\u96c6，\u8f93\u51fa\u4e00\u884c\u4e00\u4e2a\u6574\u6570，\u5373 Elbdson\\\\texttt{Elbdson}<\/span><\/span>Elbdson<\/span><\/span><\/span><\/span><\/span><\/span> \u9700\u8981\u7684\u6700\u5c11\u521d\u59cb\u519b\u961f\u6570，\u4f7f\u5f97\u5f81\u670d\u6982\u7387\u5927\u4e8e 0.50.5<\/span><\/span>0.5<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

    \u6837\u4f8b<\/h3>\n

    \u8f93\u5165<\/p>\n

    1
    \n2
    \n3
    \n0<\/p>\n

    \u8f93\u51fa<\/p>\n

    3
    \n4
    \n6<\/p>\n

    \u9898\u76ee\u5206\u6790<\/h3>\n

    \u672c\u9898\u662f\u4e00\u4e2a\u6982\u7387\u8ba1\u7b97\u95ee\u9898，\u6838\u5fc3\u5728\u4e8e\u6a21\u62df\u968f\u673a\u6218\u6597\u8fc7\u7a0b，\u5e76\u6c42\u51fa\u7ed9\u5b9a\u9632\u5b88\u65b9\u519b\u961f\u6570 XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u65f6，\u6700\u5c0f\u7684\u8fdb\u653b\u65b9\u521d\u59cb\u519b\u961f\u6570 NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>，\u4f7f\u5f97\u4ece\u72b6\u6001 (N,X)(N, X)<\/span><\/span>(<\/span>N<\/span>,<\/span><\/span>X<\/span>)<\/span><\/span><\/span><\/span><\/span> \u51fa\u53d1，\u6700\u7ec8\u5f81\u670d（\u9632\u5b88\u65b9\u519b\u961f\u964d\u4e3a 00<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>）\u7684\u6982\u7387\u5927\u4e8e 0.50.5<\/span><\/span>0.5<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

    \u6218\u6597\u8fc7\u7a0b\u662f\u9a6c\u5c14\u53ef\u592b\u94fe，\u72b6\u6001\u7531 (a,d)(a, d)<\/span><\/span>(<\/span>a<\/span>,<\/span><\/span>d<\/span>)<\/span><\/span><\/span><\/span><\/span> \u8868\u793a，\u5176\u4e2d aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u4e3a\u8fdb\u653b\u65b9\u519b\u961f\u6570，dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u4e3a\u9632\u5b88\u65b9\u519b\u961f\u6570，\u4e14 d>0d > 0<\/span><\/span>d<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3002\u6bcf\u8f6e\u6218\u6597：<\/p>\n

      \n
    • \u8fdb\u653b\u65b9\u53ef\u4f7f\u7528\u7684\u9ab0\u5b50\u6570 A=min\u2061(3,a\u22121)A = \\\\min(3, a – 1)<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>min<\/span>(<\/span>3<\/span>,<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>（\u5fc5\u987b\u4fdd\u7559 11<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e2a\u5728\u539f\u9886\u5730）；<\/li>\n
    • \u9632\u5b88\u65b9\u53ef\u4f7f\u7528\u7684\u9ab0\u5b50\u6570 D=min\u2061(3,d)D = \\\\min(3, d)<\/span><\/span>D<\/span><\/span>=<\/span><\/span><\/span><\/span>min<\/span>(<\/span>3<\/span>,<\/span><\/span>d<\/span>)<\/span><\/span><\/span><\/span><\/span>；<\/li>\n
    • \u5b9e\u9645\u6bd4\u8f83\u6b21\u6570 m=min\u2061(A,D)m = \\\\min(A, D)<\/span><\/span>m<\/span><\/span>=<\/span><\/span><\/span><\/span>min<\/span>(<\/span>A<\/span>,<\/span><\/span>D<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n<\/ul>\n

      \u6295\u63b7 AA<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u4e2a\u8fdb\u653b\u9ab0\u5b50\u548c DD<\/span><\/span>D<\/span><\/span><\/span><\/span><\/span> \u4e2a\u9632\u5b88\u9ab0\u5b50\u540e，\u6839\u636e\u6bd4\u8f83\u7ed3\u679c，\u7edf\u8ba1\u8fdb\u653b\u65b9\u83b7\u80dc\u7684\u6b21\u6570 kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>（0\u2264k\u2264m0 \\\\le k \\\\le m<\/span><\/span>0<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>k<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>）\u3002\u5219：<\/p>\n

        \n
      • \u9632\u5b88\u65b9\u635f\u5931 kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f；<\/li>\n
      • \u8fdb\u653b\u65b9\u635f\u5931 m\u2212km – k<\/span><\/span>m<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f（\u56e0\u4e3a\u6bcf\u4e00\u5bf9\u6bd4\u8f83\u5fc5\u6709\u4e00\u65b9\u635f\u5931）\u3002<\/li>\n<\/ul>\n

