'Why are we curious?' has been among the central puzzles of neuroscience and psychology in the past decades. A popular hypothesis is that curiosity is driven by intrinsically generated reward signals, which have evolved to support survival in complex environments. To formalize and test this hypothesis, we need to understand the enigmatic relationship between (i) intrinsic rewards (as drives of curiosity), (ii) optimality conditions (as objectives of curiosity), and (iii) environment structures. Here, we demystify this relationship through a systematic simulation study. First, we propose an algorithm to generate environments that capture key abstract features of different real-world situations. Then, we simulate artificial agents that explore these environments by seeking one of six representative intrinsic rewards: novelty, surprise, information gain, empowerment, maximum occupancy principle, and successor-predecessor intrinsic exploration. We evaluate the exploration performance of these simulated agents regarding three potential objectives of curiosity: state discovery, model accuracy, and uniform state visitation. Our results show that the comparative performance of each intrinsic reward is highly dependent on the environmental features and the curiosity objective; this indicates that 'optimality' in top-down theories of curiosity needs a precise formulation of assumptions. Nevertheless, we found that agents seeking a combination of novelty and information gain always achieve a close-to-optimal performance on objectives of curiosity as well as in collecting extrinsic rewards. This suggests that novelty and information gain are two principal axes of curiosity-driven behavior. These results pave the way for the further development of computational models of curiosity and the design of theory-informed experimental paradigms.