Thanks, everyone! Now I can get to my real question: The curve shown in http://math.berkeley.edu/publications/newsletter/2002/2002web%20photo%20scan... is a noncontractible simple closed curve in the triply-punctured plane with the property that if you fill in any one of the three punctures, the curve becomes contractible. For what n does there exist a noncontractible simple closed curve in the n-times punctured plane with the property that if you fill in any one of the n punctures, the curve becomes contractible? This question arose in connection with the work of Erik Demaine et al. on "picture-hanging puzzles". Jim Propp