        \u65b0\u7684\u72b6\u6001\u4e3a (a\u2212(m\u2212k),\u2005\u200ad\u2212k)(a – (m – k),\\\\; d – k)<\/span><\/span>(<\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>(<\/span>m<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span>)<\/span>,<\/span><\/span><\/span>d<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u82e5 d\u2212k=0d – k = 0<\/span><\/span>d<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u5219\u8fdb\u653b\u65b9\u83b7\u80dc；\u82e5 a\u2212(m\u2212k)\u22641a – (m – k) \\\\le 1<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>(<\/span>m<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span>)<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e14 d\u2212k>0d – k > 0<\/span><\/span>d<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u5219\u8fdb\u653b\u65b9\u5931\u8d25（\u65e0\u6cd5\u7ee7\u7eed\u653b\u51fb）\u3002<\/p>\n

        \u7531\u4e8e\u9ab0\u5b50\u7ed3\u679c\u72ec\u7acb\u4e14\u7b49\u6982\u7387，\u6211\u4eec\u53ef\u4ee5\u9884\u5148\u8ba1\u7b97\u51fa\u5bf9\u4e8e\u6bcf\u5bf9 (A,D)(A, D)<\/span><\/span>(<\/span>A<\/span>,<\/span><\/span>D<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u8fdb\u653b\u65b9\u83b7\u80dc\u6b21\u6570 kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u7684\u6982\u7387\u5206\u5e03 prob[A][D][k]\\\\textit{prob}[A][D][k]<\/span><\/span>prob<\/span><\/span>[<\/span>A<\/span>]<\/span>[<\/span>D<\/span>]<\/span>[<\/span>k<\/span>]<\/span><\/span><\/span><\/span><\/span>\u3002\u7136\u540e\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u4ece\u72b6\u6001 (a,d)(a, d)<\/span><\/span>(<\/span>a<\/span>,<\/span><\/span>d<\/span>)<\/span><\/span><\/span><\/span><\/span> \u8f6c\u79fb\u5230\u66f4\u5c0f\u7684\u72b6\u6001（\u56e0\u4e3a aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u548c dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u90fd\u662f\u975e\u589e\u7684，\u4e14\u81f3\u5c11\u6709\u4e00\u4e2a\u4e25\u683c\u51cf\u5c11），\u6700\u7ec8\u6c42\u51fa\u83b7\u80dc\u6982\u7387\u3002<\/p>\n

        \u76f4\u63a5\u5bf9\u6bcf\u4e2a\u6d4b\u8bd5\u7528\u4f8b\u72ec\u7acb DP\\\\texttt{DP}<\/span><\/span>DP<\/span><\/span><\/span><\/span><\/span><\/span>，\u6700\u5927 XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u4e3a 10001000<\/span><\/span>1000<\/span><\/span><\/span><\/span><\/span>，\u800c\u7b54\u6848 NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span> \u53ef\u80fd\u8fbe\u5230\u6570\u5343，\u91cd\u590d\u8ba1\u7b97\u4f1a\u8d85\u65f6\u3002\u56e0\u6b64\u6211\u4eec\u91c7\u7528\u4e00\u6b21\u6027\u9884\u5904\u7406\u6240\u6709 XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u7684\u7b56\u7565，\u6309\u8fdb\u653b\u65b9\u519b\u961f\u6570 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u9012\u589e\u7684\u987a\u5e8f，\u6eda\u52a8\u6570\u7ec4\u8ba1\u7b97\u6240\u6709 dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u7684\u83b7\u80dc\u6982\u7387，\u5e76\u5728\u8fc7\u7a0b\u4e2d\u8bb0\u5f55\u6bcf\u4e2a XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u9996\u6b21\u6ee1\u8db3\u6982\u7387 >0.5> 0.5<\/span><\/span>><\/span><\/span><\/span><\/span>0.5<\/span><\/span><\/span><\/span><\/span> \u65f6\u7684 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u503c\u3002<\/p>\n

