# Amusements in Mathematics

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).
NON-ATTACKING CHESSBOARD ARRANGEMENTS.
We know that n queens may always be placed on a square board of n2 squares (if n be
greater than 3) without any queen attacking another queen. But no general formula for
enumerating the number of different ways in which it may be done has yet been
discovered; probably it is undiscoverable. The known results are as follows:—
Where n = 4 there is 1 fundamental solution and 2 in all.
Where n = 5 there are 2 fundamental solutions and 10 in all.