紫书第五章习题 5-2 Ducci 序列（Uva 1594)

A Ducci sequence is a sequence of n-tuples of integers. Given an n-tuple of integers (a1, a2, · · · , an),the next n-tuple in the sequence is formed by taking the absolute differences of neighboring integers:
(a1, a2, · · · , an) → (|a1 − a2|, |a2 − a3|, · · · , |an − a1|)
Ducci sequences either reach a tuple of zeros or fall into a periodic loop. For example, the 4-tuple sequence starting with 8,11,2,7 takes 5 steps to reach the zeros tuple:
(8, 11, 2, 7) → (3, 9, 5, 1) → (6, 4, 4, 2) → (2, 0, 2, 4) → (2, 2, 2, 2) → (0, 0, 0, 0).
The 5-tuple sequence starting with 4,2,0,2,0 enters a loop after 2 steps:
(4, 2, 0, 2, 0) → (2, 2, 2, 2, 4) → (0,0,0,2,2) → (0, 0, 2, 0, 2) → (0, 2, 2, 2, 2) → (2, 0, 0, 0, 2) →
(2, 0, 0, 2, 0) → (2, 0, 2, 2, 2) → (2, 2, 0, 0, 0) → (0, 2, 0, 0, 2) → (2, 2, 0, 2, 2) → (0, 2, 2, 0, 0) →
(2, 0, 2, 0, 0) → (2, 2, 2, 0, 2) → (0, 0, 2, 2, 0) → (0, 2, 0, 2, 0) → (2, 2, 2, 2, 0) → (0,0,0,2,2) → · · ·
Given an n-tuple of integers, write a program to decide if the sequence is reaching to a zeros tuple
or a periodic loop.

Input

Your program is to read the input from standard input. The input consists of T test cases. The number
of test cases T is given in the first line of the input. Each test case starts with a line containing an
integer n (3 ≤ n ≤ 15), which represents the size of a tuple in the Ducci sequences. In the following
line, n integers are given which represents the n-tuple of integers. The range of integers are from 0 to
1,000. You may assume that the maximum number of steps of a Ducci sequence reaching zeros tuple
or making a loop does not exceed 1,000.

Output

Your program is to write to standard output. Print exactly one line for each test case. Print ‘LOOP’ if
the Ducci sequence falls into a periodic loop, print ‘ZERO’ if the Ducci sequence reaches to a zeros tuple.

Sample Input

4
4
8 11 2 7
5
4 2 0 2 0
7
0 0 0 0 0 0 0
6
1 2 3 1 2 3

Sample Output

ZERO
LOOP
ZERO
LOOP

对于一个n元组（a1, a2, … , an），可以对于每个数求出它和下一个数的差的绝对值，得到一个新的 n 元组 （|a1-a2|, |a2-a3|, … , |an-a1|）。重复这个过程，得到的序列称之为 Ducci 序列。求Ducci数列如果在1000步以内循环了，就输出 LOOP，否则输出 ZERO
-
思路：就是一个简单的模拟，需要注意的一点是记得保存a1的值。

#include<bits/stdc++.h>
using namespace std;
int a[15]={0};
int main()
{
    int t,n;
    cin>>t;
    while(t--)
    {
        scanf("%d",&n);
        int cont=0;
        int flag=0;
        for(int i=0;i<n;i++)
        {
            cin>>a[i];
        }
        for(int k=1;k<=1000;k++)
        {
            int tmp=a[0];
            cont=0;
            for(int j=0;j<n-1;j++)
            {
                a[j]=abs(a[j]-a[j+1]);
            }
            a[n-1]=abs(a[n-1]-tmp);
            for(int i=0;i<n;i++)
            {
                if(a[i]==0)
                {
                    cont++;
                }
            }
            if(cont==n)
            {
                cout<<"ZERO"<<endl;
                flag=1;
                break;
            }
        }
        if(flag==0)
        cout<<"LOOP"<<endl;
    }
    return 0;
}