The debate in philosophy and cognitive science about the Chinese Room Argument has focused on whether it shows that machines can have minds. We present a quantitative argument which shows that Searle's thought experiment is not relevant to Turing's Test for intelligence. Instead, we consider a narrower form of Turing's Test, one that is restricted to the playing of a chess endgame, in which the equivalent of Searle's argument does apply. An analysis of time/space trade-offs in the playing of chess endgames shows that Michie's concept of Human Window offers a hint of what a machine's mental representations might need to be like to be considered equivalent to human cognition.