From 1984 through 1988, the authors worked with teachers using an inquiry approach to teach high school geometry courses with the aid of the "Geometric Supposers." Problems are a critical component of the approach because they focus attention and energy and guide students in the application, integration, and extension of knowledge. Inquiry problems differ from traditional, single-answer textbook problems in that they must leave room for student initiative and creativity. The observations presented in this paper about the delicate balance between specifying too much instruction and too little, which is part of creating and posing inquiry problems, are based on careful examination of students' inquiry problem papers. The paper also discusses speculations on whether these observations suggest general lessons for those seeking practical and successful strategies to introduce student inquiry into classrooms, with the hope of stimulating interest in and discussion of such strategies. Appendices include solutions to 12 problems and examples of students' answers. (Author/YP)