Subcontests
(4)Jbmo 2015 problem 4
An L-shape is one of the following four pieces, each consisting of three unit squares:[asy]
size(300);
defaultpen(linewidth(0.8));
path P=(1,2)--(0,2)--origin--(1,0)--(1,2)--(2,2)--(2,1)--(0,1);
draw(P);
draw(shift((2.7,0))*rotate(90,(1,1))*P);
draw(shift((5.4,0))*rotate(180,(1,1))*P);
draw(shift((8.1,0))*rotate(270,(1,1))*P);
[/asy]
A 5×5 board, consisting of 25 unit squares, a positive integer k≤25 and an unlimited supply of L-shapes are given. Two players A and B, play the following game: starting with A they play alternatively mark a previously unmarked unit square until they marked a total of k unit squares.We say that a placement of L-shapes on unmarked unit squares is called <spanclass=′latex−italic′>good</span> if the L-shapes do not overlap and each of them covers exactly three unmarked unit squares of the board.
B wins if every <spanclass=′latex−italic′>good</span> placement of L-shapes leaves uncovered at least three unmarked unit squares. Determine the minimum value of k for which B has a winning strategy.