Strategies for a game where players alternate placing $2\times 2$ squares and three-square "L" shapes on a $100\times100$ board

Question

This is a question for a homework assignment that I need to solve: please only provide advice, I need to understand it. I also want to solve it on my own.

The question is as follows:

On a $100\times100$ board, two people, Art and Bart take turns putting pieces on the board.

Art's pieces are $2\times2$ squares.

Bart's pieces are "L" shapes made out of $3$ squares (a $2\times2$ square without one corner).

The player who can no longer fit one of their pieces on the board loses.

a) If Art starts, does either player have a winning strategy?

b) If Bart starts, does either player have a winning strategy?

Answer 1

Part $1$:

If Bart starts, he can first place the L-shape tile in the center $2\times2$ area first with bottom right corner empty. (since the board is $100\times100$ it has a center $2\times2$).

Then whatever Art does, Bart always try to "mirror Art's move" with respect to the center point. Note that this is always possible, I will show some hints and let you prove the details.

Hint: try to break into two cases, one case where Art places the $2\times2$ tile that contains an "empty corner of an L shape"(think about the situation on the mirrored side when is this possible), and the other case otherwise.

Notice that, in the first case you also need to show that if Art places the $2\times2$ tile in the center area "bottom-right empty corner", Bart can also mirror his move.

There is a minor point that you need to prove Bart is always able to avoid the case where he places four L-shape tiles in a circular pattern (i.e. there is a $2\times 2$ space in the middle) by using only two orientations. I will let you think about why and how.

Once this is proven, Bart has his winning strategy because he always has a move no matter Art does.

Part $2$:

Partial Answer:

If Art starts anything other than filling the $2\times2$ center area with his first move, Bart can use similar strategy as in part $1$ to win. This means Art has to start by filling the center $2\times 2$ square.

I think Bart has a winning strategy by carefully leading the situation into the mirror game, but have thought for a while without being able to prove it. I hope this helps.