The "Method Actors" approach to prompt engineering involves thinking of large language models (LLMs) like actors, where prompts are scripts and responses are performances. This approach helps improve the performance of LLMs in solving complex reasoning tasks, like the New York Times Connections puzzle. The idea is to decompose complex tasks into smaller, more manageable sub-tasks that the LLM can imitate, like brainstorming potential solutions based on patterns from past puzzles. By carefully crafting prompts with vivid language and specific instructions, we can guide the LLM to reason more effectively. This method has proven successful, with LLMs using this approach surpassing human expert performance in solving Connections puzzles perfectly.