What is the formula for winning at Pentago?

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Discussion Overview

The discussion revolves around finding a formula or strategy for winning at the board game Pentago, particularly for the first player. Participants explore the concept of perfect play, comparisons to tic-tac-toe, and the feasibility of a general formula applicable at all stages of the game.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants compare Pentago to tic-tac-toe, suggesting that similar strategies might apply, but the complexity of Pentago is acknowledged.
  • One participant claims that the first player can always win in Pentago, similar to tic-tac-toe, but later corrects this to acknowledge that perfect play in tic-tac-toe results in a draw.
  • Another participant references a website that stores results of all positions with fewer than 18 stones, indicating that every first move is winning except for the corners.
  • There is a question about what constitutes perfect play in Pentago, with one participant suggesting that an algorithm exists to determine it.
  • A mathematical approach is proposed, involving encoding the board position with variables and constructing a polynomial to determine optimal moves, though it is noted that this is impractical for human players.
  • One participant expresses doubt about the possibility of simplifying the polynomial approach for practical use.

Areas of Agreement / Disagreement

Participants generally agree that Pentago is more complex than tic-tac-toe and that perfect play is a challenging concept. However, there is no consensus on a specific formula or strategy that can be universally applied by human players.

Contextual Notes

The discussion highlights the limitations of current strategies and the complexity of deriving a usable formula for Pentago, particularly for human players, as indicated by the reliance on supercomputers for solving the game.

greswd
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Pentago is a board game and you can think of it as a highly advanced version of tic-tac-toe.

With the aid of supercomputers, it has been strongly solved. Just like tic-tac-toe, it is possible for the player who starts first to always in.

I'm looking for a formula to always win at Pentago if I'm the first player. For tic-tac-toe, always mark the central square. For Pentago, never touch the 4 corners.

Tic-tac-toe is simple enough, but what is a formula for Pentago that can be applied by humans anytime?

Just like once someone memorises the algorithm, he can solve any Rubik's cube problem.
 
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greswd said:
Just like tic-tac-toe, it is possible for the player who starts first to always in.

Tic-tac-toe is impossible to lose for either player unless they make mistakes. If both players know what they are doing, it will end in a draw.
 
Oops. My bad.

So in Tic-Tac-Toe perfect play, it will always end in a draw. What constitutes perfect play for Pentago?
 
look here:

https://perfect-pentago.net/

They store the result of all positions with less than 18 stones. Every first move is winning, except for the corners.
 
willem2 said:
look here:

https://perfect-pentago.net/

They store the result of all positions with less than 18 stones. Every first move is winning, except for the corners.

omg i didnt see that earlier. looks like they've just updated the site
 
but anyway, is there a general formula that can be applied to all stages of the game? what constitutes perfect play?
 
greswd said:
but anyway, is there a general formula that can be applied to all stages of the game? what constitutes perfect play?
Depends on what you mean by "formula".

Is there an algorithm to determine perfect play? Yes.

Is there a polynomial formula that returns an optimal move for any possible position. Yes, trivially. Encode the board position as 36 variables x1 through x36 with values 0 (no stone), 1 (black stone) or -1 (white stone). Then a polynomial with 336 terms exists which will return the required result. That polynomial could, in principle, be constructed as a sum of 336 products like the following one that embodies the starting board position:

k * (x1 - 1 )(x1+1)(x2-1)(x2+1) ... (x36-1)(x36+1)

Is there a formula that a human can use in reasonable time with pencil and paper? The fact that the Perfect Pentago site used a supercomputer to accomplish the task suggests that no such formula is known.
 
perhaps it could be simplified? that would be very hard though
 

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