The aim of this article is to present a set of simple exercises that help the students of a second course in quantum chemistry to understand that some molecular properties do not improve even when the molecular energy improves with an increasing number of basis functions in the expansion of the wavefunction. The essential idea is to create a dilemma for the students by showing that a wavefunction that gives a better energy (i.e., lower, according to the variational principle) may not yield a better value for another molecular property. The first exercise is to calculate the ground-state energy of the hydrogen atom for the contracted 1s STO-2G basis function; the second is to compute the ground-state energy for the STO-2G basis function reformulated as a linear variational function; the third is to calculate the expectation value of the electron--nuclear distance for the two approximate wavefunctions. Comparison of the approximate results with the exact one reveals starkly the dilemma of the "best wavefunction". The set of exercises presented here are interesting to any quantum mechanics course.