Find the maximum number of bishops that can be placed on an chessboard such that
no two attack each other. The answer is (Dudeney 1970, Madachy 1979), giving the sequence 2, 4,
6, 8, ... (the even numbers) for , 3, .... One maximal solution for is illustrated above. The numbers of distinct maximal arrangements
for ,
2, ... bishops are 1, 4, 26, 260, 3368, ... (OEIS A002465).
The numbers of rotationally and reflectively distinct solutions on an board for is
(1)
for
(Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent
formula also valid for
is
(2)
where
is the floor function, giving the sequence for
, 2, ... as 1, 1, 2, 3, 6, 10, 20,
36, ... (OEIS A005418).
The minimum number of bishops needed to occupy or attack all squares on an chessboard is , arranged as illustrated above.
that can be placed on an 
chessboard such that
no two attack each other. The answer is 
(Dudeney 1970, Madachy 1979), giving the sequence 2, 4,
6, 8, ... (the even numbers) for 
, 3, .... One maximal solution for 
is illustrated above. The numbers of distinct maximal arrangements
for 
,
2, ... bishops are 1, 4, 26, 260, 3368, ... (OEIS A002465).
The numbers of rotationally and reflectively distinct solutions on an 
board for 
is
![B(n)={2^((n-4)/2)[2^((n-2)/2)+1] for n even; 2^((n-3)/2)[2^((n-3)/2)+1] for n odd](/images/equations/BishopsProblem/NumberedEquation1.svg)
(Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent
formula also valid for 
is
is the floor function, giving the sequence for
, 2, ... as 1, 1, 2, 3, 6, 10, 20,
36, ... (OEIS A005418).
chessboard is 
, arranged as illustrated above.
Queens Problem." §C18 in