Question

Figure P can be divided into four identical pieces. Assuming that the pieces can be rotated and flipped, which of the options will fit?

• A. A
• B. B
• C. C
• D. D

Hint:

In dissection puzzles, the first and most important step is often to calculate the area of the constituent pieces. This can quickly eliminate many of the options without having to perform complex mental rotations or trial-and-error tiling.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a dissection puzzle. We need to determine the shape of the identical pieces that can form the larger shape P. The pieces are polyominoes (shapes made of connected squares).

Step 2: Key Formula or Approach:


Count the total number of unit squares in the target shape P.
Divide the total number of squares by the number of pieces (four) to find the number of squares in each identical piece.
Examine the options to see which ones have the correct number of squares and could potentially tile the shape P.

Step 3: Detailed Explanation:
1. Count the squares in Shape P:
By carefully counting the dashed squares that make up shape P, we can find the total area.

Topmost row: 2 squares
Second row: 4 squares
Third row: 3 squares
Bottommost row: 3 squares
Total number of squares = 2 + 4 + 3 + 3 = 12 squares. 2. Calculate squares per piece:
The shape P is to be divided into four identical pieces. Number of squares per piece = Total squares / Number of pieces = 12 / 4 = 3 squares. 3. Analyze the options:
Each piece must be a tromino (a polyomino of order 3).

Option (A): This shape is made of 4 squares (a tetromino). It cannot be the correct piece.
Option (B): This shape is made of 3 squares in an 'L' formation. This is an L-tromino. It has the correct number of squares.
Option (C): This shape is also made of 3 squares in an 'L' formation (it is a rotated version of B). It is also an L-tromino.
Option (D): This shape is made of 4 squares (a tetromino). It cannot be the correct piece.

Step 4: Final Answer:
Since each of the four identical pieces must contain 3 squares, only the trominoes are valid options. Options A and D are tetrominoes (4 squares) and are therefore incorrect. Options B and C both depict the L-tromino, which is the correct shape of the piece. Therefore, both B and C are correct. For completeness, a possible tiling of shape P with four L-trominoes is shown below.