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:--
1 queen guards 2² board in 1 fundamental way.
1 queen guards 3² board in 1 fundamental way.
2 queens guard 4² board in 3 fundamental ways (protected).
3 queens guard 4² board in 2 fundamental ways (not protected).
3 queens guard 5² board in 37 fundamental ways (protected).
3 queens guard 5² board in 2 fundamental ways (not protected).
3 queens guard 6² board in 1 fundamental way (protected).
4 queens guard 6² board in 17 fundamental ways (not protected).
4 queens guard 7² board in 5 fundamental ways (protected).
4 queens guard 7² board in 1 fundamental way (not protected).
