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I am trying to find out if there are any 'optimal' moves in the game, so as to trap my enemy to put his pawns in the place where I need them

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    $\begingroup$ Connect Four is a solved game. The first player can force a win by going in the middle column. Wikipedia describes it in en.wikipedia.org/wiki/Connect_Four#Mathematical_solution. $\endgroup$ – Kendall Frey Jun 3 '14 at 11:31
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    $\begingroup$ As a heads-up, I've flagged this for a suggested move; this question would likely be a better fit for boardgames.SE rather than puzzling.SE, since it concerns a game rather than a puzzle. $\endgroup$ – Steven Stadnicki Jun 4 '14 at 0:38
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forcing only happens when you can either make a 4 in a row next turn (forcing him to put his piece there to remain in the game) or

a 2 in a row with 2 spots one way and 1 spot the other:

2  1  B  B  2

you could place a piece in 1 while the 2s are free then you have a guaranteed win, he needs to place in any of the free spots to be able to block you

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In addition to the answers above, another key concept for Connect-4 is the notion of Zugzwang: having to make a move that one doesn't want to. Often parity is a way to force this; eventually a player is forced to play in a column that they don't want to. A very simple example of this concept is the following structure:

   x
  xo
  oo
__xx

Suppose it's X's turn to move here (the underscores represent an empty column). Then X can force a win by playing on the left hand side:

   x
  xo
  oo
x_xx

Now, o has to move into the second column to block the four-across - but doing so lets x win on the diagonal instead, by playing above o's piece. This particular configuration is obviously highly contrived - but 'locking' columns so that it's suicidal for one player to play there is often easier than it might seen; once that's done, it becomes a question of leaving that player with no other good moves.

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    $\begingroup$ Your point is a valid one, but this is not zugzwang, it's a double threat. If player A were in zugzwang, that would mean A has the move, and would be able to get a better result if he were allowed to pass this turn, than the best he can actually get. Here, if o were allowed to pass, x could still win along the row. $\endgroup$ – Rosie F Jul 29 '16 at 11:53
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In addition to @ratchetfreak's answer, it is possible to indirectly force a move in another way: when you can place a piece on your next turn such that there are multiple slots on which you can win (branching). This is a generalization of the "two free slots" case presented in his answer.

For example,

x
oxx
oxox

If x moves on row 2 (counting from the left), he is guaranteed a win on the subsequent turn; therefore, o must place a piece to block this branching first.

A more complex example of this could be multi-stage branching, which could look something like this:

oxoxo
oxxxo

o is forced to move in one of rows 1-5 in order to avoid losing; try playing it out to see why. (Needless to say, this would be very rare in a real game, but it's theoretically possible.)

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  • $\begingroup$ In your second scenario, if o plays 6 or 7, then the game will go 2 2 4 4 3 and then either 1 5 or 5 1 wins the game for x. $\endgroup$ – Joe Z. Jun 4 '14 at 2:00

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