RIOPLATENSE 2015 Problem 4

You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is good if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice. Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a good coloring.