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.
As always, if you are reading this on FB, do not post your answer on Comments section. Sent me a message.
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….
Ramu has cracked it already
Ramu has cracked it already
So has Deepak
So has Deepak
Balaji and Dilesh cracked it too
Balaji and Dilesh cracked it too
And Pradeep !!
And Pradeep !!
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
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