        \u89e3\u9898\u601d\u8def<\/h3>\n

        1. \u9884\u8ba1\u7b97\u8f6c\u79fb\u6982\u7387<\/h4>\n

        \u5bf9\u4e8e\u6240\u6709\u53ef\u80fd\u7684 A,D\u2208{1,2,3}A, D \\\\in \\\\{1, 2, 3\\\\}<\/span><\/span>A<\/span>,<\/span><\/span>D<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>{<\/span>1<\/span>,<\/span><\/span>2<\/span>,<\/span><\/span>3<\/span>}<\/span><\/span><\/span><\/span><\/span>，\u679a\u4e3e\u6240\u6709\u9ab0\u5b50\u7ed3\u679c\u7ec4\u5408\u3002\u8fdb\u653b\u65b9\u6709 6A6^A<\/span><\/span>6<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u79cd\u7ed3\u679c，\u9632\u5b88\u65b9\u6709 6D6^D<\/span><\/span>6<\/span><\/span>D<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u79cd\u7ed3\u679c，\u603b\u5171\u6700\u591a 63\u00d763=466566^3 \\\\times 6^3 = 46656<\/span><\/span>6<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>6<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>46656<\/span><\/span><\/span><\/span><\/span> \u79cd\u7ec4\u5408\u3002\u5bf9\u6bcf\u79cd\u7ec4\u5408，\u5206\u522b\u6392\u5e8f\u540e\u6bd4\u8f83\u524d m=min\u2061(A,D)m = \\\\min(A, D)<\/span><\/span>m<\/span><\/span>=<\/span><\/span><\/span><\/span>min<\/span>(<\/span>A<\/span>,<\/span><\/span>D<\/span>)<\/span><\/span><\/span><\/span><\/span> \u5bf9，\u7edf\u8ba1\u8fdb\u653b\u65b9\u80dc\u5229\u7684\u6b21\u6570 kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u3002\u6700\u540e\u5c06\u9891\u6570\u9664\u4ee5\u603b\u6570，\u5f97\u5230\u6982\u7387 prob[A][D][k]\\\\textit{prob}[A][D][k]<\/span><\/span>prob<\/span><\/span>[<\/span>A<\/span>]<\/span>[<\/span>D<\/span>]<\/span>[<\/span>k<\/span>]<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

        \u8fd9\u4e00\u6b65\u53ea\u9700\u505a\u4e00\u6b21，\u53ef\u4ee5\u5728\u7a0b\u5e8f\u5f00\u59cb\u65f6\u5b8c\u6210\u3002<\/p>\n

        2. \u52a8\u6001\u89c4\u5212\u72b6\u6001\u5b9a\u4e49<\/h4>\n

        \u4ee4 dp[a][d]\\\\textit{dp}[a][d]<\/span><\/span>dp<\/span><\/span>[<\/span>a<\/span>]<\/span>[<\/span>d<\/span>]<\/span><\/span><\/span><\/span><\/span> \u8868\u793a\u5f53\u8fdb\u653b\u65b9\u6709 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f\u3001\u9632\u5b88\u65b9\u6709 dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u4e2a\u519b\u961f\u65f6，\u8fdb\u653b\u65b9\u6700\u7ec8\u5f81\u670d\u7684\u6982\u7387\u3002\u8fb9\u754c\u6761\u4ef6：<\/p>\n

          \n
        • \u82e5 d=0d = 0<\/span><\/span>d<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u5219\u5df2\u5f81\u670d，\u6982\u7387\u4e3a 11<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>（\u4f46\u6211\u4eec\u7684\u72b6\u6001\u59cb\u7ec8\u4fdd\u8bc1 d>0d > 0<\/span><\/span>d<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u8f6c\u79fb\u65f6\u5904\u7406）；<\/li>\n
        • \u82e5 a\u22641a \\\\le 1<\/span><\/span>a<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e14 d>0d > 0<\/span><\/span>d<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u5219\u65e0\u6cd5\u653b\u51fb，\u6982\u7387\u4e3a 00<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n<\/ul>\n

          \u5bf9\u4e8e a\u22652,d\u22651a \\\\ge 2, d \\\\ge 1<\/span><\/span>a<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>2<\/span>,<\/span><\/span>d<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，\u8f6c\u79fb\u516c\u5f0f\u4e3a：<\/p>\n

          dp[a][d]=\u2211k=0mprob[A][D][k]\u00d7{1,d\u2212k=0,0,a\u2212(m\u2212k)\u22641\u00a0\u4e14\u00a0d\u2212k>0,dp[a\u2212(m\u2212k)][d\u2212k],otherwise.
          \n\\\\textit{dp}[a][d] = \\\\sum_{k=0}^{m} \\\\textit{prob}[A][D][k] \\\\times
          \n\\\\begin{cases}
          \n1, & d – k = 0,\\\\\\\\
          \n0, & a – (m-k) \\\\le 1 \\\\text{ \u4e14 } d – k > 0,\\\\\\\\
          \n\\\\textit{dp}[a – (m-k)][d – k], & \\\\text{otherwise}.
          \n\\\\end{cases}
          \n<\/span><\/span>dp<\/span><\/span>[<\/span>a<\/span>]<\/span>[<\/span>d<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>k<\/span>=<\/span>0<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>prob<\/span><\/span>[<\/span>A<\/span>]<\/span>[<\/span>D<\/span>]<\/span>[<\/span>k<\/span>]<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span><\/span><\/span>0<\/span>,<\/span><\/span><\/span><\/span>dp<\/span><\/span>[<\/span>a<\/span><\/span>\u2212<\/span><\/span>(<\/span>m<\/span><\/span>\u2212<\/span><\/span>k<\/span>)]<\/span>[<\/span>d<\/span><\/span>\u2212<\/span><\/span>k<\/span>]<\/span>,<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span>=<\/span><\/span>0<\/span>,<\/span><\/span><\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span>(<\/span>m<\/span><\/span>\u2212<\/span><\/span>k<\/span>)<\/span><\/span>\u2264<\/span><\/span>1<\/span>\u00a0<\/span>\u4e14<\/span>\u00a0<\/span><\/span>d<\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span>><\/span><\/span>0<\/span>,<\/span><\/span><\/span><\/span>otherwise<\/span><\/span>.<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

