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We explore the creative problem-solving capabilities of modern large language models (LLMs) in a constrained setting. The setting requires circumventing a cognitive bias known in psychology as ''functional fixedness'' to use familiar objects in innovative or unconventional ways. To this end, we create MacGyver, an automatically generated dataset consisting of 1,600 real-world problems that deliberately trigger functional fixedness and require thinking 'out-of-the-box'. We then present our collection of problems to both LLMs and humans to compare and contrast their problem-solving abilities. We show that MacGyver is challenging for both groups, but in unique and complementary ways. For example, humans typically excel in solving problems that they are familiar with but may struggle with tasks requiring domain-specific knowledge, leading to a higher variance. On the other hand, LLMs, being exposed to a variety of highly specialized knowledge, attempt broader problems but are prone to overconfidence and propose actions that are physically infeasible or inefficient. We also provide a detailed error analysis of LLMs, and demonstrate the potential of enhancing their problem-solving ability with novel prompting techniques such as iterative step-wise reflection and divergent-convergent thinking. This work provides insight into the creative problem-solving capabilities of humans and AI and illustrates how psychological paradigms can be extended into large-scale tasks for comparing humans and machines.