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?
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Each tile covers 4 cells; check which option's total cell count is not a multiple of 4.
Every tile used, whether the square block or the line piece, contributes exactly 4 cells to the final figure, and no partial tile can ever be used since the problem states there are no overlaps and construction uses only whole copies of the two given patterns. So if a target shape has area $A$ and we use $n$ tiles total to build it, we must have $A = 4n$, which forces $n = A/4$ to be a whole number. Testing the four given areas: for (A) $8/4=2$ tiles needed; for (B) $12/4=3$ tiles needed; for (C) $15/4$ is not an integer, so no whole number of tiles can ever sum to 15; for (D) $20/4=5$ tiles needed. Since (A), (B) and (D) each give an integer tile count, a valid arrangement can be attempted and indeed found (for example, (D)'s 5x4 rectangle splits cleanly into five 1x4 vertical strips, each strip being one line tile), confirming they are constructible. For (C), because $15/4=3.75$ is fractional, there is no possible combination of whole tiles, each contributing 4 cells, that can sum to 15 cells; some part of the region would always be left uncovered, or a tile would have to overhang the boundary or overlap another tile, both of which are disallowed. Hence the 5x3 rectangle of 15 cells is the one pattern that can never be built, matching option (C). $\boxed{C}$
