# Queens on Board

### Problem Statement :

```Queens on Board

You have an N * M chessboard on which some squares are blocked out. In how many ways can you place one or more queens on the board, such that, no two queens attack each other? Two queens attack each other, if one can reach the other by moving horizontally, vertically, or diagonally without passing over any blocked square. At most one queen can be placed on a square. A queen cannot be placed on a blocked square.

Input Format

The first line contains the number of test cases T. T test cases follow. Each test case contains integers N and M on the first line. The following N lines contain M characters each, and represent a board. A '#' represents a blocked square and a '.' represents an unblocked square.

Constraints

1 <= T <= 100
1 <= N <= 50
1 <= M <= 5

Output Format

Output T lines containing the required answer for each test case. As the answers can be really large, output them modulo 109+7.```

### Solution :

```                            ```Solution in C :

In C++ :

#include<stdio.h>
#include<string.h>
#define MOD 1000000007
int n,m,bit[1 << 10] ;
char g ;

int memo2[1 << 15] ;
{

for(int i = 0;i < m;i++)
{
if(mask & 1 << 3 * i) if(i > 0) nmask |= 1 << 3 * i - 3 ;
if(mask & 1 << 3 * i + 1) nmask |= 1 << 3 * i + 1 ;
if(mask & 1 << 3 * i + 2) if(i + 1 < m) nmask |= 1 << 3 * i + 5 ;
}
}

int good[1 << 8],szg,block ;
int memo[1 << 15] ;
{
if(x == n) return 1 ;

int ret = 0 ;
for(int i = 0;i < szg[x];i++) if(!(good[x][i] & mask))
{
ret += cret ;
if(ret >= MOD) ret -= MOD ;
}
}

int solve()
{
for(int i = 0;i < n;i++)
{
block[i] = 0 ;
for(int j = 0;j < m;j++) if(g[i][j] == '#')
{
cmask |= 1 << j ;
block[i] |= 7 << 3 * j ;
}

szg[i] = 0 ;
for(int j = 0;j < 1 << m;j++) if((j & cmask) == 0)
{
bool valid = true ;
for(int k = 0;k < m;k++) if(j & 1 << k)
for(int kk = k + 1;kk < m && g[i][kk] != '#';kk++)
if(j & 1 << kk)
valid = false ;
if(!valid) continue ;

int sp = 0 ;
for(int k = 0;k < m;k++) if(j & 1 << k) sp |= 7 << 3 * k ;
good[i][szg[i]] = sp ;
szg[i]++ ;
}
}
memset(memo,255,sizeof memo) ;
memset(memo2,255,sizeof memo2) ;
int ret = solve(0,0) ;
return ret ;
}

int main()
{
for(int i = 1;i < 1 << 10;i++) bit[i] = bit[i >> 1] + (i & 1) ;

int runs ;
scanf("%d",&runs) ;
while(runs--)
{
scanf("%d%d",&n,&m) ;
for(int i = 0;i < n;i++) scanf("%s",g[i]) ;
int ret = solve() ;
ret = (ret - 1 + MOD) % MOD ;
printf("%d\n",ret) ;
}
return 0 ;
}

In Java :

import java.io.*;
import java.util.*;
import java.text.*;
import java.math.*;
import java.util.regex.*;

class Solution {
static int NS = 50;
static int MS = 5;
static int K = 32;
static int[][][][]s = new int[NS][K][K][K];
static int AK;
static int[][] mp= new int[NS][MS];

static int n;
static int m;

static int dp(int c, int b1, int b2, int b3) {
if (c == n) return 1;
if (s[c][b1][b2][b3] >= 0) return s[c][b1][b2][b3];
/*
System.out.print(c);System.out.print(' ');
System.out.print(b1);System.out.print(' ');
System.out.print(b2);System.out.print(' ');
System.out.print(b3);System.out.print(' ');
System.out.println();
*/
int sum = 0;
for (int i = 0; i < AK; i++) {
if (check(c,i,b1,b2,b3)){
/*
System.out.print(c);
System.out.println(i);
*/
}
}
s[c][b1][b2][b3] = sum;
return sum;
}

static boolean check(int c, int i, int b1, int b2, int b3) {
int[] loc = {1,2,4,8,16};
boolean selfblock = false;

