---

Problem 147

---

Rectangles in cross-hatched grids

In a 3x2 cross-hatched grid, a total of 37 different rectangles could be situated within that grid as indicated in the sketch.

There are 5 grids smaller than 3x2, vertical and horizontal dimensions being important, i.e. 1x1, 2x1, 3x1, 1x2 and 2x2. If each of them is cross-hatched, the following number of different rectangles could be situated within those smaller grids:

1x1: 1
2x1: 4
3x1: 8
1x2: 4
2x2: 18

Adding those to the 37 of the 3x2 grid, a total of 72 different rectangles could be situated within 3x2 and smaller grids.

How many different rectangles could be situated within 47x43 and smaller grids?

---

交叉对角线方格中的长方形

在一个3x2的标记了所有交叉对角线的方格中，一共有37种不同的长方形其各边是方格内的线段，如下图所示。

从长和宽两方面考虑，有五个方格比3x2要小，大小分别是1x1，2x1，3x1，1x2和2x2。 如果将它们都标记上交叉对角线，则在这些更小的方格中有如下数目的不同长方形：

1x1: 1
2x1: 4
3x1: 8
1x2: 4
2x2: 18

把这些数和37加起来，可知对于3x2或更小的交叉对角线方格，一共有72种不同的长方形。

对于47x43或更小的交叉对角线方格，一共有多少种不同的长方形？

---