题意：
变形的约瑟夫环，最初为每个人编号1到n，第i次删去报号为i的人，然后从它的下一个人开始重新从1开始报号，问最终剩下第几号人？
分析：
首先看一下裸的约瑟夫环问题：
共n个人，从1开始报数，报到k的人从环中退出，问最后剩下的一个人的编号是多少？
为取模方便，假设下标从0开始，倒推分析：
假设该轮有n个人，那么上一轮(n+1)人，编号为0的人上一轮编号为k，也即编号为f[n]的人上一轮编号为(f[n]+k)%(n+1)。
我们知道最后剩下的人在最后一轮编号肯定为0，那么这样不断倒推就可以推出其在第一轮的编号，也即他本来的编号。
#include<iostream>
using namespace std;
const int maxn = 105;
int f[maxn];
int main (void)
{
f[1] = 0;
int n; cin>>n;
int k; cin>>k;
for(int i = 2; i <= n; i++){
f[i] = (f[i - 1] + k)%i;
}
cout<<f[n] + 1<<endl;
}
那么对于这道题，只要不断改变k就好啦~
代码：
题目代码1：
写了一个超级超级慢的模拟，直接离线打表。
#include <cstdio>
#include<iostream>
using namespace std;
const int maxm = 2005, mod = 1e9+7;
int ans[5010] = {0,1,2,2,2,4,5,4,8,8,7,11,8,13,4,11,12,8,12,2,13,7,22,2,8,13,26,4,26,29,17,27,26,7,33,20,16,22,29,4,13,22,25,14,22,37,18,46,42,46,9,41,12,7,26,42,24,5,44,53,52,58,29,22,12,48,27,30,58,52,49,57,13,14,32,24,75,8,67,56,40,61,51,77,35,57,74,63,75,12,72,89,11,32,32,59,61,62,89,22,31,67,33,101,101,34,74,90,89,61,73,18,26,72,96,25,53,93,42,11,80,42,94,32,80,12,7,97,22,82,119,94,88,60,123,88,119,76,128,34,116,57,140,103,133,107,44,100,53,20,51,12,74,59,146,111,13,119,39,9,142,31,154,99,26,5,99,110,135,155,99,86,2,8,49,16,126,149,76,123,101,168,128,94,51,38,107,119,70,64,149,4,62,94,99,61,2,39,164,149,73,109,83,81,169,26,181,166,109,140,89,64,62,94,30,168,27,31,192,82,215,7,184,199,158,198,215,62,157,58,18,51,200,57,226,2,236,50,106,188,22,128,149,162,206,228,150,26,205,18,9,107,68,188,165,184,238,48,95,92,62,143,164,238,156,64,86,134,130,245,258,252,62,191,138,71,243,102,111,187,25,95,59,186,185,274,111,261,72,272,42,137,235,62,86,144,193,77,146,265,156,54,188,40,49,125,110,16,29,218,219,68,138,268,70,223,296,142,146,247,49,133,310,131,63,60,94,315,59,240,177,14,226,332,164,94,53,227,33,130,331,97,23,316,114,271,67,339,330,247,299,189,278,158,283,54,215,49,219,157,31,344,128,37,213,335,57,298,102,9,356,99,362,342,133,14,121,30,64,314,102,318,209,41,34,57,247,77,219,96,248,177,238,169,268,342,347,256,31,278,118,369,33,395,229,16,218,158,208,250,9,335,75,93,146,399,230,116,357,372,282,257,193,152,269,131,233,258,7,376,170,222,244,242,393,407,154,47,286,191,422,340,217,83,209,233,374,355,144,87,266,238,44,363,197,240,377,297,167,453,282,279,387,358,254,107,414,427,117,220,342,333,177,199,396,280,113,22,348,14,424,162,86,339,219,141,408,432,117,108,400,436,236,195,145,456,295,371,122,69,360,312,197,108,326,396,259,161,197,20,385,369,225,161,8,182,358,215,224,167,124,42,225,263,96,100,26,129,330,524,214,247,187,31,380,334,183,251,254,366,336,56,482,491,206,278,39,162,107,14,441,13,189,330,297,238,104,298,552,491,456,480,310,396,73,205,59,177,208,75,447,520,394,402,332,507,157,268,196,388,249,179,498,493,471,577,501,85,316,580,508,502,458,488,241,316,363,395,259,589,113,255,256,364,240,288,27,304,172,182,73,319,126,152,172,379,170,420,110,151,156,199,217,258,560,383,132,379,290,537,76,360,281,329,389,568,425,440,487,553,466,531,318,619,541,23,25,246,27,205,157,226,212,445,193,420,338,249,457,250,439,594,12,285,256,298,68,265,290,287,391,47,443,524,650,193,104,202,344,234,449,639,385,98,519,237,591,228,305,457,441,518,631,73,112,346,74,298,500,539,673,588,670,248,316,262,513,705,405,140,229,260,339,639,477,637,706,125,218,609,196,600,89,285,377,472,508,697,485,24,142,535,421,76,176,320,358,597,641