[math-fun] How Many Crossword Puzzles Are There?

My friend Bob Baillie <rjbaillie@frii.com> asks

Has anyone ever estimated how many crossword puzzles of size N by N there are, where the entries are words in English?

I assume that the answer for very large N must be exponential in the area of the crossword. Assume we can make two KxK tiles with the same boundary conditions but differeing internally, and that all the necessary fixups can be made on the outer edge of the pattern. Similar questions could be asked for a randomly generated language with English statistics, and for other real languages. But this is not really the question he's asking. Rich