# Winning strategy in game

Context: I played this game at one point (and lost) and now I'm wondering whether it was possible to win or not.

We have a hexagonal board like this: I'm defining the following terms:

1. Each hexagon is called a piece.
2. A wall is a border of the board. There are four walls- left, right, top and bottom.
The entire top border is a wall, i.e. the border shaped /¯\_/¯\_/¯\ is the top wall.
Similarly for the other walls.

The rules are as follows:

1. Player A starts from the piece marked A, and Player B starts from the piece marked B.
2. In a turn, a player can "claim" any one piece for himself and colour it (blue for A, pink for B).
3. A piece once claimed cannot be claimed by the opposing player.
4. To win, a player must create a continuous sequence of pieces from his starting hexagon to a non-adjacent wall (thus A must create a continuous sequence to the bottom or right wall, and B to the top or left wall).

If player A starts the game, can B have a winning strategy? Please explain in detail.

If player A starts then player B

cannot

have a winning strategy because

if they did, then player A could follow the same strategy and win sooner. That is: player A could pick a cell, pretend that player B has claimed it as B's first move, and then follow that winning strategy rotated through 180 degrees. If at some point B actually does pick the cell A pretended they started with, A should pick another yet-unclaimed cell and pretend B claimed that one. If B's winning strategy worked, then A would end up winning in this way -- which means that B can't have had a winning strategy after all.

• The classic strategy-stealing argument! – Jaap Scherphuis Apr 6 at 15:54
• Oh, man! I need to contact the maths teacher that made us play this game :) – Righter Apr 6 at 15:57
• But is there a winning strategy for A? – justhalf Apr 6 at 19:46
• That question wasn't asked :-). Often the answer is "yes, because if there's no B-path from B's starting place to another side then necessarily there is an A-path from A's starting place to another side". But I don't think that's true here, because you can draw parallel (SW-to-NE-ish) lines of A-cells and B-cells that block both A's and B's starting place from the non-adjacent sides. A might still have a win, but it's not so obvious. – Gareth McCaughan Apr 6 at 21:41
• @Righter Comment because I haven't checked all the lines, but if pieces are labelled like a chessboard (so A is on A5, B is on E1), as far as I can tell, B4 wins. It threatens D4, so B must play in C5,C4,D4,E5,or E4. If C4, then B3 threatens A2 and C2, so B must play B2, but then A wins with C3. If B plays anything else, A plays B2 and I think is just too fast. – alphacapture Apr 7 at 3:09