Problem Description

A ring is compose of n circles as shown in diagram. Put natural number 1, 2, ..., n into each circle separately, and the sum of numbers in two adjacent circles should be a prime.

Note: the number of first circle should always be 1.

Input

n (0 < n < 20).

Output

The output format is shown as sample below. Each row represents a series of circle numbers in the ring beginning from 1 clockwisely and anticlockwisely. The order of numbers must satisfy the above requirements. Print solutions in lexicographical order.

You are to write a program that completes above process.

Print a blank line after each case.

Sample Input

Sample Output

Source

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#include <iostream>
#include<cstdio>
#include<cstring>
#include<cmath>
using namespace std;

int n,ans[22];
bool isp[45];
bool vis[45];
void dfs(int p,int x)   //p為當前第幾個可行數 x為可行的數
{
    ans[p]=x;           
    vis[x]=1;
    if(p==n)             //當P等於n證明最後一個數與倒數第二個數相加為素數 隻需判斷是否跟1相加為素數
    {
        if(isp[ans[p]+1]) //判斷最後一個數是否與1相加為素數 如果是那麼該環成立 輸出結果
        {
            for(int i=1; i<=n; i++)
            {
                printf("%d",ans[i]);
                if(i<n)
                    printf(" ");
                else
                    printf("\n");
            }
            return ;
        }
    }
    for(int i=1; i<=n; i++)
    {
        if(!vis[i]&&isp[ans[p]+i])
        {
            dfs(p+1,i);//符合值進入下一層搜索
            vis[i]=0;  //不符合 回溯上一層
        }
    }
    return ;
}


int main()
{
    int s=1;
    memset(isp,1,sizeof(isp));
    isp[1]=0;
    isp[0]=0;
    for(int i=2; i<=int(sqrt(40)); i++)//打出素數表
    {
        if(isp[i])
        {
            for(int j=2*i; j<=40; j+=i)
                isp[j]=0;
        }
    }
    while(scanf("%d",&n)!=EOF)
    {
        memset(vis,0,sizeof(vis));
        cout<<"Case "<<s<<':'<<endl;
        dfs(1,1);                        
        s++;
        cout<<endl;
    }

    return 0;
}