Research on insight problem solving has mainly been treated within the scope of Gestalt psychology in the past. And now, insight problems are being investigated in cognitive psychology. However, little is known about how the process develops. In this study, we investigated the process of insight problem solving using a discovery task. The purpose of this research is to answer the following two points: First, how do constraint relaxation and the swith of a problem space search progress in insight problem solving? Second, what differences lie between successful subjects who reached the solution and unsuccessful subjects who did not? In this research, we analyzed the process of subjects searching their hypothesis space based on hypotheses that the subjects verbally reported. Additionally, we also measured the subjects' eye movement when they solved the problem. The following results were obtained. First, insight was not suddenly obtained in an instant, but rather gained through a relatively gradual mental relaxation process. Second, we found that the subjects who found the correct target reacted more sensitively to the irregular experimental results, as negative feedback caused rejection of the subjects' hypotheses. Our study detected insight problem solving processes in more detail by using not only subjects' verbal reports, but also psychological data obtained through eye movement measurements.