suppose an ordinary game of tic-tac-toe, two players take turn placing a cross or a circle. You win by getting three in a row (horizontally, vertically or diagonally). There is no winning strategy for any of the players.
If you now add an extra square as the picture below. Does any player have a winning strategy?
I believe there is now winning strategy for any player. However I am having a big problem trying show that no player has a winning strategy... Anyone who can help?
[ I am also supposed to construct a game, with similar rules as tic-tac-toe, with as few squares as possible (does not have to be rectangular), where the player that starts will have a winning strategy... I've managed to get to seven squares, but I am not sure if there is any other possibilities. ]
Lulu's comment hits the nail on the head. The first player can force a win. Player 1 puts a $\times$ here: $$\matrix{- & - & - \\ - & - & - \\ - & - & \times & -}$$ If the second player does not put a $\Large\circ$ somewhere on the bottom row, then Player 1 plays like so: $$\matrix{? & ? & ? \\ ? & ? & ? \\ - & \times & \times & -}$$ Now, player 1 has two winning moves, which cannot be blocked. Thus, Player 2 needs to play somewhere on the bottom row.
Let's consider the case where player 2 plays like so:
$$\matrix{- & - & - \\ - & - & - \\ \Large\circ & - & \times & -}$$
Then player 1 plays:
$$\matrix{- & - & - \\ - & - & - \\ \Large\circ & \times & \times & -}$$
which forces:
$$\matrix{- & - & - \\ - & - & - \\ \Large\circ & \times & \times & \Large\circ}$$
Then player 1 plays:
$$\matrix{- & - & - \\ - & \times & - \\ \Large\circ & \times & \times & \Large\circ}$$
which gives player 1 two possible winning moves, and player 2 will lose after player 1's next turn.
Otherwise, if player 2 did not respond with that move, then the board looks like this:
$$\matrix{- & - & - \\ - & - & - \\ - & ? & \times & ?}$$
Player 1's next move is $$\matrix{- & - & - \\ - & \times & - \\ - & ? & \times & ?}$$ which forces $$\matrix{\Large\circ & - & - \\ - & \times & - \\ - & ? & \times & ?}$$ At this point, player 1's next move is $$\matrix{\Large\circ & - & - \\ - & \times & \times \\ - & ? & \times & ?}$$ Player 2 has no way to prevent player 1's victory.