# Finding the number of rectangles

Another group of questions commonly missed in the 2nd Philippine National Vedic Mathematics Olympiad was finding the number of rectangles in the given grid.

We can of course, count the number of rectangles by using the sutra By Alternate Elimination and Retention. We can first retain(count) all rectangles of a certain size and eliminate (momentarily disregard) the other sizes. Then proceed with the other sizes until all possible sizes are considered.

In this question in the beginners category, we have a rectangle composed of 9 smaller rectangles arranged in 3 columns and 3 rows.

We can also count 2 – 2 x 1 and 1 – 3 x 1 rectangles in each row for a total of 6 – 2 x 1 and 3 – 3 x 1 rectangles in the 3 rows.

Similarly, we can count 2 – 1 x 2 and 1 – 1 x 3 rectangles in each column for a total of 6 – 1 x 2 and 3 – 1 x 3 rectangles in the 3 columns.

We can also count 4 – 2 x 2, 2 – 2 x 3 and 2 – 3 x 2 rectangles.

Finally we have 1 – 3 x 3 rectangle.

To summarize, we have:

1 x 1   –           9

2 x 1   –           6

3 x 1   –           3

1 x 2   –           6

1 x 3   –           3

2 x 2   –           4

2 x 3   –           2

3 x 2   –           2

3 x 3   –           1

Total              36

We can have a direct solution to this type of problem by considering that any pair of horizontal lines combined with any pair of vertical lines can form a rectangle.

In the figure above, we can see that we can choose first any of the 4 vertical lines and pair it with any of the remaining 3 vertical lines. And since the order in which the two lines were chosen would not affect the figure formed, the number of possible way to choose the vertical lines would be (4 x 3)/2 = 6

Now there are also 4 horizontal lines in the figure, so we also have 6 possible pairs. The number of rectangles is 6 x 6 or 36.

In the primary category we have a 4 x 4 grid:

So the total number of pairs of vertical lines is (5 x 4)/2 or 10. Since we also have 10 possible pairs of horizontal lines, we have a total of 10 x 10 or 100 possible rectangles in the figure.

In the Junior category, we have 25 small triangles arranged in 5 rows inside a big rectangle as shown in the figure below: In this case we have 6 vertical and 6 horizontal lines. The number of possible rectangles is [(6 x 5)/2] x [(6 x 5)/2] = 15 x 15 = 225.In the Intermediate category, we have a 5 x 7 grid, so the total number of rectangles is [(6 x 5)/2] x [(8 x 7)/2] = 15 x 28 = 420.What do you think is the answer to the question in the senior category where we have a 5 x 9 grid?