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As we all know in standard tic-tac-toe the game can't be won if both players play optimally. So:

  1. Is Quarto a game that will end in a draw if both players play optimally?
  2. If yes, what are the most common drawing strategies?
  3. If no, are there one or more strategies that guarantee a player's win, as in games like Nim for example?

I'm a math teacher creating a course on math games and mathemagic tricks, so I'm looking for games that have optimal strategies that are explained with mathematical arguments, and stuff like that. I'm open to suggestions, but will most likely post a separate question when the project gets more concrete.

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  • What is your definition of 'fair'?
    – ire_and_curses
    Oct 28, 2012 at 4:43
  • I guessit would be a game that ends in a draw if both player play optimaly, or (equivalently?) a game where any player can win.
    – Jean-Sébastien
    Oct 28, 2012 at 5:35
  • While it's outside the scope of this particular question, I certainly recommend Nim (not the 'up to N' version that most people are familiar with, but the multiple-piles remove-any-number version that forms the basis of the mathematical theory) as a 'teaching game'. From there you can go to something like Kayles as a game whose theory revolves around Nimbers and show how they make it solvable, or even to something like Dots and Boxes as a game that can be partially analyzed through the theory...
    – Steven Stadnicki
    Oct 29, 2012 at 17:33
  • Great, I'll look the last two up
    – Jean-Sébastien
    Oct 29, 2012 at 17:48

1 Answer 1

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Quarto has been evaluated as draw if both players play optimally. There's a presentation on the result here:

http://www.cs.rhul.ac.uk/~wouter/Talks/quarto.pdf

Quarto seems to have been originally solved by Luv Goossens in 1998, although I'm not sure whether this was published or not. The presentation links to this old web page by Goossens:

http://web.archive.org/web/20041012023358/http://ssel.vub.ac.be/Members/LucGoossens/quarto/quartotext.htm

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  • Do you know if this conclusion takes into account the advanced rule ? Advanced rule adds winning position when a 2*2 square is made with 4 pieces sharing a unique trait. If not, maybe this advanced rule makes the game winnable in all situation.
    – Kii
    Sep 1, 2016 at 15:24
  • This conclusion doesn't consider the advanced rules. I saw an AI implementation that claimed to be unbeatable in both versions, but I didn't see any more details than just the assertion. I think it would be interesting as a separate question, although it would be worth defining exactly what counts as a 2x2 square in the question. Eg. Do the four corners count? How about squares not parallel to the grid?
    – tttppp
    Sep 2, 2016 at 7:16
  • I created the question. To answer your question, the 2*2 square must be "side by side". There are nine 2*2 squares in the 4*4 board.
    – Kii
    Sep 2, 2016 at 11:00
  • Great! When I was looking for research about this earlier I found some confusion about the definition of the 2x2 square here: mathpages.com/home/kmath352.htm
    – tttppp
    Sep 2, 2016 at 18:36

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