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Ingy döt Net 2013-10-27 23:48:49 +00:00
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import Control.Monad
import Data.List
-- given n, "queens n" solves the n-queens problem, returning a list of all the
-- safe arrangements. each solution is a list of the columns where the queens are
-- located for each row
queens :: Int -> [[Int]]
queens n = map fst $ foldM oneMoreQueen ([],[1..n]) [1..n] where
-- foldM :: (Monad m) => (a -> b -> m a) -> a -> [b] -> m a
-- foldM folds (from left to right) in the list monad, which is convenient for
-- "nondeterminstically" finding "all possible solutions" of something. the
-- initial value [] corresponds to the only safe arrangement of queens in 0 rows
-- given a safe arrangement y of queens in the first i rows, and a list of
-- possible choices, "oneMoreQueen y _" returns a list of all the safe
-- arrangements of queens in the first (i+1) rows along with remaining choices
oneMoreQueen (y,d) _ = [ (x:y, d\\[x]) | x <- d, safe x y 1]
-- "safe x y n" tests whether a queen at column x is safe from previous
-- queens as recorded in y, at the distance n rows away
safe x [] n = True
safe x (c:y) n = and [ x /= c , x /= c + n , x /= c - n , safe x y (n+1)]
-- prints what the board looks like for a solution; with an extra newline
printSolution y = let n = length y in
do mapM_ (\x -> putStrLn [if z == x then 'Q' else '.' | z <- [1..n]]) y
putStrLn ""
-- prints all the solutions for 6 queens
main = mapM_ printSolution $ queens 6