Question

The figure shows two 4-tile patterns.

Either one or both of the patterns can be used any number of times and in any orientation to construct a new pattern. Which one of the options below cannot be constructed by using only these two 4-tile patterns, assuming there are no overlaps among them?

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

Each piece always covers exactly 4 tiles, so check whether the total number of tiles in each option is a multiple of 4.

Answer 1

The Correct Option is C

Step 1: Identify the two building blocks.
The first pattern is a \(2\times2\) square made of 4 tiles. The second pattern is a straight bar of 4 tiles in a row, and it can be placed either horizontally or vertically since it can be rotated. Both pieces always cover exactly 4 unit tiles each.

Step 2: Note the key restriction.
Every piece used, whether the square block or the straight bar, covers exactly 4 tiles. So any shape built out of any number of these two pieces, with no overlaps and no gaps left uncovered, must have a total tile count that is a multiple of 4.

Step 3: Count the tiles in option (A).
Option (A) is a grid of 4 columns and 2 rows, giving \(4\times2=8\) tiles. Since 8 is a multiple of 4, it can be built, for example using two \(2\times2\) square blocks placed side by side.

Step 4: Count the tiles in option (B).
Option (B) is a grid of 4 columns and 3 rows, giving \(4\times3=12\) tiles. Since 12 is a multiple of 4, it can be built using three of the pieces, such as three \(2\times2\) blocks stacked, or a mix of blocks and bars.

Step 5: Count the tiles in option (C).
Option (C) is a grid of 5 columns and 3 rows, giving \(5\times3=15\) tiles. Since 15 is not divisible by 4 (\(15=4\times3+3\)), it is impossible to tile this shape using only 4-tile pieces with no overlaps and no gaps. No matter how the square blocks and bars are arranged, they can never add up to 15 tiles.

Step 6: Count the tiles in option (D).
Option (D) is a grid of 5 columns and 4 rows, giving \(5\times4=20\) tiles. Since 20 is a multiple of 4, it can be built using five of the pieces.

Final Answer:
Only option (C), with 15 tiles, fails the basic requirement that the total tile count be a multiple of 4, so it cannot be constructed. \[ \boxed{\text{Option (C)}} \]