Flip Game （高斯消元 || dfs）

Flip game is played on a rectangular 4x4 field with two-sided pieces placed on each of its 16 squares. One side of each piece is white and the other one is black and each piece is lying either it's black or white side up. Each round you flip 3 to 5 pieces, thus changing the color of their upper side from black to white and vice versa. The pieces to be flipped are chosen every round according to the following rules:

1. Choose any one of the 16 pieces.
2. Flip the chosen piece and also all adjacent pieces to the
   left, to the right, to the top, and to the bottom of the chosen piece
   (if there are any).

Consider the following position as an example:

bwbw

wwww

bbwb

bwwb

Here "b" denotes pieces lying their black side up and "w"
denotes pieces lying their white side up. If we choose to flip the 1st
piece from the 3rd row (this choice is shown at the picture), then the
field will become:

bwbw

bwww

wwwb

wwwb

The goal of the game is to flip either all pieces white side
up or all pieces black side up. You are to write a program that will
search for the minimum number of rounds needed to achieve this goal.

Input

The input consists of 4 lines with 4 characters "w" or "b" each that denote game field position.

Output

Write to the output file a single integer number - the minimum
number of rounds needed to achieve the goal of the game from the given
position. If the goal is initially achieved, then write 0. If it's
impossible to achieve the goal, then write the word "Impossible"
(without quotes).

Sample Input

bwwb
bbwb
bwwb
bwww

#include<stdio.h>
#include<string.h>
int c[][]= {,,,,-,,,-,,},min1,t;
char Map[][];

void turn(int x,int y)//改变状态；
{
    for(int i=; i<; i++)
    {
        int dx=x+c[i][];
        int dy=y+c[i][];
        if(dx<||dy<||dx>=||dy>=)continue;
        if(Map[dx][dy]=='b')
            Map[dx][dy]='w';
        else Map[dx][dy]='b';
    }
}

int check()//检查是否达到目标状态；
{
    int i,j;
    for(i=; i<; i++)
        for(j=; j<; j++)
            if(Map[i][j]!=Map[][])
                return ;
    return ;
}

void dfs(int x,int y,int s)
{
    if(check())//检查是否达到目标状态；
    {
        if(s<min1)//更新结果；
            min1=s;
        return ;
    }
    if(x==)return ;
    turn(x,y);//改变状态/回溯
    //翻转；
    if(y==)dfs(x+,,s+);//换行；
    else dfs(x,y+,s+);
    //不翻转；
    turn(x,y);//改变状态/回溯
    if(y==)dfs(x+,,s);
    else dfs(x,y+,s);
}

int main()
{
    int i;
    min1=;
    for(i=; i<; i++)
        scanf("%s",Map[i]);
    dfs(,,);
    if(min1<=)
        printf("%d\n",min1);
    else printf("Impossible\n");
}

// 高斯消元

#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std ;
#define INF 0x3f3f3f3f
int Map[][] , a[] , min1 , x[] , freex[] ;
char str[][] ;
void getMap(int k)
{
    int i , j ;
    memset(Map,,sizeof(Map)) ;
    for(i =  ; i <  ; i++)
    {
        for(j =  ; j <  ; j++)
        {
            a[i*+j] = (str[i][j] - '') ^ k ;
            Map[i*+j][i*+j] =  ;
            if( i >  )
                Map[i*+j][i*+j-] =  ;
            if( i <  )
                Map[i*+j][i*+j+] =  ;
            if( j >  )
                Map[i*+j][i*+j-] =  ;
            if( j <  )
                Map[i*+j][i*+j+] =  ;
        }
    }
    return ;
}
void swap1(int p,int q)
{
    int j , temp ;
    temp = a[p] ;
    a[p] = a[q] ;
    a[q] = temp ;
    for(j =  ; j <  ; j++)
    {
        temp = Map[p][j] ;
        Map[p][j] = Map[q][j] ;
        Map[q][j] = temp ;
    }
    return ;
}
int solve()
{
    int i , j , k , t =  , num1 =  ;
    for(i =  ; i <  && t <  ; t++ , i++)
    {
        for(j = i ; j <  ; j++)
            if( Map[j][t] )
                break ;
        if( j ==  )
        {
            freex[num1++] = t ;
            i-- ;
            continue ;
        }
        if( i != j )
            swap1(i,j) ;
        for(j = i+ ; j <  ; j++)
        {
            if( Map[j][t] )
            {
                a[j] = a[j]^a[i] ;
                for(k = t ; k <  ; k++)
                    Map[j][k] = Map[j][k] ^ Map[i][k] ;
            }
        }
    }
    for( ; i <  ; i++)
        if( a[i] )
            return - ;
    if( num1 ) return num1 ;
    for(i =  ; i >=  ; i--)
    {
        x[i] = a[i] ;
        for(j = i+ ; j <  ; j++)
            x[i] = x[i]^(Map[i][j]*x[j]) ;
    }
    return num1 ;
}
int f(int s)
{
    int i , j , k , key , ans , min1 = INF ;
    getMap(s) ;
    key = solve() ;
    ans =  ;
    if( key ==  )
    {
        ans =  ;
        for(i =  ; i <  ; i++)
            ans += x[i] ;
        min1 = min(min1,ans) ;
    }
    else if( key >  )
    {
        int temp = <<key ;
        for(int t =  ; t < temp ; t++)
        {
            memset(x,,sizeof(x)) ;
            ans =  ;
            for(j =  ; j < key ; j++)
                if( t & (<<j) )
                {
                    x[ freex[j] ] =  ;
                    ans++ ;
                }
            for(i =  ; i >=  ; i--)
            {
                for(k =  ; k <  ; k++)
                    if( Map[i][k] )
                        break ;
                x[k] = a[i] ;
                for(j = k+ ; j <  ; j++)
                    x[k] ^= (Map[i][j]*x[j]) ;
                ans += x[k] ;
            }
            min1 = min(min1,ans) ;
        }
    }
    return min1 ;
}
int main()
{
    int i , j , min1 = INF , k , ans , temp ;
    for(i =  ; i <  ; i++)
    {
        scanf("%s", str[i]) ;
        for(j =  ; j <  ; j++)
        {
            if( str[i][j] == 'b' )
                str[i][j] = '' ;
            else
                str[i][j] = '' ;
        }
    }
    ans = f() ;
    min1 = min(min1,ans) ;
    ans = f() ;
    min1 = min(min1,ans) ;
    if( min1 == INF )
        printf("Impossible\n") ;
    else
        printf("%d\n", min1) ;
    return ;
}