TITLE: Is there a good argument for why you can't place 4 queens which cover a chessboard?
QUESTION [31 upvotes]: It has been known since the 1850's (or even much earlier) that 5 queens could be placed on an 8*8 
chessboard so that every square on the board lies in the same row, column, or diagonal as at least one
of the queens. It was also "known" that this could not be done with 4 queens. But I have not been able to
obtain or track down any rigorous mathematical proof of this that could be (or could have been) carried out
in a reasonable time by a human being with pencil and paper. There are altogether 635376 ways of placing
4 queens on an 8*8 chessboard. Does anyone know of a combinatorial algorithm, exploiting the symmetries of
the chessboard, which would reduce the number of cases to be considered to the hundreds (or, at most, the
low thousands.)?
                  This is, of course, a trivial problem for modern computers, which have many times since
verified that 5 queens are needed.

REPLY [5 votes]: I also wonder whether the formation came before or after 1990, because that is when Spencer came up with the lower bound for n=11.  I know first-hand that formations can be searched for by hand for say, up to n=20 with reasonable results (note that is much different than actual proof that the formation is minimal, for if we find the formation we've only found an upper bound).  So it's quite possible that someone found the formation first, and then Spencer might have been the first to observe it.  Of course it's also possible that Spencer's bound came first.
I found Dr Chatham's comment interesting enough to vote it up, although seeing as how I'm stuck at one reputation, I can't.