          \u5176\u4e2d A=min\u2061(3,a\u22121)A = \\\\min(3, a-1)<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>min<\/span>(<\/span>3<\/span>,<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>，D=min\u2061(3,d)D = \\\\min(3, d)<\/span><\/span>D<\/span><\/span>=<\/span><\/span><\/span><\/span>min<\/span>(<\/span>3<\/span>,<\/span><\/span>d<\/span>)<\/span><\/span><\/span><\/span><\/span>，m=min\u2061(A,D)m = \\\\min(A, D)<\/span><\/span>m<\/span><\/span>=<\/span><\/span><\/span><\/span>min<\/span>(<\/span>A<\/span>,<\/span><\/span>D<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

          3. \u8ba1\u7b97\u987a\u5e8f\u4e0e\u6eda\u52a8\u6570\u7ec4<\/h4>\n

          \u6ce8\u610f\u5230\u8f6c\u79fb\u540e\u7684\u72b6\u6001 (a\u2032,d\u2032)(a', d')<\/span><\/span>(<\/span>a<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>d<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u6ee1\u8db3 a\u2032\u2264aa' \\\\le a<\/span><\/span>a<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>，d\u2032\u2264dd' \\\\le d<\/span><\/span>d<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>d<\/span><\/span><\/span><\/span><\/span>，\u4e14\u81f3\u5c11\u4e00\u4e2a\u4e25\u683c\u51cf\u5c0f（\u56e0\u4e3a kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u548c m\u2212km-k<\/span><\/span>m<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u4e0d\u80fd\u540c\u65f6\u4e3a 00<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>）\u3002\u56e0\u6b64，\u6211\u4eec\u53ef\u4ee5\u6309 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u4ece\u5c0f\u5230\u5927\u9012\u589e，\u5bf9\u4e8e\u6bcf\u4e2a aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>，\u518d\u6309 dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u4ece\u5c0f\u5230\u5927\u9012\u589e\u8ba1\u7b97，\u8fd9\u6837\u6240\u6709\u4f9d\u8d56\u7684\u72b6\u6001 a\u2032<aa' < a<\/span><\/span>a<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u6216 a\u2032=aa' = a<\/span><\/span>a<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u4e14 d\u2032<dd' < d<\/span><\/span>d<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><<\/span><\/span><\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u90fd\u5df2\u7ecf\u8ba1\u7b97\u5b8c\u6bd5\u3002<\/p>\n

          \u56e0\u4e3a aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u53ef\u80fd\u8fbe\u5230\u6570\u5343，\u4f46\u8f6c\u79fb\u53ea\u4f9d\u8d56\u6700\u8fd1 44<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span> \u4e2a aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u503c（\u56e0\u4e3a a\u2212(m\u2212k)a – (m-k)<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>(<\/span>m<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span>)<\/span><\/span><\/span><\/span><\/span> \u4e0e aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u7684\u5dee\u6700\u5927\u4e3a m\u22643m \\\\le 3<\/span><\/span>m<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>），\u6240\u4ee5\u53ef\u4ee5\u7528\u6eda\u52a8\u6570\u7ec4 dp[4][d]\\\\textit{dp}[4][d]<\/span><\/span>dp<\/span><\/span>[<\/span>4<\/span>]<\/span>[<\/span>d<\/span>]<\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d\u7b2c\u4e00\u7ef4\u4e3a a\u200amod\u200a4a \\\\bmod 4<\/span><\/span>a<\/span><\/span><\/span>mod<\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