,643,590,320,407,554,518,250,194,459,434,110,660,203,475,530,719,313,686,275,411,735,727,285,146,572,627,160,369,82,398,320,169,421,508,187,123,356,384,747,64,449,101,26,118,228,410,684,540,179,299,726,16,385,44,737,97,391,344,88,740,197,436,713,114,433,308,38,140,377,785,223,65,422,505,192,333,771,645,169,263,679,245,389,396,823,64,458,671,117,57,795,16,477,422,96,75,343,501,418,359,230,552,274,590,77,392,773,650,464,419,783,159,20,584,223,424,843,111,708,590,127,483,793,164,628,510,479,408,633,159,635,708,173,553,125,162,76,841,369,716,324,529,638,852,445,669,31,481,243,814,752,90,632,813,209,546,662,196,810,894,761,700,219,629,892,535,779,714,706,806,272,578,157,233,866,297,786,123,576,793,312,396,57,264,223,829,639,219,508,853,527,560,116,501,866,200,933,292,573,867,575,885,454,607,199,329,69,428,199,229,892,257,611,770,636,787,213,635,420,790,479,134,164,386,145,475,33,882,585,107,428,867,766,669,540,967,559,855,526,696,973,592,184,635,321,262,237,474,60,715,183,35,678,261,941,244,131,192,656,335,609,154,669,947,635,952,589,79,825,891,569,63,685,918,850,64,377,58,526,75,268,754,535,925,696,116,879,999,708,1019,880,984,71,923,810,394,127,450,198,409,835,348,143,524,194,178,54,545,274,840,608,604,14,819,417,787,509,965,322,1013,661,310,867,983,900,194,815,238,1061,262,1063,714,316,613,559,725,239,899,447,87,151,862,902,327,110,612,118,212,123,626,407,816,832,994,705,59,943,529,193,437,1068,660,410,982,377,838,728,1017,738,375,1023,489,81,758,433,774,938,936,734,281,1036,567,1010,477,194,830,309,939,902,984,1059,619,212,591,480,818,483,951,811,129,212,170,307,1020,781,112,958,62,125,402,181,676,726,1149,718,222,1091,506,329,988,634,843,639,436,1133,509,295,921,668,89,1023,376,96,829,471,966,1136,238,1135,283,184,1124,870,997,1024,479,1109,970,841,1009,62,387,264,875,668,118,789,320,46,765,891,692,819,614,295,1004,520,1013,914,473,280,610,775,1148,982,680,478,1017,1141,269,1093,686,550,1000,672,50,133,871,538,82,117,365,466,984,712,127,517,490,482,112,798,634,342,504,778,184,1153,656,567,748,1157,451,546,1168,1224,1252,104,836,566,321,185,322,184,282,2,681,263,836,956,389,63,1062,1195,176,65,797,737,195,360,678,394,1202,843,932,747,906,11,786,586,41,1214,244,209,1132,872,742,785,1209,961,481,183,1187,889,63,1235,1112,680,364,359,1073,1027,530,236,76,82,580,135,1057,776,424,835,960,994,860,574,341,1307,743,730,151,82,298,1133,46,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030,577,1290,980,1001,2819,2914,473,3393,2279,2135,1514,693,590,1766,1445,208,2974,2578,3451,3031,2797,2546,1145,1712,2681,1665,1465,1188,195,2766,402,270,2259,2024,652,2511,2712,2766,1879,1737,3543,417,1518,243,271,2267,2078,3575,2509,3482,2558,1744,804,733,1076,1748,1152,654,1917,3311,1115,285,3085,2865,1993,1219,3464,2937,1320,1411,116,128,1457,1304,3417,3406,2237,1932,667,193,2773,2282,728,2288,2873,2823,1519,1572,3365,2778,1375,1164,366,3313,1682,2001,3530,271,3355,2823,748,874,2642,2512,650,425,3494,101,627,1772,1098,809,2807,2231,83,918,2063,2187,944,1139,3634,3660,2325,2362,1054,1029,2627,1988,805,3672,2932,2631,1065,1386,3184,3319,2137,1881,130,115,2726,2092,1623,1889,3022,2852,1720,1626,433,3559,1669,2475,791,173,3413,3539,1026,2070,196,3208,1957,2030,972,3165,3162,2702,1712,2137,3323,3491,2029,2083,977,433,2638,2154,1249,1