What is the thinnest soap film (excluding the case of zero thickness) that appears black when illuminated with light with a wavelength of 500 nm ? the index of refraction of the film is 1.32, and there is air on both sides of the film?

This is easy! you just have to realise that it is the cancellation of waves (the light waves are perfectly out of phase) that create this 'black' phenomenon. Therefore to cancel out one wave, you need another wave exactly superimposed (in superposition) and out of phase with the original wave by HALF a wavelength. Put another way, the second photon needs to travel exactly half a wavelength more distance than the first photon.

Since the photons are reflected, the minimum thickness must be 1/4 of a wavelength (since the the photon must travel TWICE through the 1/4 thickness to make 1/2 a wavelength in difference).