checkerboard problem

[Ranjusree Sinha]

You are given a 4 × 4 chessboard, and asked to fill it with five 3 × 1 pieces and one 1 × 1 piece. Then, over all such fillings, the number of squares that can be occupied by the 1 × 1 piece is
(A) 4 (B) 8 (C) 12 (D) 16

[Saumik Karfa]

will get back to you in 24 hours

[Saumik Karfa]

Answer is 4.

3 cases may arise

Case 1 :

Placing the $1\times 1$ piece at any corner of the board.

then there we can place  $3\times 1 $ piece in that row or column .

then 7 square are filled and a $3\times 3 $ square left to fill which can be easily filled by 3 $3\times 1 $ pieces.

Case 2 :

Placing $1 \times 1 $ piece at any square on the edge except the corner squares.

then it is dividing that edge in $2 boxes : 1 box $ ratio. then this edge can not be filled with $3\times 1$ pieces.

Let us divide the board in two rectangular parts of dimensions $4\times 2$  and $4\times 1$ by the row (or column) of  $1\times 1$ piece . Those two rectangular parts can not be covered with $3\times 1$ pieces.

Case 3 :

$1\times 1$ square is places in any middle square.

then the row (or column) is divided in the ratio $2 squares : 1 squares$ which can not be covered with $3\times 1 $ pieces.

Let us divided the board into two rectangular parts dimensions $4\times 2$  and $4\times 1$ by the row (or column) of $1\times 1$ piece.

Those rectangles can not be covered with $3\times 1 $ pieces.

 

 

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