//check other block
for (int li = 0; li < m; li++) {
if ((i&loc[li]) != 0) {
if (mp[c][li] == 0) return false;
if (selfblock == true) return false;

if ((b1&loc[li]) !=0 || (b2&loc[li]) !=0|| (b3&loc[li]) !=0) return false;

selfblock = true;
}

if (mp[c][li] == 0) selfblock = false;
}

return true;
}

static int[] mask(int c, int i, int b1, int b2, int b3){
int[] loc = {1,2,4,8,16};
mask = ((b2 << 1) % K) | ((i << 1) % K);
mask = (b3 >> 1) | (i >> 1);
for (int li = 0; li < m; li++){
if(mp[c][li] == 0) {
}
}

}

public static void main(String[] args) {
Scanner in = new Scanner(System.in);
int TN = in.nextInt();

for (int ti = 0; ti < TN; ti++) {
n = in.nextInt();
m = in.nextInt();
String str[] = new String[n];
for(int i=0; i<n; i++)
str[i] = in.next();
for (int i = 0; i < n; i++)
for (int j = 0; j < m; j++) {
mp[i][j] = 1;
if (str[i].charAt(j) == '#')
mp[i][j] = 0;
}

for (int i1 = 0; i1 < n; i1++)
for (int i2 = 0; i2 < K; i2++)
for (int i3 = 0; i3 < K; i3++)
for (int i4 = 0; i4 < K; i4++)
s[i1][i2][i3][i4] = -1;

AK = 1;
for (int i =0; i< m; i++)
AK = AK *2;

int r = dp(0,0,0,0);
System.out.println((r+1000000007-1) % 1000000007);
}
}
}

In C:

#include <stdio.h>

#define MAX_N 50
#define MAX_M 5
#define MAX_CACHE 1000

typedef struct cache_data cache_data;
struct cache_data {
char b[MAX_N * MAX_M + 1];
int s, v; /* safe cells, value */
cache_data *next;
};
cache_data cache[MAX_CACHE];

int solv(char *b, int n, int m) {
}

void init_cache() {
int i;
for (i = 0; i < MAX_CACHE; i++) {
cache[i].next = NULL;
}
}

void clear_cache() {
int i;
cache_data *p, *next;
for (i = 0; i < MAX_CACHE; i++) {
for (p = cache[i].next; p; p = next) {
next = p->next;
free(p);
}
cache[i].next = NULL;
}
}

unsigned int calc_cache_key(char *b) {
unsigned int k = 0;
char *p = b;
for (; *p; p++) {
k *= 23;
if (*p == '.') k += 1;
else k += 2;
}
return k % MAX_CACHE;
}

int get_cache(char *b, int s) {
cache_data *p = cache[calc_cache_key(b)].next;
for (; p; p = p->next) {
if (s == p->s && strcmp(b, p->b) == 0) return p->v;
}
return -1;
}

int main() {
int t, n, m, i, j;
char b[MAX_N * MAX_M + 1];

scanf("%d", &t);
init_cache();
for (i = 0; i < t; i++) {
scanf("%d %d", &n, &m);
for (j = 0; j < n; j++) {
scanf("%s\n", b + j * m);
}
printf("%d\n", solv(b, n, m) % 1000000007);
clear_cache();
}

return 0;
}

In Python3 :

from __future__ import print_function

import collections

QUEEN = 'q'
BLOCK = '#'

SigBlock = collections.namedtuple('SigBlock', 'left middle right')

