Omni-MATH / Let n be a positive integer. Initially, a 2n×2n grid has k black cells and the rest white cells. The following two operations are allowed : (1) If a 2×2 square has exactly three bl…

Let n be a positive integer. Initially, a 2n×2n grid has k black cells and the rest white cells. The following two operations are allowed :
(1) If a 2×2 square has exactly three black cells, the fourth is changed to a black cell; 
(2) If there are exactly two black cells in a 2×2 square, the black cells are changed to white and white to black.
 Find the smallest positive integer k such that for any configuration of the 2n×2n grid with k black cells, all cells can be black after a finite number of operations.

Let $n$ be a positive integer. Initially, a $2n \times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :
(1) If a $2\times 2$ square has exactly three black cells, the fourth is changed to a black cell; 
(2) If there are exactly two black cells in a $2 \times 2$ square, the black cells are changed to white and white to black.
 Find the smallest positive integer $k$ such that for any configuration of the $2n \times 2n$ grid with $k$ black cells, all cells can be black after a finite number of operations.