[Haskell-cafe] xkcd #287 "NP-Complete"

[Albert Y. C. Lai]

http://xkcd.com/c287.html

import Data.Array
import Control.Monad

-- exactly n v
-- items in v that sum to exactly n
-- returns list of solutions, each solution list of items
exactly :: (Real a) => a -> Array Int a -> [[a]]
exactly 0 v = return []
exactly n v = do
   i <- indices v
   guard (v!i <= n)
   liftM (v!i :) (exactly (n - v!i) (v `without` i))
-- for solutions that use items multiple times,
-- change (v `without` i) to v

-- v `without` i
-- new array like v except one shorter with v!i missing
without :: Array Int a -> Int -> Array Int a
without v i = ixmap (lo, hi-1) f v
     where (lo, hi) = bounds v
           f j | j >= i = j+1
               | otherwise = j

play = exactly 1505 menu
menu = listArray (1,6) [215, 275, 335, 355, 420, 580]

test = exactly 10 (listArray (1,5) [1,1,2,3,4])

It disappoints me that there is no solution if each item is used at most 
once. However, do change the code to allow multiple uses, then there are 
many solutions.