Apart from the overarching Westing game, chess is the most important game in Raskin's book. Several characters play it really well – we hear of the Judge, Theo, and Turtle all going up against Westing, and each of them is good enough to make it fairly far in games with him, even though Turtle's the only one who ever wins. In a way, the other heirs also play, because they make the same strategic mistake playing the Westing game that Theo and the Judge make playing chess with Westing: sacrificing the queen.
For those of you who don't know that much about chess, it's a game of strategy between two sides, which are usually seen as black and white. Significantly, each side begins the game with sixteen playing pieces, including a king and queen (it's not a coincidence that there are sixteen players in the Westing game). The object of the game is to capture your opponent's king. If we extend the idea of chess to the book as a whole, this idea would probably transfer to capturing the murderer (or identifying the true Sam Westing).
So, the idea of chess reverberates throughout the text. Moving pieces on a chessboard is related explicitly to the book's structure: each time the heirs go over to the Westing house, Theo makes a move on the chessboard there, only to find his opponent's next move when he returns. The Judge calls herself a pawn, which means she feels like she's being manipulated and doesn't have any power. Remember, too, that Otis calls himself and Crow the king and queen, and Crow does wind up as the person/piece who's temporarily sacrificed at the end of the game. Theo sees Sydelle's black and white dress as a chessboard, and wanders around asking people whether they know how to play. He's right that the answer to that question is an important clue, but unfortunately when he asks the right person, Sandy doesn't tell the truth.