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Quarto has a set of basic rules which describes the winner as the one who align 4 pieces sharing a common trait. The alignment can be horizontal, vertical or diagonal.

An advanced rule states that if 4 pieces share a trait and are disposed to form a 2*2 square then it is a winning position.

The board being 4*4, the advanced rule adds 9 winning positions.

Is a draw possible with this advance rule ?

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A draw is definitely possible. For example here is a game I played against myself on an Android app.

However I've not seen any results about whether this version of the game is a draw with perfect play or not (in contrast to the basic game which is known to be a draw). The extra victory condition might mean that perfect play results in a win for one of the two players.

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