A closed form solution for n-queens, from "Explicit Solutions to the N-Queens Problem for all N"

N-Queens.hs

type Queen = (Int,Int)

solveNot6kplus2 :: Int -> [Queen]
solveNot6kplus2 n = firstHalf ++ secondHalf
    where 
        firstHalf = [(j, 2 * j) | j <- range]
        secondHalf = [(n `div` 2 + j, 2 * j - 1) | j <- range]
        range = [1 .. n `div` 2]

solveNot6k :: Int -> [Queen]
solveNot6k n = firstHalf ++ secondHalf
    where 
        firstHalf = [(n+1 - j, n - getY j) | j <- range]
        secondHalf = [(j, 1 + getY j) | j <- range]
        getY j = mod (2 * (j-1) + n `div` 2 - 1) n
        range = [1 .. n `div` 2]

solve n 
    | mod n 2 /= 0 = (n,n) : solve (n-1)
    | mod n 6 /= 0 = solveNot6k n
    | mod (n - 2) 6 /= 0 = solveNot6kplus2 n


isValid :: [Queen] -> Bool
isValid stack = foldl validate True pairs
    where pairs = [(s1,s2) | s1 <- stack, s2 <- stack, s1 /= s2]
          validate False _ = False
          validate True ((x1,y1), (x2,y2)) =
              x1 /= x2 && y1 /= y2 && (abs $ y2 - y1) /= (abs $ x2 - x1)

check = all isValid [solve n | n <- [4..1000]]

main = print $  length $ solve 1000000
--main = print check

{--
> ghc --make -O3 -fforce-recomp nqueens.hs; time ./nqueens
[1 of 1] Compiling Main             ( nqueens.hs, nqueens.o )
Linking nqueens ...
1000000

real	0m0.074s
user	0m0.064s
sys	0m0.008s
--}