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?
Show Hint
Each piece always covers exactly 4 tiles, so check whether the total number of tiles in each option is a multiple of 4.
Step 1: Work out the area covered by one piece.
Both given shapes, the small square and the straight bar, are made of exactly 4 unit tiles. This means every time we place one piece down, we add exactly 4 tiles to our picture, never more and never less.
Step 2: Turn this into a rule about totals.
If we use $k$ pieces in total, a mix of squares and bars in any orientation, the finished shape must contain exactly $4k$ tiles. So a valid shape always has a tile count that leaves remainder 0 when divided by 4.
Step 3: Work out the tile count of each option first, before worrying about shape.
Option (A) has $4\times2=8$ tiles. Option (B) has $4\times3=12$ tiles. Option (C) has $5\times3=15$ tiles. Option (D) has $5\times4=20$ tiles.
Step 4: Apply the divide by 4 test.
$8\div4=2$, exact. $12\div4=3$, exact. $20\div4=5$, exact. But $15\div4=3$ remainder $3$, not exact.
Step 5: Interpret the failing case.
Since option (C) needs 15 tiles and no combination of 4-tile pieces can ever sum to a number that is not a multiple of 4, option (C) fails before we even try to fit the pieces together. The other three options pass this test, and each can indeed be filled: squares stacked for (B), a row of squares for (A), and a mix for (D).
Step 6: State the conclusion.
The shape that cannot be built is the one whose total tile count is not a multiple of 4.
\[ \boxed{\text{Option (C)}} \]
