UVA Josephus Problem 11089 （约瑟夫环） 数学好题

Josephus Problem

Time Limit: 2000ms

Memory Limit: 32768KB

This problem will be judged on  LightOJ. Original ID:  1179
64-bit integer IO format:  %lld      Java class name:  Main

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The historian Flavius Josephus relates how, in the Romano-Jewish conflict of 67 A.D., the Romans took the town of Jotapata which he was commanding. Escaping, Josephus found himself trapped in a cave with 40 companions. The Romans discovered his whereabouts and invited him to surrender, but his companions refused to allow him to do so. He therefore suggested that they kill each other, one by one, the order to be decided by lot. Tradition has it that the means for affecting the lot was to stand in a circle, and, beginning at some point, count round, every third person being killed in turn. The sole survivor of this process was Josephus, who then surrendered to the Romans. Which begs the question: had Josephus previously practiced quietly with 41 stones in a dark corner, or had he calculated mathematically that he should adopt the 31^st position in order to survive?

Now you are in a similar situation. There are n persons standing in a circle. The persons are numbered from 1 to n circularly. For example, 1 and n are adjacent and 1 and 2 are also. The count starts from the first person. Each time you count up to k and the k^th person is killed and removed from the circle. Then the count starts from the next person. Finally one person remains. Given n and k you have to find the position of the last person who remains alive.

Input

Input starts with an integer T (≤ 200), denoting the number of test cases.

Each case contains two positive integers n (1 ≤ n ≤ 10⁵) and k (1 ≤ k < 2³¹).

Output

For each case, print the case number and the position of the last remaining person.

Sample Input

Sample Input	Output for Sample Input
6 2 1 2 2 3 1 3 2 3 3 4 6   #include #include using namespace std; int f[100010]; int main() { int t,T=1; int n,k,i; scanf("%d",&t); while(t--) { scanf("%d%d",&n,&k); f[1] = 0; for(i = 2; i <= n; i ++) //从零开始报数 { f[i] = (f[i-1] + k)%i; //数学推导当i个人数数时，赢家f[i]与f[i-1]有数学关系 } printf("Case %d: %d\n",T++,f[n]+1); } return 0; }	Case 1: 2 Case 2: 1 Case 3: 3 Case 4: 3 Case 5: 2 Case 6: 3