302,3495,3505,2234,2162,633,627,3212,332,1635,360,380,3293,2364,2856,3637,411,3168,3488,2739,1945,3311,196,2285,2161,648,1590,2348,3073,2983,1103,1362,540,1307,2544,1011,1300,3756,49,2142,2060,1065,710,3180,333,1415,1255,508,3037,183,769,3586,256,3770,399,2888,2130,3594,454,3394,3290,1733,2823,623,1463,2950,3661,1982,2869,3096,3576,3550,2225,2434,2035,2451,3795,2886,2272,1011,969,3057,3201,1956,1877,1347,1950,2864,2743,2510,2037,971,115,2042,3063,1964,1124,391,748,2690,3084,1932,1789,1105,1057,3140,2322,1543,612,3388,1405,1635,1974,2464,2176,1137,3726,2773,2785,1736,1672,3623,1384,2366,2625,2409,1679,474,840,2641,2560,1136,2281,336,199,1669,2202,1762,1542,386,3777,3226,3015,1762,2110,192,522,3387,2643,2482,2219,295,3845,2276,2874,1892,1441,1234,764,3026,598,3014,3345,1090,2088,3419,3375,1420,1648,1935,2587,3098,374,3779,2570,2196,2164,3532,793,2842,2715,2274,2688,1097,1312,266,3640,2051,3376,1321,981,568,3813,3100,3424,941,2310,493,42,3918,3354,2460,2178,750,1145,3862,710,2646,1839,1614,2081,526,895,2207,2453,1629,1715,1324,1018,2608,391,2843,2785,1936,2082,1239,2925,3268,3195,2478,3126,3031,984,887,3743,49,2579,1304,2396,23,486,2966,73,1701,2257,2142,2199,891,1525,2586,2910,2256,2745,860,880,2521,3649,3471,3135,3084,2671,546,87,3544,2944,2971,575,1254,497,411,1038,3763,3959,1760,2713,498,1205,1339,1644,2578,12,2565,2782,1885,1327,3883,3883,80,8,3006,3551,647,1441,785,847,783,3555,2198,2160,2129,1306,1503,791,2592,3550,3199,3227,2478,2271,1042,3768,3473,3914,2527,37,1166,2612,787,958,1434,1595,2681,2743,2720,1875,691,2034,3763,620,186,3762,3400,2421,1356,1926,208,907,3186,797,2295,3121,2616,2045,1013,2442,72,3794,3316,3594,3046,93,597,1409,691,1350,1015,419,3286,3372,1284,2695,1241,2618,63,4059,694,890,3176,3960,2007,1574,1131,2361,1053,455,2968,260,2705,3252,1734,2375,658,1138,1143,763,3584,68,2511,2870,2289,3144,821,1334,3415,620,3650,3180,3332,3775,1691,2199,1886,787,3845,839,3900,3276,2458,2303,2409,3539,3682,1952,4098,870,728,79,2596,2670,1842,1664,939,2654,20,1246,1087,1154,811,11,2771,1423,1841,1948,2463,1463,3199,647,4074,4020,3674,292,2572,3634,3097,2811,422,2618,3713,3582,76,3203,3993,3272,1769,3043,1527,1571,947,1793,2909,1163,4184,431,3312,4128,2322,2243,1069,1782,1175,3326,2361,939,3918,1193,3507,3662,1494,2574,1085,2075,1818,2938,1054,1155,970,1067,3656,892,2098,2244,2874,1917,2738,3069,713,1514,473,221,723,294,2253,3056,2751,2958,2440,3053,1460,1145,384,1357,1645,383,3860,3743,2912,3216,2235,3720,973,3041,1816,2418,1744,2229,581,3733,4135,999,3392,1921,1724,3551,3161,3118,3401,2998,1247,2306,890,1374,398,2199,3007,119,3842,4228,2784,3435,2949,2875,2117,2254,1693,1870,3784,445,438,4266,229,1513,3454,83,3448,3470,1784,3899,2620,2477,1759,2552,3017,1698,130,982,54,197,413,968,3227,339,3834,206,2824,3149,1923,1848,1411,2344,1446,3215,430,992,1051,920,354,1519,3404,769,4051,3301,2578,62,2792,3354,1989,3405,1569,2228,2170,2149,1306,884,889,2392,3738,1147,791,3867,4046,49,3032,3806,3368,4290,2067,94,1623,2950,1658,1835,2689,1938,629,1445,896,255,624,1508,51,1278,900,254,4071,862,3624,2468,3137,3690,3487,4292,941,2471,3659,2508,2303,2513,1561,1562,871,1010,554,1022,3432,4382,1188,1806,3953,1954,4104,19,2724,3839,3749,93,1861,3775,3650,3640,2611,2906,605,2845,1897,2804,2184,3946,2436,2058,968,1543,1332,1663