Ducci序列(Ducci Sequence)

2023-05-16

Ducci Sequence

Time limit: 3.000 seconds

A Ducci sequence is a sequence of n-tuples of integers. Given an n-tuple of integers (a₁, a₂,^..., a_n), the next n-tuple in the sequence is formed by taking the absolute differences of neighboring integers:

( a ₁, a ₂, ^..., a _n) $\displaystyle \rightarrow$ (| a ₁ - a ₂|,| a ₂ - a ₃|, ^...,| a _n - a ₁|)

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) $\displaystyle \rightarrow$ (3, 9, 5, 1) $\displaystyle \rightarrow$ (6, 4, 4, 2) $\displaystyle \rightarrow$ (2, 0, 2, 4) $\displaystyle \rightarrow$ (2, 2, 2, 2) $\displaystyle \rightarrow$ (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) $\displaystyle \rightarrow$ (2, 2, 2, 2, 4) $\displaystyle \rightarrow$ ( 0, 0, 0, 2, 2) $\displaystyle \rightarrow$ (0, 0, 2, 0, 2) $\displaystyle \rightarrow$ (0, 2, 2, 2, 2) $\displaystyle \rightarrow$ (2, 0, 0, 0, 2) $\displaystyle \rightarrow$

(2, 0, 0, 2, 0) $\displaystyle \rightarrow$ (2, 0, 2, 2, 2) $\displaystyle \rightarrow$ (2, 2, 0, 0, 0) $\displaystyle \rightarrow$ (0, 2, 0, 0, 2) $\displaystyle \rightarrow$ (2, 2, 0, 2, 2) $\displaystyle \rightarrow$ (0, 2, 2, 0, 0) $\displaystyle \rightarrow$

(2, 0, 2, 0, 0) $\displaystyle \rightarrow$ (2, 2, 2, 0, 2) $\displaystyle \rightarrow$ (0, 0, 2, 2, 0) $\displaystyle \rightarrow$ (0, 2, 0, 2, 0) $\displaystyle \rightarrow$ (2, 2, 2, 2, 0) $\displaystyle \rightarrow$ ( 0, 0, 0, 2, 2) $\displaystyle \rightarrow$ ^...

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 $\le$ n $\le$ 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.

The following shows sample input and output for four test cases.

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

【分析】

由于输入保证最多1000步就会变成0或者循环，那么我们只需要在1005步内（当然只要大于1000步就可以，但也不能太大，因为考虑到效率问题）判断是否变成0即可，如果在1005步内没有变成0的话说明已经进入了循环。如何判断是否全变为0，只要将所有的数相加判断他们的总和是否等于0即可。

用java语言编写程序，代码如下：

import java.util.Scanner;

public class Main {
	public static void main(String[] args) {
		Scanner input = new Scanner(System.in);
		int[] ds = new int[20];
		int t = input.nextInt();
		for(int i = 0; i < t; i++) {
			int n = input.nextInt();
			
			int sum = 0;
			for(int j = 0; j < n; j++) {
				ds[j] = input.nextInt();
				sum += ds[j];
			}
			if(sum == 0) {
				System.out.println("ZERO");
				continue;
			}
			
			int temp = 0;
			boolean isLoop = true;
			for(int k = 0; k < 1005; k++) {
				temp = ds[0];
				sum = 0;
				for(int x = 0; x < n; x++) {
					if(x == n - 1)
						ds[x] = (int)(Math.abs(ds[x] - temp));
					else
						ds[x] = (int)(Math.abs(ds[x] - ds[x + 1]));
					
					sum += ds[x];
				}
				
				if(sum == 0) {
					isLoop = false;
					break;
				}
			}
			
			if(isLoop)
				System.out.println("LOOP");
			else
				System.out.println("ZERO");
		}
	}
}

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Ducci

Sequence