We identify novel mechanisms of general intelligence involving activation patterns of largescale brain networks. During hard, cognitively demanding tasks, the fronto-parietal network differentially activates relative to the default mode network, creating greater "separation" between the networks, while during easy tasks, network separation is reduced. In 920 adults in the Human Connectome Project dataset, we demonstrate that these network separation patterns across hard and easy task conditions are strongly associated with general intelligence, accounting for 21% of the variance in intelligence scores across individuals. Moreover, we identify the presence of a crossover relationship in which FPN-DMN separation profiles that strongly predict higher intelligence in hard task conditions reverse direction and strongly predict lower intelligence in easy conditions, helping to resolve conflicting findings in the literature. We further clarify key properties of FPN-DMN separation: It is a mediator, and not just a marker, of general intelligence, and FPN-DMN separation profiles during the task state can be reliably predicted from connectivity patterns during rest. We demonstrate the robustness of our results by replicating them in a second task and in an independent large sample of youth. Overall, our results establish FPN-DMN separation as a major locus of individual differences in general intelligence, and raise intriguing new questions about how FPN-DMN separation is regulated in different cognitive tasks, across the lifespan, and in health and disease.