          4. \u4e00\u6b21\u6027\u9884\u5904\u7406\u6240\u6709\u7b54\u6848<\/h4>\n

          \u8bfb\u5165\u6240\u6709\u67e5\u8be2\u7684 XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span>，\u8bb0\u5f55\u6700\u5927\u503c maxX\\\\textit{maxX}<\/span><\/span>maxX<\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u7136\u540e\u4ece a=2a = 2<\/span><\/span>a<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span> \u5f00\u59cb\u9012\u589e，\u5bf9\u6bcf\u4e2a aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u8ba1\u7b97\u6240\u6709 d\u2208[1,maxX]d \\\\in [1, \\\\textit{maxX}]<\/span><\/span>d<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>[<\/span>1<\/span>,<\/span><\/span>maxX<\/span><\/span>]<\/span><\/span><\/span><\/span><\/span> \u7684 dp[a\u200amod\u200a4][d]\\\\textit{dp}[a \\\\bmod 4][d]<\/span><\/span>dp<\/span><\/span>[<\/span>a<\/span><\/span><\/span>mod<\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>]<\/span>[<\/span>d<\/span>]<\/span><\/span><\/span><\/span><\/span>\u3002\u6bcf\u8ba1\u7b97\u5b8c\u4e00\u4e2a aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>，\u68c0\u67e5\u6240\u6709\u5c1a\u672a\u627e\u5230\u7b54\u6848\u7684\u67e5\u8be2 XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span>，\u82e5 dp[a\u200amod\u200a4][X]>0.5\\\\textit{dp}[a \\\\bmod 4][X] > 0.5<\/span><\/span>dp<\/span><\/span>[<\/span>a<\/span><\/span><\/span>mod<\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>]<\/span>[<\/span>X<\/span>]<\/span><\/span>><\/span><\/span><\/span><\/span>0.5<\/span><\/span><\/span><\/span><\/span>，\u5219\u8bb0\u5f55\u7b54\u6848 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>，\u5e76\u6807\u8bb0\u8be5\u67e5\u8be2\u5df2\u89e3\u51b3\u3002\u5f53\u6240\u6709\u67e5\u8be2\u90fd\u627e\u5230\u7b54\u6848\u65f6，\u505c\u6b62\u5faa\u73af\u3002<\/p>\n

          5. \u6b63\u786e\u6027\u8bf4\u660e<\/h4>\n
            \n
          • \u9884\u8ba1\u7b97\u7684\u6982\u7387\u5206\u5e03\u51c6\u786e\u53cd\u6620\u4e86\u5355\u8f6e\u6218\u6597\u7684\u968f\u673a\u6027\u3002<\/li>\n
          • \u52a8\u6001\u89c4\u5212\u8f6c\u79fb\u65b9\u7a0b\u5b8c\u6574\u63cf\u8ff0\u4e86\u6240\u6709\u53ef\u80fd\u7684\u6218\u6597\u7ed3\u679c\u53ca\u5176\u540e\u7eed\u6982\u7387\u3002<\/li>\n
          • \u56e0\u4e3a\u72b6\u6001\u7a7a\u95f4\u968f\u7740 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u548c dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u5355\u8c03\u51cf\u5c0f，\u5faa\u73af\u987a\u5e8f\u6b63\u786e，\u6eda\u52a8\u6570\u7ec4\u4fdd\u7559\u4e86\u5fc5\u8981\u7684\u5386\u53f2\u4fe1\u606f\u3002<\/li>\n
          • \u5f53 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u589e\u5927\u65f6，\u83b7\u80dc\u6982\u7387\u5355\u8c03\u4e0d\u51cf（\u56e0\u4e3a\u66f4\u591a\u519b\u961f\u603b\u662f\u66f4\u6709\u5229），\u56e0\u6b64\u7b2c\u4e00\u4e2a\u6ee1\u8db3\u6761\u4ef6\u7684 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u5c31\u662f\u6700\u5c0f\u503c\u3002<\/li>\n<\/ul>\n

