A rectangular room is completely tiled by 1-foot square tiles. All the adjacent to a door or wall are purple, and the rest of the tiles are white. If exactly 2/7 of the tiles are purple, then what is the smallest possible area of the room, in square feet?
Let $a$ be the number of white tiles, and let $p$ be the number of purple tiles. Then $a + p$ is the total number of tiles, equal to the area of the room. Since the room is completely tiled, and the tiles are 1-foot square, $a + p$ is an integer. Note that each purple tile has at most four neighbors, so each purple tile has at most 4 white neighbors. Hence,
\[\frac{2}{7} p \leq \frac{4}{7} (a + p) = \frac{4}{7} (a + \tfrac{2}{7} p).\] Multiplying this inequality by 7, we get
\[2p \leq 4a + 2p,\] which simplifies to $a \geq 0.$
Hence, the smallest possible area is $\boxed{0}.$