We analyze the problem-solving strategies of physics professors in a case where their physical intuition fails. A nonintuitive introductory-level problem was identified and posed to twenty physics professors. The problem placed the professors in a situation often encountered by students, and their response highlights the importance of intuition and experience in problem solving. Although professors had difficulty in solving the problem under the time constraint, they initially employed a systematic approach, for example, visualizing the problem, considering various conservation laws, and examining limiting cases. After finding that familiar techniques were not fruitful, they made incorrect predictions based on one of two equally important factors. In contrast, other more familiar problems that require the consideration of two important principles (for example, conservation of both energy and momentum for a ballistic pendulum) were quickly solved by the same professors. The responses of students who were given the same problem reflected no overarching strategies or systematic approaches, and a much wider variety of incorrect responses were given. This investigation highlights the importance of teaching effective problem-solving heuristics, and suggests that instructors assess the difficulty of a problem from the perspective of beginning students.