# N-Queens Problem

The **eight queens puzzle** is the problem of placing eight chess queens 
on an `8×8` chessboard so that no two queens threaten each other. 
Thus, a solution requires that no two queens share the same row, 
column, or diagonal. The eight queens puzzle is an example of the 
more general *n queens problem* of placing n non-attacking queens 
on an `n×n` chessboard, for which solutions exist for all natural 
numbers `n` with the exception of `n=2` and `n=3`.

For example, following is a solution for 4 Queen problem.

![N Queens](https://cdncontribute.geeksforgeeks.org/wp-content/uploads/N_Queen_Problem.jpg)

The expected output is a binary matrix which has 1s for the blocks 
where queens are placed. For example following is the output matrix 
for above 4 queen solution.

```
{ 0,  1,  0,  0}
{ 0,  0,  0,  1}
{ 1,  0,  0,  0}
{ 0,  0,  1,  0}
```

## Naive Algorithm

Generate all possible configurations of queens on board and print a 
configuration that satisfies the given constraints.

```
while there are untried configurations
{
   generate the next configuration
   if queens don't attack in this configuration then
   {
      print this configuration;
   }
}
```

## Backtracking Algorithm

The idea is to place queens one by one in different columns, 
starting from the leftmost column. When we place a queen in a 
column, we check for clashes with already placed queens. In 
the current column, if we find a row for which there is no 
clash, we mark this row and column as part of the solution. 
If we do not find such a row due to clashes then we backtrack 
and return false.

```
1) Start in the leftmost column
2) If all queens are placed
    return true
3) Try all rows in the current column.  Do following for every tried row.
    a) If the queen can be placed safely in this row then mark this [row, 
        column] as part of the solution and recursively check if placing  
        queen here leads to a solution.
    b) If placing queen in [row, column] leads to a solution then return 
        true.
    c) If placing queen doesn't lead to a solution then umark this [row, 
        column] (Backtrack) and go to step (a) to try other rows.
3) If all rows have been tried and nothing worked, return false to trigger 
    backtracking.
```

## References

- [Wikipedia](https://en.wikipedia.org/wiki/Eight_queens_puzzle)
- [GeeksForGeeks](https://www.geeksforgeeks.org/backtracking-set-3-n-queen-problem/)
- [On YouTube by Abdul Bari](https://www.youtube.com/watch?v=xFv_Hl4B83A&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)
- [On YouTube by Tushar Roy](https://www.youtube.com/watch?v=xouin83ebxE&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)