Question 168
Given a positive integer $n$, find the smallest positive integer $k$ with the following property: After coloring any $k$ unit squares black and the remaining squares white in a $2n \times 2n$ grid, it is always possible to make all squares black using a finite number of the following two operations:
(Operation 1) Select a $2 \times 2$ square that contains exactly 3 black squares and 1 white square, and change the color of the white square to black.
(Operation 2) Select a $2 \times 2$ square that contains exactly 2 black squares and 2 white squares, and change the color of all squares within the square (black squares become white, and white squares become black).
*Note*: The same $2 \times 2$ square can be operated on multiple times, and each square can change color multiple times.
(Operation 1) Select a $2 \times 2$ square that contains exactly 3 black squares and 1 white square, and change the color of the white square to black.
(Operation 2) Select a $2 \times 2$ square that contains exactly 2 black squares and 2 white squares, and change the color of all squares within the square (black squares become white, and white squares become black).
*Note*: The same $2 \times 2$ square can be operated on multiple times, and each square can change color multiple times.