Two players play a game on $21 \times 2010$ grid. The first player starts. On each turn, the current player should choose some square and color it in black. Initially all the cells are white, also, all the cells of the chosen square should be white. The player who can't make a move, loses. What is the winning strategy for the first player?
P.S If dimensions make a sense, then consider $21 \times 2100$, since I don't remember exactly which one was mentioned in the original problem.
Thanks.
There is a very simple strategy. If the size was $21$ times an odd number $\geq 21$, then you pick a $21\times21$ square in the middle, leaving two equal unconnected areas. Then you copy every move of the opponent in the other area until everything is filled.
If the size is $21 \times$ some even number, you put a $20\times20$ square in the middle. You are left with two equal areas and a $1 \times 20$ stripe that can only be filled with single squares. Again, you can copy all your opponents moves until you fill the last white square.
The strategy doesn’t work for even height and odd width.
A similar problem: Given are identically sized eggs and a round table. Place the eggs on the table. Whoever can’t fit another egg loses. To solve, google for “egg of Columbus” then use this answer.