皇后相斥问题（UVA 11538）

http://blog.sina.com.cn/s/blog_77dc9e080101hwtz.html

 

解题思路:

      1. 计数问题, 有三种相对摆放方式: 水平, 竖直, 对角线. 根据加法原理即可, 并且没有交集.

         水平和竖直是一样的, 只要n*m矩形旋转90度. 所以结果是: n*m*(m-1)+n*m*(n-1);

      2. 对角线复杂些, 先来确定对角线的长度: 1,2,3,...,n-2,n-1,n,n,n,...,n,n,n-1,n-2,...,2,1;

         其中n的个数是m-n+1 (其中假设m>n);

         结果: 2*(2*∑i*(i-1) + (m-n+1)*n*(n-1))  其中累加的范围是(1<=i<=n-1);

         化简得: 2*n*(n-1)*(3*m-n-1)/3

      3. 综上所述: n*m*(n+m-2)+2*n*(n-1)*(3*m-n-1)/3

 　　4.平方和公式：1 ^ 2 + 2 ^ 2 + ... + n ^ 2 == n * (n + 1) * (2 * n + 1) / 6

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

public classMain {
	
	long n, m;
	
	void run() {
		while (true) {
			n = cin.nextInt();
			m = cin.nextInt();
			if (n == 0 && m == 0) break;
			if (n > m) {
				long t = n;
				n = m;
				m = t;
			}
			long ans = n * m * (n + m - 2);
//			ans += 2 * n * (n - 1) * (3 * m - n - 1) / 3;
			ans += 2 * (n * (n - 1) * (2 * n - 1) / 3 - n * (n - 1) + (m - n + 1) * (n - 1) * n);
			System.out.println(ans);
		}
	}               
	
	public static void main(String[] args) {
		Mainsolved = new Main();
		solved.run();
	}
	
	static InputStreaminputStream = System.in;
	static InputReadercin = new InputReader(inputStream);
	
//	Scanner cin = new Scanner(new BufferedInputStream(System.in));
	
}


classInputReader { 

    private InputStreamstream; 
    private byte[] buf = new byte[1024]; 
    private int curChar; 
    private int numChars; 

    public InputReader(InputStreamstream) { 
        this.stream = stream; 
    } 

    public int read() { 
        if (numChars == -1) 
            return -1; 
            //throw new InputMismatchException(); 
        if (curChar >= numChars) { 
            curChar = 0; 
            try { 
                numChars = stream.read(buf); 
            } catch (IOExceptione) { 
                throw new InputMismatchException(); 
            } 
            if (numChars <= 0) 
                return -1; 
        } 
        return buf[curChar++]; 
    } 

    public int nextInt() { 
        int c = read(); 
        if (c == -1) 
            return -1; 
        while (isSpaceChar(c)) 
            c = read(); 
        int sgn = 1; 
        if (c == '-') { 
            sgn = -1; 
            c = read(); 
        } 
        int res = 0; 
        do { 
            if (c < '0' || c > '9') 
                throw new InputMismatchException(); 
            res *= 10; 
            res += c - '0'; 
            c = read(); 
        } while (!isSpaceChar(c)); 
        return res * sgn; 
    } 
     
    public Stringnext() {   
        StringBuilderstr = new StringBuilder();   
        int ch;   
        while (isSpaceChar(ch = read()));   
        if (ch == -1)   
            return null;   
        do {   
            str.appendCodePoint(ch);   
        } while (!isSpaceChar(ch = read()));   
        return str.toString();   
    }  
     
    public static boolean isSpaceChar(int c) { 
        return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1; 
    } 

    public char nextCharacter() { 
        int c = read(); 
        while (isSpaceChar(c)) 
            c = read(); 
        return (char) c; 
    } 

}