The ability to represent oneself and others as agents with knowledge, intentions, and belief states that guide their behavior - Theory of Mind - is a human universal that enables us to navigate - and manipulate - the social world. It is supported by our ability to form mental models of ourselves and others. Its ubiquity in human affairs entails that LLMs have seen innumerable examples of it in their training data and therefore may have learned to mimic it, but whether they have actually learned causal models that they can deploy in arbitrary settings is unclear. We therefore develop a novel experimental paradigm that requires that subjects form representations of the mental states of themselves and others and act on them strategically rather than merely describe them. We test a wide range of leading open and closed source LLMs released since 2024, as well as human subjects, on this paradigm. We find that 1) LLMs released before mid-2025 fail at all of our tasks, 2) more recent LLMs achieve human-level performance on modeling the cognitive states of others, and 3) even frontier LLMs fail at our self-modeling task - unless afforded a scratchpad in the form of a reasoning trace. We further demonstrate cognitive load effects on other-modeling tasks, offering suggestive evidence that LLMs are using something akin to limited-capacity working memory to hold these mental representations in mind during a single forward pass. Finally, we explore the mechanisms by which reasoning models succeed at the self- and other-modeling tasks, and show that they readily engage in strategic deception.