            6. \u590d\u6742\u5ea6\u5206\u6790<\/h4>\n
              \n
            • \u9884\u8ba1\u7b97：\u679a\u4e3e 3\u00d733 \\\\times 3<\/span><\/span>3<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span> \u5bf9 (A,D)(A, D)<\/span><\/span>(<\/span>A<\/span>,<\/span><\/span>D<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u6bcf\u5bf9\u679a\u4e3e 6A\u00d76D\u2264466566^A \\\\times 6^D \\\\le 46656<\/span><\/span>6<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>6<\/span><\/span>D<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>46656<\/span><\/span><\/span><\/span><\/span> \u79cd\u7ec4\u5408，\u6392\u5e8f\u548c\u6bd4\u8f83\u5f00\u9500\u5f88\u5c0f，\u603b\u65f6\u95f4\u5e38\u6570\u3002<\/li>\n
            • \u52a8\u6001\u89c4\u5212：\u5047\u8bbe\u6700\u5927\u6240\u9700 aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u4e3a MM<\/span><\/span>M<\/span><\/span><\/span><\/span><\/span>（\u5b9e\u6d4b\u4e0d\u8d85\u8fc7 1000010000<\/span><\/span>10000<\/span><\/span><\/span><\/span><\/span>），\u5219\u72b6\u6001\u6570\u4e3a M\u00d7maxXM \\\\times \\\\textit{maxX}<\/span><\/span>M<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>maxX<\/span><\/span><\/span><\/span><\/span><\/span>，\u6bcf\u4e2a\u72b6\u6001\u8f6c\u79fb\u6700\u591a 44<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span> \u79cd kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>（\u56e0\u4e3a m\u22643m \\\\le 3<\/span><\/span>m<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>），\u603b\u65f6\u95f4\u590d\u6742\u5ea6 O(M\u22c5maxX\u22c54)O(M \\\\cdot \\\\textit{maxX} \\\\cdot 4)<\/span><\/span>O<\/span>(<\/span>M<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>maxX<\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>4<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d MM<\/span><\/span>M<\/span><\/span><\/span><\/span><\/span> \u7ea6\u4e3a 2000020000<\/span><\/span>20000<\/span><\/span><\/span><\/span><\/span> \u4ee5\u5185，maxX\u22641000\\\\textit{maxX} \\\\le 1000<\/span><\/span>maxX<\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>1000<\/span><\/span><\/span><\/span><\/span>，\u5b8c\u5168\u53ef\u884c\u3002<\/li>\n
            • \u7a7a\u95f4\u590d\u6742\u5ea6：\u6eda\u52a8\u6570\u7ec4 O(4\u22c5maxX)O(4 \\\\cdot \\\\textit{maxX})<\/span><\/span>O<\/span>(<\/span>4<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>maxX<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n<\/ul>\n

              \u4ee3\u7801\u5b9e\u73b0<\/h3>\n

              \/\/ Playing War<\/span>
              \n\/\/ UVa ID: 11061<\/span>
              \n\/\/ Verdict: Accepted<\/span>
              \n\/\/ Submission Date: 2026-06-21<\/span>
              \n\/\/ UVa Run Time: 0.640s<\/span>
              \n\/\/<\/span>
              \n\/\/ \u7248\u6743\u6240\u6709（C）2026，\u90b1\u79cb\u3002metaphysis # yeah dot net<\/span><\/p>\n

              #<\/span>include<\/span> <bits\/stdc++.h><\/span><\/span>
              \nusing<\/span> namespace<\/span> std;<\/span><\/p>\n

              double<\/span> prob[<\/span>4<\/span>]<\/span>[<\/span>4<\/span>]<\/span>[<\/span>4<\/span>]<\/span>;<\/span> \/\/ prob[A][D][k]：A\u4e2a\u653b\u51fb\u9ab0\u5b50\u3001D\u4e2a\u9632\u5fa1\u9ab0\u5b50\u65f6，\u653b\u51fb\u65b9\u80dck\u6b21\u7684\u6982\u7387<\/span><\/p>\n

              \/\/ \u751f\u6210\u6240\u6709\u957f\u5ea6\u4e3anum\u7684\u9ab0\u5b50\u70b9\u6570（1~6）\u7ec4\u5408<\/span>
              \nvoid<\/span> genRolls<\/span>(<\/span>int<\/span> num,<\/span> vector<<\/span>int<\/span>><\/span>&<\/span> cur,<\/span> vector<<\/span>vector<<\/span>int<\/span>>><\/span>&<\/span> all)<\/span> {<\/span>
              \n if<\/span> (<\/span>(<\/span>int<\/span>)<\/span>cur.<\/span>size<\/span>(<\/span>)<\/span> ==<\/span> num)<\/span> {<\/span>
              \n all.<\/span>push_back<\/span>(<\/span>cur)<\/span>;<\/span>
              \n return<\/span>;<\/span>
              \n }<\/span>
              \n for<\/span> (<\/span>int<\/span> i =<\/span> 1<\/span>;<\/span> i <=<\/span> 6<\/span>;<\/span> ++<\/span>i)<\/span> {<\/span>
              \n cur.<\/span>push_back<\/span>(<\/span>i)<\/span>;<\/span>
              \n genRolls<\/span>(<\/span>num,<\/span> cur,<\/span> all)<\/span>;<\/span>
              \n cur.<\/span>pop_back<\/span>(<\/span>)<\/span>;<\/span>
              \n }<\/span>
              \n}<\/span><\/p>\n

