Well, if we were to delve into a more serious analysis of what might be going on in the brain during the act of trying to fit a round object into a square hole, we could explore it from a cognitive and psychological perspective:

1. **Perseveration**: This term refers to the tendency to persist with a specific behavior or thought pattern even when it's unproductive. In this scenario, perseveration might manifest as an individual fixating on the task at hand without considering alternative solutions.

2. **Functional Fixedness**: This cognitive bias limits a person's ability to see an object's alternative uses beyond its traditional function. Attempting to force-fit the round peg into the square hole could be indicative of functional fixedness at play.

3. **Impaired Problem-Solving Skills**: The struggle with mismatched shapes could highlight difficulties in problem-solving strategies such as abstraction, visualization, and spatial reasoning.

4. **Lack of Flexibility**: A rigid mindset that refuses to adapt or consider different approaches may hinder successful completion of tasks like fitting geometric shapes together.

5. **Executive Function Challenges**: Individuals facing executive function deficits may struggle with tasks requiring planning, organization, and problem-solving—traits crucial for successfully navigating spatial puzzles like inserting round objects into square openings.

Ultimately, while trying to fit that round peg into a square hole might evoke lighthearted amusement on one level, exploring the underlying cognitive processes can shed light on nuanced aspects of human cognition and behavior patterns related to problem-solving and mental flexibility.