by the way for 45 & 135 degree ,there are two equations:

• y-c = 1*(x-r) ⇒ x+y == r+c
• y-c = -1*(x-r) ⇒ x-y == r-c

class Solution {
public:
    vector<vector<string>> solveNQueens(int n) {
        vector<vector<string>> res;
        vector<string> q(n,string(n,'.'));
        solve(res,q,0,n);
        return res;
    }
    void solve(vector<vector<string>>& res,vector<string>& q,int r,int n){
        if(r==n){
            res.push_back(q);
            return ;
        }                   
        for(int j=0;j<n;j++){
            if(isValid(q,r,j,n)){
                q[r][j] = 'Q';
                solve(res,q,r+1,n);
                q[r][j] = '.';
            }
        }      
    }
    bool isValid(vector<string>& q,int r,int c,int n){
        //row&col
        for(int i=0;i<n;i++){
            if(q[r][i]=='Q'||q[i][c]=='Q'){
                return false;
            }
        }
        //45 degree & 135 degree
        for(int x=0;x<n;x++){
            for(int y=0;y<n;y++){
                if(x-y==r-c||x+y==r+c){
                    if(q[x][y]=='Q'){
                        return false;
                    }
                }
            }
        }
        return true;
    }
};

Solution B: Use flag vectors as bitmask, 4ms:

The number of columns is n, the number of 45° diagonals is 2 * n - 1, the number of 135° diagonals is also 2 * n - 1. When reach [row, col], the column No. is col, the 45° diagonal No. is row + col and the 135° diagonal No. is n - 1 + col - row. We can use three arrays to indicate if the column or the diagonal had a queen before, if not, we can put a queen in this position and continue.

NOTE: Please don’t use vector flag to replace vector flag in the following C++ code. In fact, vector is not a STL container. You should avoid to use it. You can also get the knowledge from here and here.

class Solution {
public:
    vector<vector<string>> solveNQueens(int n) {
        vector<vector<string>> res;
        vector<string> q(n,string(n,'.'));
        vector<int> flag_col(n,1),flag_45(2*n-1,1),flag_135(2*n-1,1);      
        solve(res,q,0,n,flag_col,flag_45,flag_135);
        return res;
    }
    void solve(vector<vector<string>>& res,vector<string>& q,int r,int n,vector<int>& flag_col,vector<int>& flag_45,vector<int>& flag_135){
        if(r==n){
            res.push_back(q);
            return ;
        }                   
        for(int j=0;j<n;j++){
            if(flag_col[j]&&flag_45[r+j]&&flag_135[n-1+j-r]){
                flag_col[j]=flag_45[r+j]=flag_135[n-1+j-r]=0;
                q[r][j] = 'Q';
                solve(res,q,r+1,n,flag_col,flag_45,flag_135);
                q[r][j] = '.';
                flag_col[j]=flag_45[r+j]=flag_135[n-1+j-r]=1;
            }
        }      
    }
};