MathDB

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\times 5

board, consisting of

25

unit squares, a positive integer

k\leq 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

<span class='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

<span class='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.

4

Problems(1)