# Number of possible scenarios of tic-tac-toe

1. Nov 28, 2005

### hypermonkey2

I was debating with a friend whether it is easy to know the number of possible scenarios of tic-tac-toe possible. what do we think?

2. Nov 28, 2005

### matt grime

If you mean the number of possible final board positions in a game of tic tac toe then yes it is easy to find. Exercise: try doing it not talking about it. Start with the 2x2 case if it is easier.

3. Nov 28, 2005

### hypermonkey2

the easy answer would be 9!. however this cannot be true since it is possible for a game to fiish without having the board filled, no?

4. Nov 28, 2005

### hypermonkey2

no, i think we meant the entire game notation. as in, total game developments. how many possible ways are there for a tic tac toe game to develop?

5. Nov 29, 2005

### Werg22

Well if you are the first player to play, there is a minimum of 5 turns before you win. Considering a possibility, there is always 6 cases where there is no tic tac tow. If n is number of turn, that means that n-3 tokens can be anywhere on those 6 cases. This result is given by
$$\prod_{k=0}^{n-4} 6-k$$
Since there is 8 possibilies, you multiply this result by 8. There for turns above 5, you have to substract the possibilies that implies there is another tictactoe. To find this you'll have to find how many possibilty there is with 6 cases. And you will also have to find how many possibilities there is with 3 cases to find that, then you can calculate the final result.

Last edited: Nov 29, 2005