Question

The grid of squares shown in the figure is to be tiled (covered with tiles) with the tiles shown in the options. The covering tiles must not overlap and should not have gaps around them. Only four squares in the middle are not to be tiled. Once a tile is chosen, other tiles must be of that type only. Tiles can be flipped and rotated if required. Which of the tiles can be used to tile the given grid?

• A. A
• B. B
• C. C
• D. D

Hint:

For tiling puzzles, always first check divisibility of the area by the tile size, then visualize symmetry around special regions like holes.

Answer 1

The Correct Option is C

Step 1: Analyze the grid.
The grid is a square with a central 2×2 portion that must remain uncovered. The rest of the grid must be fully covered with identical tiles.
Step 2: Check tile sizes.
Each option (A, B, C, D) consists of exactly 5 unit squares (pentomino-like tiles). The tiling is possible only if the total number of squares to be covered is divisible by 5.
Step 3: Count squares.
The grid is 8×8 = 64 squares. The central 2×2 = 4 must remain empty. So, total squares to cover = 64 – 4 = 60. Since 60 ÷ 5 = 12, tiling is possible with 12 tiles of the same type.
Step 4: Match tile shapes with central cavity.
- (A) and (B): Their arrangement does not allow them to wrap around the central 2×2 hole without leaving gaps.
- (C): The “T”-shaped tile can perfectly tile around the central cavity and fill the outer portions symmetrically.
- (D): The “zig-zag” arrangement creates mismatches and gaps around the central area.
Step 5: Conclude.
Only tile (C) works. Final Answer: \[ \boxed{\text{C}} \]