Two players take turns placing dominoes on an $n \times 1$ board of squares, where each domino covers two squares and dominoes cannot overlap. The last player to play wins.
(a) Where would you place the first domino when $n = 11$?
(b) Prove that for $n$ even and positive, the first player can guarantee a win.
My attempt. I am trying to solve this problem, for the question (a), I figure it out that we should place it in the third or forth position to make sure we can win. However, for question (b), I think the first player should place the domino in the middle of the square and am trying to use complete induction to solve this question but I got stuck.
How to prove that for $n$ even and positive, the first player can guarantee a win?
As regards (b), you are right, if $n>0$ is even, the first player should place the domino in the middle of the board leaving two identical boards $(n/2-1)\times 1$ and then he follows a mirror strategy, that is he replicates each second player's move on the other half of the board.
For (a), read MJD's answer HERE. If $n=11$ then putting a domino on the third and fourth squares we leave two boards: $2\times 1$ and $7\times 1$. Since $r_2=1$ and $r_7=1$ and $r_2\oplus r_7=0$, this is a winning move. Also putting a domino on the first and second squares is a nice move because $r_9=0$. One the other hand, putting a domino on the fourth and fifth squares is not good because $r_3\oplus r_6=1\oplus 3=2\not=0$.