              \/\/ \u9884\u8ba1\u7b97\u6240\u6709(A,D)\u7ec4\u5408\u4e0b\u7684\u6982\u7387\u5206\u5e03<\/span>
              \nvoid<\/span> precompute<\/span>(<\/span>)<\/span> {<\/span>
              \n for<\/span> (<\/span>int<\/span> A =<\/span> 1<\/span>;<\/span> A <=<\/span> 3<\/span>;<\/span> ++<\/span>A)<\/span> {<\/span>
              \n for<\/span> (<\/span>int<\/span> D =<\/span> 1<\/span>;<\/span> D <=<\/span> 3<\/span>;<\/span> ++<\/span>D)<\/span> {<\/span>
              \n vector<<\/span>vector<<\/span>int<\/span>>><\/span> att,<\/span> def;<\/span>
              \n vector<<\/span>int<\/span>><\/span> cur;<\/span>
              \n genRolls<\/span>(<\/span>A,<\/span> cur,<\/span> att)<\/span>;<\/span>
              \n genRolls<\/span>(<\/span>D,<\/span> cur,<\/span> def)<\/span>;<\/span>
              \n int<\/span> total =<\/span> (<\/span>int<\/span>)<\/span>att.<\/span>size<\/span>(<\/span>)<\/span> *<\/span> (<\/span>int<\/span>)<\/span>def.<\/span>size<\/span>(<\/span>)<\/span>;<\/span>
              \n int<\/span> m =<\/span> min<\/span>(<\/span>A,<\/span> D)<\/span>;<\/span>
              \n vector<<\/span>int<\/span>><\/span> cnt<\/span>(<\/span>m +<\/span> 1<\/span>,<\/span> 0<\/span>)<\/span>;<\/span>
              \n for<\/span> (<\/span>auto<\/span>&<\/span> a :<\/span> att)<\/span> {<\/span>
              \n sort<\/span>(<\/span>a.<\/span>begin<\/span>(<\/span>)<\/span>,<\/span> a.<\/span>end<\/span>(<\/span>)<\/span>,<\/span> greater<\/span><<\/span>int<\/span>><\/span><\/span><\/span>(<\/span>)<\/span>)<\/span>;<\/span>
              \n for<\/span> (<\/span>auto<\/span>&<\/span> d :<\/span> def)<\/span> {<\/span>
              \n sort<\/span>(<\/span>d.<\/span>begin<\/span>(<\/span>)<\/span>,<\/span> d.<\/span>end<\/span>(<\/span>)<\/span>,<\/span> greater<\/span><<\/span>int<\/span>><\/span><\/span><\/span>(<\/span>)<\/span>)<\/span>;<\/span>
              \n int<\/span> win =<\/span> 0<\/span>;<\/span>
              \n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> m;<\/span> ++<\/span>i)<\/span>
              \n if<\/span> (<\/span>a[<\/span>i]<\/span> ><\/span> d[<\/span>i]<\/span>)<\/span> ++<\/span>win;<\/span>
              \n ++<\/span>cnt[<\/span>win]<\/span>;<\/span>
              \n }<\/span>
              \n }<\/span>
              \n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <=<\/span> m;<\/span> ++<\/span>k)<\/span>
              \n prob[<\/span>A]<\/span>[<\/span>D]<\/span>[<\/span>k]<\/span> =<\/span> (<\/span>double<\/span>)<\/span>cnt[<\/span>k]<\/span> \/<\/span> total;<\/span>
              \n }<\/span>
              \n }<\/span>
              \n}<\/span><\/p>\n

              int<\/span> main<\/span>(<\/span>)<\/span> {<\/span>
              \n precompute<\/span>(<\/span>)<\/span>;<\/span><\/p>\n

              vector<<\/span>int<\/span>><\/span> queries;<\/span>
              \n int<\/span> x;<\/span>
              \n while<\/span> (<\/span>cin >><\/span> x &&<\/span> x !=<\/span> 0<\/span>)<\/span>
              \n queries.<\/span>push_back<\/span>(<\/span>x)<\/span>;<\/span>
              \n if<\/span> (<\/span>queries.<\/span>empty<\/span>(<\/span>)<\/span>)<\/span> return<\/span> 0<\/span>;<\/span><\/p>\n

              int<\/span> maxX =<\/span> *<\/span>max_element<\/span>(<\/span>queries.<\/span>begin<\/span>(<\/span>)<\/span>,<\/span> queries.<\/span>end<\/span>(<\/span>)<\/span>)<\/span>;<\/span>
              \n vector<<\/span>int<\/span>><\/span> ans<\/span>(<\/span>maxX +<\/span> 1<\/span>,<\/span> 0<\/span>)<\/span>;<\/span>
              \n vector<<\/span>bool<\/span>><\/span> found<\/span>(<\/span>maxX +<\/span> 1<\/span>,<\/span> false<\/span>)<\/span>;<\/span><\/p>\n

