 However, it has not been proven that no computationally cheap way of determining the best move in a chess position exists, nor even that a traditional alpha-beta searcher running on present-day computing hardware could not solve the initial position in an acceptable amount of time.The difficulty in proving the latter lies in the fact that, while the number of board positions that could happen in the course of a chess game is huge (on the order of at least 1043 to 1047), it is hard to rule out with mathematical certainty the possibility that the initial position allows either side to force a mate or a threefold repetition after relatively few moves, in which case the search tree might encompass only a very small subset of the set of possible positions.It has been mathematically proven that "generalized chess" (chess played with an arbitrarily large number of pieces on an arbitrarily large chessboard) is EXPTIME-complete, meaning that determining the winning side in an arbitrary position of generalized chess provably takes exponential time in the worst case; however, this theoretical result gives no lower bound on the amount of work required to solve ordinary 8x8 chess.