On an ordinary chessboard, 8 by 8, every square can be guarded--that is,
either occupied or attacked--by 5 queens, the fewest possible. There are
exactly 91 fundamentally different arrangements in which no queen
attacks another queen. If every queen must attack (or be protected by)
another queen, there are at fewest 41 arrangements, and I have recorded
some 150 ways in which some of the queens are attacked and some not, but
this last case is very difficult to enumerate exactly.
On an ordinary chessboard every square can be guarded by 8 rooks (the
fewest possible) in 40,320 ways, if no rook may attack another rook, but
it is not known how many of these are fundamentally different. (See
solution to No. 295, "The Eight Rooks.") I have not enumerated the ways
in which every rook shall be protected by another rook.
On an ordinary chessboard every square can be guarded by 8 bishops (the
fewest possible), if no bishop may attack another bishop. Ten bishops
are necessary if every bishop is to be protected. (See Nos. 297 and 298,
"Bishops unguarded" and "Bishops guarded.")
On an ordinary chessboard every square can be guarded by 12 knights if
all but 4 are unprotected. But if every knight must be protected, 14 are
necessary. (See No. 319, "The Knight-Guards.")
Dealing with the queen on n² boards generally, where n is less
than 8, the following results will be of interest:--