Find the maximum number of bishops which 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 number of distinct
maximal arrangements of bishops for , 2, ... are 1, 4, 26, 260, 3368, ... (Sloane's A002465). The number of
rotationally and reflectively distinct solutions on an board for is

(Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent formula is

where is the Floor Function, giving the sequence for , 2, ... as 1, 1, 2, 3, 6, 10, 20, 36, ... (Sloane's A005418).

The minimum number of bishops needed to occupy or attack all squares on an Chessboard is , arranged as illustrated above.

**References**

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, 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 ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-26