ROW_QUEENS = {
'.....': [[], , , , , ],
'#....': [[], , , , ],
'.#...': [[], , , , , [0, 2], [0, 3], [0, 4]],
'..#..': [[], , , , , [0, 3], [0, 4], [1, 3], [1, 4]],
'...#.': [[], , , , , [0, 4], [1, 4], [2, 4]],
'....#': [[], , , , ],
'##...': [[], , , ],
'#.#..': [[], , , , [1, 3], [1, 4]],
'#..#.': [[], , , , [1, 4], [2, 4]],
'#...#': [[], , , ],
'.##..': [[], , , , [0, 3], [0, 4]],
'.#.#.': [[], , , , [0, 2], [0, 4], [2, 4], [0, 2, 4]],
'.#..#': [[], , , , [0, 2], [0, 3]],
'..##.': [[], , , , [0, 4], [1, 4]],
'..#.#': [[], , , , [0, 3], [1, 3]],
'...##': [[], , , ],
'###..': [[], , ],
'##.#.': [[], , , [2, 4]],
'##..#': [[], , ],
'#.##.': [[], , , [1, 4]],
'#.#.#': [[], , , [1, 3]],
'#..##': [[], , ],
'.###.': [[], , , [0, 4]],
'.##.#': [[], , , [0, 3]],
'.#.##': [[], , , [0, 2]],
'..###': [[], , ],
'####.': [[], ],
'###.#': [[], ],
'##.##': [[], ],
'#.###': [[], ],
'.####': [[], ],
'#####': [[]],
'....': [[], , , , ],
'#...': [[], , , ],
'.#..': [[], , , , [0, 2], [0, 3]],
'..#.': [[], , , , [0, 3], [1, 3]],
'...#': [[], , , ],
'##..': [[], , ],
'#.#.': [[], , , [1, 3]],
'#..#': [[], , ],
'.##.': [[], , , [0, 3]],
'.#.#': [[], , , [0, 2]],
'..##': [[], , ],
'###.': [[], ],
'##.#': [[], ],
'#.##': [[], ],
'.###': [[], ],
'####': [[]],
'...': [[], , , ],
'#..': [[], , ],
'.#.': [[], , , [0, 2]],
'..#': [[], , ],
'##.': [[], ],
'#.#': [[], ],
'.##': [[], ],
'###': [[]],
'..': [[], , ],
'#.': [[], ],
'.#': [[], ],
'##': [[]],
'.': [[], ],
'#': [[]],
}

def sig_str(sig):
return ''.join(
'./'[s.left] + '.|'[s.middle] + '.\\'[s.right]
for s in sig
)

def print_sig_counts(sig_counts):
for sig, count in sig_counts.items():
print('\t{0}: {1}'.format(sig_str(sig), count))

QUEEN_SIG = SigBlock(1, 1, 1)

def transform(sig, row):
m = len(sig)
def new_sig():
for i in range(m):
if row[i] == BLOCK:
yield SigBlock(0, 0, 0)
else:
middle = sig[i].middle
left = sig[i+1].left if i < m-1 else 0
right = sig[i-1].right if i > 0 else 0
yield SigBlock(left, middle, right)
base = list(new_sig())
for queen_pos in ROW_QUEENS[row]:
copy = list(base)
for i in queen_pos:
if any(base[i]): break
copy[i] = QUEEN_SIG
else:
yield tuple(copy)

def base_sig(m):
return tuple([SigBlock(0, 0, 0)] * m)

def count_ways(m, rows):
sig_counts = {base_sig(m): 1}
for row in rows:
transformed = collections.defaultdict(int)
for sig, count in sig_counts.items():
for new_sig in transform(sig, row):
transformed[new_sig] += count
sig_counts = transformed
#print('row:', repr(row))
#print_sig_counts(sig_counts)
return sum(sig_counts.values()) - 1  # no queens doesn't count

import sys
stdin = iter(sys.stdin)
line = lambda: next(stdin).strip()
t = int(line())
for _ in range(t):
n, m = [int(i) for i in line().split()]
rows = [line() for _ in range(n)]
yield m, rows

if __name__ == '__main__':
print(count_ways(m, rows) % (10 ** 9 + 7))```
```

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