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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 '12 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 '12 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 '12 at 17:33
    
Great, I'll look the last two up –  Jean-Sébastien Oct 29 '12 at 17:48

1 Answer 1

up vote 3 down vote accepted

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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