Question 182
Let $n$ be a positive integer. Compose a $3 \times 3 \times 3$ cube of 26 white unit cubes and one black unit cube by putting the black one into the center. Compose a $3n \times 3n \times 3n$ cube of $n^3$ such $3 \times 3 \times 3$ cubes. Determine the smallest number $k$ such that it is possible to paint $k$ white unit cubes red so that each white cube will have at least one common vertex with some red cube.