To start the Monthly Topic Challenge #3: Pencil and Paper Games : A simple question (not necessarily with a simple answer).
3 fanatic dots-and-boxes players come together.
Of course no one wants to wait while the others play, and no one likes king making.
So they decide on a team match: 2 versus 1
To compensate their numerical disadvantage, the single player team gets 2 advantages:
- They only need one of the boxes to win.
- They can set the player order
Assuming perfect play, will the single player team win on a 3*3 area?
Assuming perfect play, will the single player team win on a 5*5 area?
Since wikipedia mentions variants: They start with an empty board. For the rules see e.g https://en.wikipedia.org/wiki/Dots_and_Boxes
On a 3 by 3 the single player has a big advantage when choosing to play first but winning is not automatic.
For example if player1 plays orange (in picture 1), the second position will be possible at player1's next turn, and the area played can (with the help of player2) always be taken by player3, who then can play the safe spot in the other area (e.g. picture 3)
The single player can win if they can take the last square. Thus player 1 can play 'defensively': If scoring 2 boxes with 1 line can be prevented while picking the appropriate play order, player 1 will win.
This is e.g. trivially possible in 1 by 1 (by starting first).
This is not possible for 5 by 5 (and larger). The 2-player team can force the formation of a 2 by 2 or larger enclosed area and then score 2 boxes with 1 line when filling that area!
Player 1 will also win however if there are 3 or more areas in the end phase (when all moves open scoring for the next player). Thus player 1 can also play 'offensively': Create as many areas as possible. (I do not see a clear way to do this fast enough.)
For 5 by 5 a combination seems best: Choose 3rd position so that the 2-player team must force a box. Create as many areas as possible, while hindering them forming their box whenever useful. After some playing against myself, I expect that the team wins, but it is a close call, and I did not work out all possibilities