Stone by stone, you try to construct enclaves, engulfing those of your opponent, who is all the time trying to engulf your own. Adding to the complications, there are usually several skirmishes going on simultaneously in different corners of the board. If chess is like a medieval battle, it is sometimes said, Go is more like a world war. And it can be maddeningly difficult to determine who is ahead.

''In chess, if a player loses even a single pawn at world champion level, it can decide the game maybe 99 percent of the time,'' said Dr. Hans Berliner, a computer scientist at Carnegie-Mellon University in Pittsburgh who is an expert on computer chess. ''In Go, you keep hearing people say that you can lose a life-and-death battle along the edge of the board, but that is far from deciding the outcome. You can go on to other battles. It's a very different kind of game.''

From the point of view of a computer, the difference could not be more profound. Because of the tight constraints in how chess pieces can be moved, a player is faced with an average of only about 35 legal moves to consider with each turn. Computer programs like Deep Blue analyze these moves, considering the opponent's possible countermoves, and then the countermoves to the countermoves. In computer chess terminology, each move and its response is called a ply. The fastest chess programs look ahead seven or eight plies into the game.

The result is a densely proliferating tree of possibilities with the branches and twigs representing all the different ways the game could unfold. Looking ahead just seven plies (14 individual chess moves) requires examining 3514 (more than a billion trillion) leaves representing all the various outcomes.

As the computer tries to look deeper, the number of possibilities explodes. Programmers have learned clever ways to ''prune'' the trees, so that all but a fraction of the paths can be discarded without plumbing them all the way to the bottom. Even so, a chess-playing computer looking ahead seven plies might consider as many as 50 or 60 billion scenarios each time its turn comes around.

As bad as that sounds, in Go the situation is drastically worse. The tree of possible moves is so broad and dense that not even the fastest computer can negotiate it. The first player can put a stone in any of 361 places; the opponent can respond by placing a stone on any of 360 places, and so on. As the game continues, there are steadily fewer possible places to play. But, on average, a player is faced with about 200 possible moves, compared with just 35 in chess.

As a computer scientist would put it, the branching factor is much higher for Go than for chess. In chess the approximate number of possible board positions after only four moves is typically 35x35x35x35= 1,500,625. For Go, the number is 200x200x200x200=1,600,000,000 -- and far more toward the beginning of a game. Search one ply deeper and the numbers rapidly diverge: about 1.8 billion possible outcomes for chess and 64 trillion for Go.