Linguistics olympiad problems
The typical form of the problem consists of raw data and translations, and you have to figure out what morpheme corresponds to what meaning, and perhaps translate a few sentences. Although there are quirkier problems (like the one where you figure out how pieces get promoted in Japanese chess - that's in the first set of problems listed below). Problems are supposed not to require any previous knowledge of the language - and they don't. They're basically exercises in pattern-finding, though I did employ some heuristics learnt in linguistics class to speed up the process (e.g. singular nouns tend to be less marked than plurals). Most of all, they're fun! A few of them are hard, but many will take no longer than 5-10 mins to solve. And they really expand your mind as to the variety of linguistic structures available to the world's languages.
Anyway, there is now an international linguistics olympiad and there were attempts to set up a U.S. linguistics olympiad (though it doesn't seem to be running now), which means there are problems available at English to solve! Thomas Payne (of "Describing morphosyntax" fame) maintained the U.S. olympiad website; it included a description and history people may be interested in. Most of the problems seem to be on sites that are down, but thanks to the genius of Brewster Kahle and the Internet Archive, we can rescue them from oblivion. I've listed a few of them here:
Problems from the first international olympiad in linguistics (2003), incl. some sample problems
Puzzles from the 1998-2001 U.S. olympiad
Problems offered at the 27th and 28th (Russian) olympiad, sorted by "stages"
Russian-speaking readers can apparently find problems here and here, but I can't tell because I don't read Russian.
Wouldn't it be fun if there were introductory linguistics textbooks chock-full of problems integrated into the text? For example, while talking about morphology you could have a problem with Arabic words and their meanings and the task would be to figure out the principles of Arabic morphology - i.e. its root-and-pattern structure and non-concatenative nature. It would sure make more of an impression than passively reading about Semitic morphology. It would be a little like Lyle Campbell's Historical Linguistics, but synchronic.