13 September 2013

# Chessboard puzzle

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Imagine a chessboard – with your usual 8 by 8 = 64 squares. Say the squares are one inch by one inch each. Now imagine rectangular blocks that are one inch by two inches each. So, any rectangular block can cover 2 adjacent squares on the chessboard.
You have to fill up the chessboard with the rectangular blocks without any rectangle spilling outside the board. They are allowed to overlap on each other though.

What is the minimum number of rectangular blocks required to fill up the whole chessboard? That is 32 of course.

Okay, now imagine that I tore off one of the corner squares on that chessboard. And I went ahead and tore off the corner square that is diagonally on the other end of the board. So, now I have a chessboard with two farthest corner squares missing.

Now, tell me how many minimum rectangular blocks are required to cover the whole board (that has only 62 squares now). You are allowed to overlap but cannot spill outside the board.

Even trickier – what is the minimum number of blocks required if you CANNOT overlap or spill outside either?

Posted September 13, 2013 by Rajib Roy in category "Puzzles

1. By Rajib Roy (Post author) on

Start with the more difficult one first. What if we are not allowed to overlap (or spill over). Note that the corners of a chessboard has squares of the SAME color. So, you are left with 32 of white and 30 of black (or vice versa). Every rectangle will cover exactly one black and one white, any which way you position it. Thus if you have “n” rectangles, they will cover “n” white and “n” black squares. Given that we have an unequal number of black and white squares left, it is NOT POSSIBLE to ever cover the board with rectangles without overlapping or spilling over.

2. By Rajib Roy (Post author) on

The initial part was easier. The answer is same 32 as when you had a full board. You just pull the rectangle one square away and make them overlap with some other squares. It cannot be 31 anyways – because of the previous answer.

3. By Rajib Roy (Post author) on

I also believe I forgot to mention Debasish’s twin daughters Debatri and Bijetri cracked the problem!

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