TOPICS
Search

Bishops Problem


BishopsMax

Find the maximum number of bishops B(n) that can be placed on an n×n chessboard such that no two attack each other. The answer is 2n-2 (Dudeney 1970, Madachy 1979), giving the sequence 2, 4, 6, 8, ... (the even numbers) for n=2, 3, .... One maximal solution for n=8 is illustrated above. The numbers of distinct maximal arrangements for n=1, 2, ... bishops are 1, 4, 26, 260, 3368, ... (OEIS A002465). The numbers of rotationally and reflectively distinct solutions on an n×n board for n>=2 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
(1)

for n>1 (Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent formula also valid for n>1 is

 B(n)=2^(n-3)+2^(|_(n-1)/2_|-1),
(2)

where |_n_| is the floor function, giving the sequence for n=1, 2, ... as 1, 1, 2, 3, 6, 10, 20, 36, ... (OEIS A005418).

BishopsMin

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


See also

Bishop Graph, Chess, Kings Problem, Knights Problem, Queens Problem, Rooks Problem

Explore with Wolfram|Alpha

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 and 96, 1970.Guy, R. K. "The n 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.

Referenced on Wolfram|Alpha

Bishops Problem

Cite this as:

Weisstein, Eric W. "Bishops Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BishopsProblem.html

Subject classifications