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 n squared boards generally, where n is less than 8, the following results will be of interest:—­

1 queen guards 2 squared board in 1 fundamental way.

1 queen guards 3 squared board in 1 fundamental way.

2 queens guard 4 squared board in 3 fundamental ways (protected).

3 queens guard 4 squared board in 2 fundamental ways (not protected).

3 queens guard 5 squared board in 37 fundamental ways (protected).

3 queens guard 5 squared board in 2 fundamental ways (not protected).

3 queens guard 6 squared board in 1 fundamental way (protected).

4 queens guard 6 squared board in 17 fundamental ways (not protected).

4 queens guard 7 squared board in 5 fundamental ways (protected).

4 queens guard 7 squared 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 n squared 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.

Where n = 6 there is 1 fundamental solution and 4 in all.

Where n = 7 there are 6 fundamental solutions and 40 in all.

Where n = 8 there are 12 fundamental solutions and 92 in all.

Where n = 9 there are 46 fundamental solutions.

Where n = 10 there are 92 fundamental solutions.

Where n = 11 there are 341 fundamental solutions.

Obviously n rooks may be placed without attack on an n squared board in n! ways, but how many of these are fundamentally different I have only worked out in the four cases where n equals 2, 3, 4, and 5.  The answers here are respectively 1, 2, 7, and 23. (See No. 296, “The Four Lions.”)