,4048,1024,26,117,471,1273,3795,696,4172,4320,3314,449,3275,3591,3277,3134,3676,1364,1718,3057,2725,3246,3816,3371,1569,2426,2118,2721,1673,3621,1457,2444,2399,2364,2496,2741,1465,1487,436,1465,2383,2215,3778,785,597,698,327,1233,4260,4270,202,457,577,1181,3505,364,63,245,225,191,3156,434,3760,4299,3674,4011,2606,3711,3319,3609,3283,4462,3687,3806,2693,3064,2826,154,2010,3110,3191,3758,3632,4292,2444,3138,2140,3348,1858,3831,2438,2367,3168,2933,2749,4019,2292,2895,2131,2752,2217,2656,1606,2645,2175,3015,1520,3204,1970,1744,1776,2650,1921,4503,1880,1937,1760,2785,1803,1355,416,1920,1294,1548,1557,2569,973,3083,2432,3112,1502,2971,1050,1319,2197,856,2557,4053,4525,3183,1908,2915,2638,2203,1162,2446,1924,3945,1338,3064,2041,2874,2712,2581,2244,3442,2539,2589,2279,2669,2146,3609,1626,3350,2469,2625,2143,3895,2448,3163,2011,2997,2849,4487,2330,2728,3142,2977,3665,4256,2465,3838,2914,4461,3435,4116,1792,197,3590,4549,4243,567,3787,3932,3184,4386,180,1526,1786,1090,245,818,1178,884,3144,728,4091,677,4581,1832,359,1863,1044,1581,1405,2629,1031,1418,1987,1135,1561,2771,916,2319,1353,2438,1563,3414,1259,2831,1804,3032,2719,333,2761,2519,3868,3436,144,4223,1034,4124,3340,4388,44,1137,3503,1091,4075,646,207,1172,93,488,99,2003,1354,1876,4393,1069,1595,1961,2159,3238,1300,2984,2159,2737,2766,4624,1394,3286,3895,4528,3371,886,3215,3412,3977,335,4501,1205,2767,1331,664,1263,2035,2250,656,1625,1048,2255,1689,3440,1434,3102,3103,3131,3195,4114,1790,2874,3032,4176,3533,4706,3559,338,387,403,282,1874,456,991,453,1553,1375,4400,382,2266,2398,3045,3296,3311,2194,3652,3150,3991,3659,940,2998,1226,143,321,49,1594,1129,674,1748,1702,2164,2510,97,4079,2516,2562,3878,3376,2672,2967,4528,209,4473,1654,3762,51,729,1405,282,1068,413,1116,2218,2058,2042,2572,2483,4022,3495,3892,4663,1383,3523,4332,4246,542,483,2985,4166,1199,2875,2643,2786,3481,2313,3942,1062,3066,2597,333,3029,57,510,2078,810,1818,795,1142,353,2183,2621,3463,4672,2725,2648,3722,3915,199,3633,133,458,566,900,3013,920,1087,1085,2555,2990,3416,1570,3751,2494,4312,4607,1977,3911,1809,200,640,1231,2794,1527,1123,1988,2961,3272,4836,1421,33,3710,1205,1258,639,4147,1734,1347,2620,2497,4449,2519,4180,2921,4713,4871,1190,1420,462,822,1474,1569,2781,1012,2658,2043,2832,4511,4871,3378,3941,31,416,501,2912,769,1045,2162,3704,3558,3095,2971,4284,3292,4843,4483,1922,4381,2068,2575,3265,2536,3763,2632,3185,4375,770,4562,2405,3129,1049,1227,2996,2771,3151,2729,3951,3702,4562,4364,1468,4423,1111,1833,2433,2404,4365,2785,3569,3431,4820,4766,632,3885,2675,1009,2010,2261,4033,2816,3165,3747,3587,3737,2101,4452,1477,1858,3158,2898,3270,1863,3820,3496,3489,986,1809,308,1158,1893,3378,2392,3700,2532,2865,4966,4357,4816,2384,260,1850,1288,2891,4642,2998,3258,3804,4212,529,4696,3575,1443,1939,4077,3739,3503,1233,4964,3977,68};
int main (void)
{
int T;scanf("%d",&T);
while(T--){
int n;scanf("%d",&n);
printf("%d\n", ans[n]);
}
}题目代码2：
讲道理的递推做法
#include<cstdio>
const int maxn = 5050;
int f[maxn], ans[maxn];
int solve(int n)
{
f[1] = 0;
int k = n - 1;
for(int i = 2; i <= n; i++){
f[i] = (f[i - 1] + (k--))%i;
}
return f[n] + 1;
}
int main (void)
{
ans[0] = 0;
for(int i = 1; i < 5005; i++)
ans[i] = solve(i);
int T;scanf("%d",&T);
while(T--){
int t;scanf("%d",&t);
printf("%d\n",ans[t]);
}
return 0;
}
倒推思想