6 February 2015

# Chessboard puzzle

Here is a simple or complex – depending upon your perspective – puzzle. It is complex if you try to enumerate all possibilities. There is a very simple and elegant way though.

Question: In a simple 8 by 8 chessboard, how many rectangles are there?

As you can imagine a rectangle can be one square by one square – and there are 64 of them – (squares are rectangles) or two squares by one square (and there are quite a few of them) or three squares by five squares and such….

Posted February 6, 2015 by Rajib Roy in category "Puzzles

1. By Rajib Roy on

The answer to the puzzle is extremely simple.
What we have to realize is that any rectangle in a chessboard is formed by choosing two vertical lines from the 9 vertical lines available (to form 8 columns, you need 9 lines) and two horizontal lines from the 9 horizontal lines available. Any such choice will give you a rectangle and all rectangles are necessarily one of those choices.
So the answer is the ways you can pick 2 vertical lines multiplied by the ways you can pick 2 horizontal lines.
In other words, 9C2 * 9C2 = 36*36 = 1296

2. By Rajib Roy on

The answer to the puzzle is extremely simple.
What we have to realize is that any rectangle in a chessboard is formed by choosing two vertical lines from the 9 vertical lines available (to form 8 columns, you need 9 lines) and two horizontal lines from the 9 horizontal lines available. Any such choice will give you a rectangle and all rectangles are necessarily one of those choices.
So the answer is the ways you can pick 2 vertical lines multiplied by the ways you can pick 2 horizontal lines.
In other words, 9C2 * 9C2 = 36*36 = 1296

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