Number of possible scenarios of tic-tac-toe

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Discussion Overview

The discussion revolves around determining the number of possible scenarios in a game of tic-tac-toe, including both final board positions and the entire game development process. Participants explore various approaches and calculations related to this combinatorial problem.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the ease of determining the number of possible scenarios in tic-tac-toe, prompting a debate on the topic.
  • Another participant suggests that finding the number of final board positions is straightforward, proposing an exercise to explore simpler cases first.
  • A different viewpoint challenges the assumption that the number of scenarios is simply 9!, noting that games can end with unfilled boards.
  • Further clarification is provided regarding the interest in the total game developments rather than just final positions, indicating a more complex inquiry.
  • One participant proposes a mathematical approach involving the number of turns and configurations, suggesting a formula to account for different game scenarios and the need to subtract cases that lead to a win.

Areas of Agreement / Disagreement

Participants express differing views on the ease and methods of calculating the number of tic-tac-toe scenarios. There is no consensus on a definitive approach or solution, and the discussion remains unresolved.

Contextual Notes

Participants note limitations in their calculations, such as the dependence on the number of turns and the configurations of the board at various stages of the game. Some assumptions about game endings and board filling are also highlighted but not fully resolved.

hypermonkey2
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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?
 
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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.
 
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?
 
matt grime said:
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.
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?
 
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
[tex]\prod_{k=0}^{n-4} 6-k[/tex]
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 possibility 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.
 
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