Number of Rectangles in a Grid

Project Euler 85: Investigating the number of rectangles in a rectangular grid

 

Number of Rectangles in a Grid

Given a grid of size m x n, calculate the total number of rectangles contained in this rectangle. All integer sizes and positions are counted.

Examples:

numberOfRectangles(3, 2) == 18
numberOfRectangles(4, 4) == 100

Here is how the 3x2 grid works (Thanks to GiacomoSorbi for the idea):

1 rectangle of size 3x2:

[][][]
[][][]

2 rectangles of size 3x1:

[][][]

4 rectangles of size 2x1:

[][]

2 rectangles of size 2x2

[][]
[][]

3 rectangles of size 1x2:

[]
[]

6 rectangles of size 1x1:

[]

As you can see (1 + 2 + 4 + 2 + 3 + 6) = 18, and is the solution for the 3x2 grid.

There is a very simple solution to this!

 

public class Grid {
  public static int NumberOfRectangles(int m, int n) {
    // Your code here!
    int sum = 0;
            for (int i = 1; i <= m; i++)
            {
                for (int j = 1; j <= n; j++)
                {
                    sum += (m - i + 1) * (n - j + 1);
                }
            }
            return sum;
  }
}

 

作者:Chuck Lu    GitHub