              \/\/ dp[4][maxX+1] \u6eda\u52a8\u6570\u7ec4，dp[i][d] \u8868\u793a\u8fdb\u653b\u65b9\u519b\u961f\u6570\u4e3a i（\u53d6\u6a214）\u65f6\u7684\u6982\u7387<\/span>
              \n vector<<\/span>vector<<\/span>double<\/span>>><\/span> dp<\/span>(<\/span>4<\/span>,<\/span> vector<\/span><<\/span>double<\/span>><\/span><\/span><\/span>(<\/span>maxX +<\/span> 1<\/span>,<\/span> 0.0<\/span>)<\/span>)<\/span>;<\/span><\/p>\n

              int<\/span> foundCount =<\/span> 0<\/span>;<\/span>
              \n for<\/span> (<\/span>int<\/span> a =<\/span> 2<\/span>;<\/span> ;<\/span> ++<\/span>a)<\/span> {<\/span>
              \n int<\/span> A =<\/span> min<\/span>(<\/span>3<\/span>,<\/span> a –<\/span> 1<\/span>)<\/span>;<\/span>
              \n int<\/span> curIdx =<\/span> a %<\/span> 4<\/span>;<\/span><\/p>\n

              for<\/span> (<\/span>int<\/span> d =<\/span> 1<\/span>;<\/span> d <=<\/span> maxX;<\/span> ++<\/span>d)<\/span> {<\/span>
              \n int<\/span> D =<\/span> min<\/span>(<\/span>3<\/span>,<\/span> d)<\/span>;<\/span>
              \n int<\/span> m =<\/span> min<\/span>(<\/span>A,<\/span> D)<\/span>;<\/span>
              \n double<\/span> sum =<\/span> 0.0<\/span>;<\/span>
              \n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <=<\/span> m;<\/span> ++<\/span>k)<\/span> {<\/span>
              \n int<\/span> la =<\/span> m –<\/span> k;<\/span> \/\/ \u8fdb\u653b\u65b9\u635f\u5931<\/span>
              \n int<\/span> ld =<\/span> k;<\/span> \/\/ \u9632\u5b88\u65b9\u635f\u5931<\/span>
              \n double<\/span> p =<\/span> prob[<\/span>A]<\/span>[<\/span>D]<\/span>[<\/span>k]<\/span>;<\/span>
              \n if<\/span> (<\/span>d –<\/span> ld ==<\/span> 0<\/span>)<\/span> sum +=<\/span> p;<\/span>
              \n else<\/span> if<\/span> (<\/span>a –<\/span> la <=<\/span> 1<\/span>)<\/span> sum +=<\/span> 0.0<\/span>;<\/span>
              \n else<\/span> sum +=<\/span> p *<\/span> dp[<\/span>(<\/span>a –<\/span> la)<\/span> %<\/span> 4<\/span>]<\/span>[<\/span>d –<\/span> ld]<\/span>;<\/span>
              \n }<\/span>
              \n dp[<\/span>curIdx]<\/span>[<\/span>d]<\/span> =<\/span> sum;<\/span>
              \n }<\/span><\/p>\n

              \/\/ \u68c0\u67e5\u662f\u5426\u6709\u67e5\u8be2\u9996\u6b21\u6ee1\u8db3\u6761\u4ef6<\/span>
              \n for<\/span> (<\/span>int<\/span> X :<\/span> queries)<\/span> {<\/span>
              \n if<\/span> (<\/span>!<\/span>found[<\/span>X]<\/span> &&<\/span> dp[<\/span>curIdx]<\/span>[<\/span>X]<\/span> ><\/span> 0.5<\/span> +<\/span> 1e-9<\/span>)<\/span> {<\/span>
              \n found[<\/span>X]<\/span> =<\/span> true<\/span>;<\/span>
              \n ans[<\/span>X]<\/span> =<\/span> a;<\/span>
              \n ++<\/span>foundCount;<\/span>
              \n }<\/span>
              \n }<\/span>
              \n if<\/span> (<\/span>foundCount ==<\/span> (<\/span>int<\/span>)<\/span>queries.<\/span>size<\/span>(<\/span>)<\/span>)<\/span> break<\/span>;<\/span>
              \n \/\/ \u5b89\u5168\u4e0a\u9650（\u7406\u8bba\u4e0a\u8db3\u591f，\u9632\u6b62\u610f\u5916）<\/span>
              \n if<\/span> (<\/span>a ><\/span> 100000<\/span>)<\/span> break<\/span>;<\/span>
              \n }<\/span><\/p>\n

              for<\/span> (<\/span>int<\/span> X :<\/span> queries)<\/span>
              \n cout <<<\/span> ans[<\/span>X]<\/span> <<<\/span> '\\\\n'<\/span>;<\/span><\/p>\n

              return<\/span> 0<\/span>;<\/span>
              \n}<\/span><\/p>\n

              \u603b\u7ed3<\/h3>\n

              \u672c\u9898\u7684\u6838\u5fc3\u662f\u6982\u7387\u52a8\u6001\u89c4\u5212，\u7ed3\u5408\u4e86\u9884\u8ba1\u7b97\u8f6c\u79fb\u6982\u7387\u548c\u6eda\u52a8\u6570\u7ec4\u4f18\u5316\u3002\u5173\u952e\u70b9\u5305\u62ec：<\/p>\n

                \n
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