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Let's consider a game on a $3\times3$ board. The first player places a cross in any free square. The second player places a circle in another free square. This continues. The first to have $3$ shapes in a row (also in the diagonals) has won.

Show that the second player can always be sure not to lose.

I would like to find something else than a game tree.

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    I'm curious why a game tree isn't sufficient. It would be very small in this case. Do you just want an answer with more insight? – 2017-02-22
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    @TheCount Could you please show me the game tree? But yea, I am searching something more insight as you said... – 2017-02-22
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    Well, Shaun has solved things, now. – 2017-02-22
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    Duplcate of http://math.stackexchange.com/q/2156604/265466 – 2017-02-22

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The question is flawed. It's possible that the game could end in a draw.

For instance,

$$\begin{align} X O O \\ X X O \\ O X X \end{align}$$

is a complete $3\times 3$ board that ends in a draw. (If diagonals are included and the first player is X, the first player could win here too.)

The second player can force a draw.

Use a case by case analysis for the first player starting in the top right, the top middle, and the middle squares, then use the symmetry of a square to eliminate similar boards, like so:

$$\begin{align} X_1 O_1 \phi \\ X_2 \phi \phi \\ O_2 \phi \phi \end{align}$$

for the top right case,

$$\begin{align} \phi X_1 O_1 \\ \phi X_2 \phi \\ \phi O_2 \phi \end{align}$$

for the top middle case, and

$$\begin{align} \phi X_2 \phi \\ \phi X_1 O_2 \\ \phi O_1 \phi \end{align}$$

for the middle case.

(Here $\phi$ is an empty square and $\Delta_n$ is the $n$th move of the player with symbol $\Delta$, assuming $X$ comes first.)

In each case the second player will always seek to prevent the first player from getting three $X$s in a row, so, up to the symmetries of a square, the cases above suffice to show that the second player can force a draw.

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    Is a draw counted as a win? – 2017-02-22
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    I suppose drawing isn't losing. – 2017-02-22
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    Yea sorry that was what I meant, show that the second player can force a draw... – 2017-02-22
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    And what if the second move of the first player in your first case would be to place it in the center? – 2017-02-22
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    The first player wouldn't do that, @DanielCortild; or would but would soon be cut off for three in a row options. I'm not sure that's clear though. – 2017-02-22