CareerCup

Answer = No of ways of placing the pair of queens on the board (Y) - No of ways in which the pair intersects (X)

Y = N*N *(N*N -1)

class Main
{
	public static void main (String[] args) throws java.lang.Exception
	{
			int n = 8;
			int arr[][] = new int[n][n];
			long x = 0;
			for (int i=0;i<n;i++){
				for (int  j=0;j<n;j++){
					
					x +=  Math.min(n-1-i, n-1-j) + Math.min(i, j) + Math.min(n-1-i,j) + Math.min(i,n-1-j);
					
					x+= 2*n -2;
				}
			}
			
			long ans = n*n * (n*n-1)  - x;
			System.out.println(ans);
	}
}

We can do this by combinations.
Observe that we have 64 * 63 possibilities for two queens, and we can change the order.
That is : 2 * 64 * 63 ways one can arrange 2 queens : T
Now, out of them, how many won't be stable?
Observe that a configuration is unstable iff The queens are in the same :
1. vertical
2. horizontal
3. diagonal.

1. Is possible by : 8 * P(8,2) ways : H
2. Is possible by : 8 * P(8,2) ways : V
3. Is possible by : 4 * ( P(8,2) + P(7,2) + ... P(2,2) ) : D

number of ways = T - H - V - D

Answer = No of ways of placing the pair of queens on the board (Y) - No of ways in which the pair intersects (X)

Y = N*N *(N*N -1)

class Main
{
	public static void main (String[] args) throws java.lang.Exception
	{
			int n = 8;
			int arr[][] = new int[n][n];
			long x = 0;
			for (int i=0;i<n;i++){
				for (int  j=0;j<n;j++){
					
					x +=  Math.min(n-1-i, n-1-j) + Math.min(i, j) + Math.min(n-1-i,j) + Math.min(i,n-1-j);
					
					x+= 2*n -2;
				}
			}
			
			long ans = n*n * (n*n-1)  - x;
			System.out.println(ans);
	}

}

class Main
{
	public static void main (String[] args) throws java.lang.Exception
	{
			int n = 8;
			int arr[][] = new int[n][n];
			long x = 0;
			for (int i=0;i<n;i++){
				for (int  j=0;j<n;j++){
					
					x +=  Math.min(n-1-i, n-1-j) + Math.min(i, j) + Math.min(n-1-i,j) + Math.min(i,n-1-j);
					
					x+= 2*n -2;
				}
			}
			
			long ans = n*n * (n*n-1)  - x;
			System.out.println(ans);
	}

}

class Main
{
	public static void main (String[] args) throws java.lang.Exception
	{
			int n = 8;
			int arr[][] = new int[n][n];
			long x = 0;
			for (int i=0;i<n;i++){
				for (int  j=0;j<n;j++){
					
					x +=  Math.min(n-1-i, n-1-j) + Math.min(i, j) + Math.min(n-1-i,j) + Math.min(i,n-1-j);
					
					x+= 2*n -2;
				}
			}
			
			long ans = n*n * (n*n-1)  - x;
			System.out.println(ans);
	}

}

class Main
{
	public static void main (String[] args) throws java.lang.Exception
	{
			int n = 8;
			int arr[][] = new int[n][n];
			long x = 0;
			for (int i=0;i<n;i++){
				for (int  j=0;j<n;j++){
					
					x +=  Math.min(n-1-i, n-1-j) + Math.min(i, j) + Math.min(n-1-i,j) + Math.min(i,n-1-j);
					
					x+= 2*n -2;
				}
			}
			
			long ans = n*n * (n*n-1)  - x;
			System.out.println(ans);
	}
}

class Main
{
	public static void main (String[] args) throws java.lang.Exception
	{
			int n = 8;
			int arr[][] = new int[n][n];
			long x = 0;
			for (int i=0;i<n;i++){
				for (int  j=0;j<n;j++){
					
					x +=  Math.min(n-1-i, n-1-j) + Math.min(i, j) + Math.min(n-1-i,j) + Math.min(i,n-1-j);
					
					x+= 2*n -2;
				}
			}
			
			long ans = n*n * (n*n-1)  - x;
			System.out.println(ans);
	}

}

Answer = No of ways of placing the pair of queens on the board (Y) - No of ways in which the pair intersects (X)

Y = N*N *(N*N -1)

2576

if queens are exactly same each other, (64*63)/2. But if queens are different from each other, 64*63

Ans