Project Euler 85 ：Counting rectangles 数长方形

Counting rectangles

By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles:

Although there exists no rectangular grid that contains exactly two million rectangles, find the area of the grid with the nearest solution.

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数长方形

如果数得足够仔细，能看出在一个3乘2的长方形网格中包含有18个不同大小的长方形，如下图所示：

尽管没有一个长方形网格中包含有恰好两百万个长方形，但有许多长方形网格中包含的长方形数目接近两百万，求其中最接近这一数目的长方形网格的面积

解题

参考博客

有下面内容：

对于任意矩形M*N

其中1*1的矩阵有M*N个

1*2的矩阵有M*(N-1)个

2*1的矩阵有（M-1)*N个

实际上只要确定小矩阵左上角顶点在大矩形中的位置，这个矩阵的位置就唯一确定了

所有在任意矩形M*N中，矩阵i*j有（M-i+1)*(N-j+1)个

所以对于M*N的矩阵总的矩阵数量是：

        int num = 0;
        for(int i =1;i<= m;i++){
            for(int j =1;j<= n;j++){
                num += (m-i + 1)*(n - j+1);
            }
        }

更让人想不到是是直接计算矩阵的数量：

num = （m+1）*m*(n+1)*n/4

Java

package Level3;
import java.util.Random;

public class PE085{

    static void run() {
        int limit = 100;
        int close = Integer.MAX_VALUE;
        int area  = 0;
        for(int m =1;m< limit ;m++){
            for(int n = 1;n< limit ;n++){
                int num = grid_num(m,n);
                if (num>2000000)
                    break;
                if( Math.abs(num - 2000000 ) < Math.abs(close - 2000000)){
                    close = num;
                    area = n*m;
                }
            }
        }
        System.out.println(area);
    }
    public static int grid_num2(int m , int n){
        int num = 0;
        num = (m+1)*m*(n+1)*n/4;
        return num;
    }
//    2772
//    running time=0s0ms
    public static int grid_num(int m , int n){
        int num = 0;
        for(int i =1;i<= m;i++){
            for(int j =1;j<= n;j++){
                num += (m-i + 1)*(n - j+1);
            }
        }
        return num;
    }
//    2772
//    running time=0s20ms

    public static void main(String[] args){
        long t0 = System.currentTimeMillis();
        run();
        long t1 = System.currentTimeMillis();
        long t = t1 - t0;
        System.out.println("running time="+t/1000+"s"+t%1000+"ms");
    }
}

你说是不是很流氓，这个规律，我怎么那么聪慧的会发现？

Python

# coding=gbk
import time as time 

t0 = time.time()

def run():
    limit = 100
    close = 0
    area = 0
    for m in range(1,limit):
        for n in range(1,limit):
            num = grid_num(m,n)
            if num>2000000:break
            if abs(num - 2000000) < abs(close -2000000):
                close = num
                area = n*m
    print area

def grid_num(m ,n):
    count = 0
    for i in range(1,m+1):
        for j in range(1,n+1):
            count += (m-i+1)*(n-j+1)
    return count 

run()
t1 = time.time()
print "running time=",(t1-t0),"s"

#
# running time= 1.19499993324 s