How it Works › Forums › Advanced Problems (AMC 8 Problems 20-25 and beyond) › Counting Cleverly Advanced Problems › Reply To: Counting Cleverly Advanced Problems

Yash Agarwal

#1:

A 5 by 5 chess board would have 1 by 1, 1 by 2, …, 2 by 1, 2 by 2, … , 5 by 5.

1 * 1: 25

1 * 2: 20

1 * 3: 15

It’s going down 5 each time. So the numbers would be 25, 20, 15, 10, and 5 respectively.

It’s a different pattern for 2:

2 * 1: 20

2 * 2: 16

This pattern is going down 4 each time.

Then, the next goes down 3 each time, starting at 15.

The next goes down 2 each time, starting at 10.

The next goes down 1 each time, starting at 5.

If you add all these numbers up, you would get a final answer of

** 225 rectangles. **

**This response is written by Yash Agarwal**