052--N-Queens II.md

The n-queens puzzle is the problem of placing n queens on an n×n chessboard such that no two queens attack each other.

Given an integer n, return the number of distinct solutions to the n-queens puzzle.

Example:

Input: 4
Output: 2
Explanation: There are two distinct solutions to the 4-queens puzzle as shown below.
[
 [".Q..",  // Solution 1
  "...Q",
  "Q...",
  "..Q."],

 ["..Q.",  // Solution 2
  "Q...",
  "...Q",
  ".Q.."]
]

class Solution(object):
    def totalNQueens(self, n):
        """
        :type n: int
        :rtype: int
        """
        res = []
        self.dfs([-1]*n, 0, [], res)
        return len( res )
 
    # nums is a one-dimension array, like [1, 3, 0, 2] means
    # first queen is placed in column 1, second queen is placed
    # in column 3, etc.
    def dfs(self, nums, index, path, res):
        if index == len(nums):
            res.append(path)
            return  # backtracking
        for i in xrange(len(nums)):
            nums[index] = i
            if self.valid(nums, index):  # pruning
                tmp = "."*len(nums)
                self.dfs(nums, index+1, path+[tmp[:i]+"Q"+tmp[i+1:]], res)

    # check whether nth queen can be placed in that column
    def valid(self, nums, n):
        for i in xrange(n):
            if abs(nums[i]-nums[n]) == n -i or nums[i] == nums[n]:
                return False
        return True