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So, I have this problem for my homework in which I'm asked to show that the amount of strategies for player number 1 in tic-tac-toe is between

$$9*7^8*5^{48} \text{ and } 9*7^8*5^{48}*3^{192}$$ But I can't see why and I don't fully understand the answer provided in the following link: Tic-Tac-Toe Game.

As far as I know (from the Wikipedia definition of strategy), a strategy tells the player what to move for every situation in the game. So, my approach was to consider strategies as functions that map an information set (that represents a particular situation of the game) into an action. Then my problem is to count how many of such functions exist.

I can map the first information set to 9 possible values (in the first move all 9 cells are available). Then player number 2 makes his move. For each of the 9 moves of the first player there are 8 possible moves for player number 2. So, after 2 moves there are 72 possible situations (information sets). For any of those 72 situations player number 1 can take 7 actions, that is $7^{72}$. It's clear that if I continue reasoning this way I won't get the desired result but I don't see why my reasoning is wrong. Can you tell me where is my mistake, or if I'm missunderstanding the definition of strategy? Thanks in advance

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Your mistake is that after the first move is made, the $9$ possibilties for that move should not be considered again as far as the strategies go in response to player two's countermove. That is, for each of the $9$ strategies relative to the first move, player $1$ only has to consider the strategies relative to each of the $8$ countermoves, and with $7$ possible strategies for each of the $8$ countermoves, that gives $9 \cdot 7^8$ strategies to cover the first three moves of the game.

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