Bishops Problem
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
![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.gif) |
(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.
Ahrens, W. Mathematische Unterhaltungen und Spiele, Vol. 1, 3rd ed. Leipzig, Germany: Teubner, p. 271, 1921.
Dudeney, H. E. "Bishops--Unguarded" and "Bishops--Guarded." §297 and 298 in Amusements in Mathematics. New York: Dover, pp. 88-89 and 96, 1970.
Guy, R. K. "The 
Queens Problem."
§C18 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 133-135,
1994.
Madachy, J. Madachy's Mathematical Recreations. New York: Dover, pp. 36-46, 1979.
Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 74-75, 1995.
Sloane, N. J. A. Sequences A002465/M3616 and A005418/M0771 in "The On-Line Encyclopedia of Integer Sequences."
Watkins, J. Across the Board: The Mathematics of Chessboard Problems. Princeton, NJ: Princeton University Press, 2004.
Weisstein, Eric W. "Bishops Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BishopsProblem.html
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