It's a 4x4 grid with a set of pieces.
Each piece has 4 properties: {Tall or Short}, {Light or Dark}, {Unfilled top or Filled top}, {Circular or Square}.The goal of the game is to get 4 pieces in a row or a square which all share one property in common. You do not pick what piece you move, however. You select a piece for your opponent to place, and then they select a piece for you to place and so on.
Here are a few questions I was curious about:
1. How many different ways are there to tie? (A tie occurs if all 16 pieces are placed and no one has won).
2. If both players play optimally, can either the 1st or 2nd player force a win, or must it always end in a draw?
Question #2 has been answered using a computer search.
Details are here: http://wouterkoolen.info/Talks/quarto.pdf
A related analysis of the symmetries in the game: http://web.archive.org/web/20041012023358/http://ssel.vub.ac.be/Members/LucGoossens/quarto/quartotext.htm
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