Quarto is a deceptively tricky game for two players. There is a four by four board, and sixteen pieces. And all you have to do to win is to get four in a row of the same shape, or color, or size, or hollowness. Which wouldn't be that hard if you got to choose which pieces you played. But since your opponent will be choosing your pieces for you, the game gets a little trickier, and it's not just tic-tac-toe.
Familiarize yourself with the pieces. When looking at a piece you should instantly think of its four attributes. Keep looking at pieces on your own time until you can immediately see all attributes of any piece you glance at.
Hand your opponent lots of pieces of the same attribute until she places two in the same line. This will allow you to partially fill the board with your chosen attribute.
Count the number of pieces remaining that are not in your chosen attribute. For example, if you have been handing your opponent white pieces, as soon as he places two white pieces in the same line, you should count how many black pieces are left.
Hand your opponent another piece of the chosen attribute if the number of remaining opposite pieces is even. For example, if you've been handing her white pieces and there are four black pieces remaining, hand her another white piece. If you've been handing her white pieces and there are only three black pieces remaining, hand her a black piece instead.
Continue handing your opponent off-attribute pieces once he has made three in a row of your chosen attribute. If you have counted correctly, he will eventually have to hand you a piece of your chosen attribute, which you can play in the row to make four unless it becomes blocked.
Always watch the board to keep track of all attributes of pieces on the board. Remember, even if you are focusing on color, your opponent may be trying to win with shape or height.
Pay particular attention to any row or column with three pieces in it. You should immediately figure out if any type of piece could be a scoring piece in the remaining space. If so, you must be sure not to give your opponent any pieces of that type.
If you will be forced to hand an opponent a scorable piece next turn, take your current turn to block the row with your piece.
Don't forget about hollow and solid pieces!