1

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?

Robert Z
  • 145,942
addcrtff
  • 13
  • 4

1 Answers1

1

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 $r_3\oplus r_6=1\oplus 3=2\not=0$.

Robert Z
  • 145,942
  • Yes, but how can I prove it using complete induction. Since n-2/2 can be an odd number, then we can not use the induction hypothesis that is: for every even number n, the first player will always win... – addcrtff Oct 06 '18 at 15:25
  • btw, thank you for modifying my question, it looks better now... – addcrtff Oct 06 '18 at 15:25
  • For (b) there is no need to use induction. Mirror strategy means that the first player replicates the move of the second player on the other half of the board. – Robert Z Oct 06 '18 at 15:27
  • Sorry to bother you again. Could you please show me how to prove it with or without induction... Thanks in advance! – addcrtff Oct 06 '18 at 15:54
  • Recall that the last player to play wins. So by the mirror strategy, if the second player has a move then the firts can take the same move on the other side. Eventually the two boards will be full of dominoes (with some $1\times 1$ empty squares) and the second player loses the game. – Robert Z Oct 06 '18 at 16:00
  • @addcrtff Any further doubt? – Robert Z Oct 07 '18 at 17:03
  • No, thank you very much! – addcrtff Oct 07 '18 at 17:32