The Traveling Salesperson Problem (TSP) is a classic algorithmic puzzle: find the shortest possible route that visits every city in a list exactly once and returns to the starting city. It's a deceptively simple problem with profound implications for logistics, planning, and even microchip design. Now, researchers are exploring whether large language models (LLMs), like the kind that power AI chatbots, can crack this combinatorial nut. A new study investigates how GPT-3.5 Turbo, a powerful LLM, performs on the TSP. Researchers tested several approaches, including giving the model no prior examples (zero-shot), a few solved examples (few-shot), and prompting it to "think" step-by-step (chain-of-thought prompting). They also fine-tuned a version of GPT-3.5 Turbo on a set of TSP problems with a fixed number of cities. The results? Surprisingly positive. The fine-tuned model performed exceptionally well on problems similar in size to its training data and even generalized reasonably well to larger problems. This suggests that LLMs can learn the underlying principles of optimization, even for complex problems like the TSP. To further boost performance, the researchers used a "self-ensemble" method, essentially prompting the model multiple times and selecting the best solution. This technique significantly improved the quality of the routes generated, demonstrating the potential of combining LLMs with established optimization strategies. While LLMs show promise for tackling the TSP, challenges remain, especially with larger problems. The model sometimes struggles with the sheer number of possible routes, and its performance can be limited by the maximum amount of text it can process. Future research will likely focus on overcoming these limitations, perhaps by integrating LLMs with more traditional optimization algorithms or developing new prompting techniques. The prospect of using AI to solve complex planning and logistics problems is tantalizing, and this research provides a compelling glimpse into that future.