@stephengdavey: I can think of literally no reason to want to do this. Never once had a chemistry exam where this was a requirement.
@jkemsley: To be clear: IMO it's ridiculous to require students to memorize the PT. Understand? Yes. Memorize? No.I am not an expert in the pedagogy of chemistry, but it seems to me that there is a lot more to be gained from the understanding of why the periodic table is organized in the fashion that it is, as opposed to memorization of where the elements are. Understanding the trends in electronegativity, atomic number, size (or memorizing them, even) would seem to me to be far more important.
I think it's fair to expect upper-level undergraduates and graduate students to have significant chunks (transition metals, etc.) of the periodic table memorized, more as a byproduct of familiarity than an actual end goal. (Here's one professor who expects this.)
From a practical perspective, I am always under the impression that most high school and early college students have access to the periodic table for exams. Does anyone out there require memorization of the whole periodic table? It's definitely a neat parlor trick, but why?
(To a great extent, this is part and parcel of the seeming resistance of the majority of chemists to memorization. For example, there's no better way to get a bunch of organic chemists really upset than to say "o-chem is just a bunch of memorization.")