Large Language Models (LLMs) have been touted as AI models possessing advanced reasoning abilities. However, recent works have shown that LLMs often bypass true reasoning using shortcuts, sparking skepticism. To study the reasoning capabilities in a principled fashion, we adopt a computational theory perspective and propose an experimental protocol centered on 3-SAT -- the prototypical NP-complete problem lying at the core of logical reasoning and constraint satisfaction tasks. Specifically, we examine the phase transitions in random 3-SAT and characterize the reasoning abilities of LLMs by varying the inherent hardness of the problem instances. Our experimental evidence shows that LLMs are incapable of performing true reasoning, as required for solving 3-SAT problems. Moreover, we observe significant performance variation based on the inherent hardness of the problems -- performing poorly on harder instances and vice versa. Importantly, we show that integrating external reasoners can considerably enhance LLM performance. By following a principled experimental protocol, our study draws concrete conclusions and moves beyond the anecdotal evidence often found in LLM reasoning research.