6. [8 points] A rectangular room is completely tiled by 1 -foot square tiles. All the tiles adjacent to a door or wall are purple, and the rest of the tiles are white. If exactly \( \frac{2}{7} \) of the tiles are purple, then what is the smallest possible area of the room, in square feet?

Let the area of the room be denoted as \( A \) square feet, which is equal to the total number of tiles in the room since each tile covers 1 square foot. According to the problem, \( \frac{2}{7} \) of the tiles are purple. Therefore, the number of purple tiles can be expressed as: \[ \text{Number of purple tiles} = \frac{2}{7}A \] Since the tiles that are not purple are white, the number of white tiles would be: \[ \text{Number of white tiles} = A - \frac{2}{7}A = \frac{5}{7}A \] Given that \( A \) must be a multiple of both 7 (to ensure that \( \frac{2}{7}A \) is an integer) and the fact that the number of tiles adjacent to walls or doors must correlate to the dimensions of the rectangle, we can find the smallest \( A \) that satisfies this condition. Let \( A = nm \), where \( n \) and \( m \) are the dimensions of the room in terms of tiles. The number of tiles adjacent to the walls (purple tiles) can be calculated as: \[ \text{Purple tiles} = 2n + 2m - 4 \] This represents all the tiles in the first and last rows (2n), and the first and last columns (2m), with 4 tiles counted twice at the corners. Setting up the equation where the number of purple tiles equals \( \frac{2}{7}A \): \[ 2n + 2m - 4 = \frac{2}{7}(nm) \] Multiplying through by 7 to eliminate the fraction gives: \[ 7(2n + 2m - 4) = 2nm \] This simplifies to: \[ 14n + 14m - 28 = 2nm \] Rearranging this leads us to: \[ 2nm - 14n - 14m + 28 = 0 \] Factoring or rearranging further gives us the quadratic in standard form: \[ nm - 7n - 7m + 14 = 0 \] Now we can rearrange it to factor: \[ (n-7)(m-7) = 49 \] The positive divisors of 49 (which will give us pairs \( (n-7, m-7) \)) include \( (1, 49), (7, 7), (49, 1) \). Calculating the values of \( n \) and \( m \): 1. For \( (1, 49) \): \[ n = 8, \, m = 56 \quad \Rightarrow \quad A = 8 \times 56 = 448 \] 2. For \( (7, 7) \): \[ n = 14, \, m = 14 \quad \Rightarrow \quad A = 14 \times 14 = 196 \] 3. For \( (49, 1) \): \[ n = 56, \, m = 8 \quad \Rightarrow \quad A = 56 \times 8 = 448 \] From the calculations above, the smallest area occurs when \( n = m = 14 \), yielding: \[ A = 196 \text{ square feet} \] Thus, the smallest possible area of the room is \[ \boxed{196} \text{ square feet}. \]