I had to dig this out for a slashdot debate and remembered that I’m reasonably proud of it. It only relates to teaching in that I wrote it up as a handout for my AP Physics class. The basic idea is, it has been asserted that allowing an ice cube to melt in a cup of water does not change the level of the water. Is it true? Check out the handout and see.
It’s important to recognize the limits of this model, such as the assumption that the density of original water and the density of the melted water would be the same, or the fact that the melting ice will cool the existing water and so (conceivably) change its density.