I don't remember who introduced me to the game originally, but it turns out that we aren't the only ones who play it. The page's author calls it "The Incompatible Food Triad" and says it was introduced to him by a philosophy professor in the 1970s. He discusses the difficulty of the problem -- and it is hard, play around with it awhile and you'll see, I've been trying for over a decade and have yet to come up with a satisfactory solution -- and then invites people to submit their ideas. The other possibility he raises, which I have also considered, is that it might be an unsolvable problem. Maybe it's not literally possible to come up with such a combination. Of course, that seems like something you ought to be able to prove, and no one has managed that, either.
Anyway, I'm mostly tickled to learn that this thought experiment that I've been playing casually for so long turns out to have a little bit of a following -- it's even in Wikipedia. Thanks to P for the link.