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. Option A shown in the figure
• B. Option B shown in the figure
• C. Option C shown in the figure
• D. Option D shown in the figure

Hint:

Try mentally tiling each option using rotations and mirror images of the two 4-tile patterns; the option that always leaves an unfillable gap is the one that cannot be constructed.

Answer 1

The Correct Option is C

Alternate approach - try building each option directly: Attempt to physically place copies of the two given patterns (rotated through 0°, 90°, 180°, 270°, and their mirror images) onto each option, one tile-group at a time.
For Option A, Option B, and Option D, you can always find a starting corner where one pattern fits perfectly, and the leftover region after each placement is again fillable - the construction completes with zero gaps or overlaps.
For Option C, every attempted placement at any boundary corner leaves behind a leftover patch that matches neither pattern nor any of its rotations - a gap or overlap is unavoidable no matter where you start.
Since Option C cannot be fully tiled using only the two given 4-tile patterns, the answer is $\text{Option C}$.