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# A chessboard

## Question

Two diagonally opposite corner squares are removed from a regular chessboard. Now is it possible to cover all 62 squares with exactly 31 rectangles ( no rectangle should overlap each other ).

If yes then how? If no then why?

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NO it's impossible. Imagine the chess board pattern. The opposite corners are the same color. One rectangle can cover on tile of each color. That leaves two of the same tile that can't be covered with the last rectangle.

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Pretty sure it cant be done. trying to pinpoint the problem in it.

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Two diagonally opposite corner squares are removed from a regular chessboard. Now is it possible to cover all 62 squares with exactly 31 rectangles ( no rectangle should overlap each other ).

If yes then how? If no then why?

yes,because you didnt say the sides of rectagles should be aligned with the sides of the squares

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squares are considered rectangles too ..some may be 2 x 1 , 3 x 1 , 1 x 1 or 2 x 2 ?

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Two diagonally opposite corner squares are removed from a regular chessboard. Now is it possible to cover all 62 squares with exactly 31 rectangles ( no rectangle should overlap each other ).

If yes then how? If no then why?

yes,because you didnt say the sides of rectagles should be aligned with the sides of the squares

Good point.

i didnt thought that way

But if I say that all rectangles should be completely aligned and should be of same size then what would you say.

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The stipulation that all 31 rectangles are “of the same size” is still a bit ambiguous. The size could be interpreted as area. If rectangles were equal that would imply they are equal in area (2 squares each) and dimensions. However, do we need a stipulation that rectangles must be alined on square boundaries?

If rectangle's dimensions were specified as 1x2, then the problem would be solved by googon97 in post #5.

But those rectangles could be 1/3 x 6, or 1/2 x 4. Still, it is impossible to cover up the board with 31 of those rectangles.

Furthermore, can we prove that we could or could not cover the board with 31 equal area (2 squares each) rectangles of any dimensions?

Edited by Prime
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The stipulation that all 31 rectangles are “of the same size” is still a bit ambiguous. The size could be interpreted as area. If rectangles were equal that would imply they are equal in area (2 squares each) and dimensions. However, do we need a stipulation that rectangles must be alined on square boundaries?

If rectangle's dimensions were specified as 1x2, then the problem would be solved by googon97 in post #5.

But those rectangles could be 1/3 x 6, or 1/2 x 4. Still, it is impossible to cover up the board with 31 of those rectangles.

Furthermore, can we prove that we could or could not cover the board with 31 equal area (2 squares each) rectangles of any dimensions?

If you want to then You are welcome to do so

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