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Will China displace the United States as the world's leading power by the centenary of Communist rule in 2049? The outcome will depend on many unknowns including what the two countries do over the next three decades. US leaders are likely to rely on their mental maps of how the world works. The primary sources of their mental maps tend to come from historical analogies and from international relations theories. Both are highly imperfect representations of reality. Historical metaphors and analogies are rife in the debate over how to understand the current rise of China, but three are particularly salient: a Thucydides trap; a new Cold War; and 1914 sleepwalkers. This article discusses the merits and demerits of relying on each of these analogies in turn, and assesses the strengths and weaknesses of prior mental maps that guided US policy-makers during the post-Cold War era. Chinese elites expect to replace the US as the leading global power by 2049. How should the US respond? Two prevalent historical analogies are misleading: a Thucydides trap about power transition, and a new Cold War that ignores the three-dimensional nature of US–China interdependence. More promising is the cautionary narrative of sleepwalking into the First World War. A successful strategy must lay out conditions for a cooperative rivalry starting with a careful net assessment of power resources and the formulation of feasible goals.