Useful, Cool, Magical Idioms

this collection is supposed to be comprised of short, useful, cool, magical examples, which incite curiosity in the reader and (hopefully) lead him to a deeper understanding of advanced haskell concepts. at a later time i might add explanations to the more obscure solutions. i've also started providing several alternatives to give more insight into the interrelations of solutions.

whoever has any more ideas, please feel free to just add them; if you see mistakes or simpler solutions please correct my chaotic collection. i'm very interested in more "obscure" solutions, which showcase the applicability of haskell's (unique) features (i.e. monad magic, folds and unfolds, fix points, ...)

 -- splitting in twos (alternating)
 -- "1234567" -> ("1357", "246")
 foldr (\a (x,y) -> (a:y,x)) ([],[])
 (map snd *** map snd) . partition (even . fst) . zip [0..]

 -- splitting in N
 -- 2 -> "1234567" -> ["12", "34", "56", "7"]
 unfoldr (\a -> if null a then Nothing else Just $ splitAt 2 a) "1234567"

 -- split at whitespace
 -- "hello world" -> ["hello","world"]
 words

 unfoldr (\a -> if null a then Nothing else Just . (second $ drop 1) . break (==' ') $ a)

 -- combinations
 -- "12" -> "45" -> ["14", "15", "24", "25"]
 sequence ["12", "45"]
 [[x,y] | x <- "12", y <- "45"]
 do { x <- "12"; y <- "45"; return [x,y] }
 "12" >>= \a -> "45" >>= \b -> return [a,b]

 -- factorial
 -- 6 -> 720
 product [1..6]
 foldl1 (*) [1..6]
 (!!6) $ unfoldr (\(n,f) -> Just (f, (n+1,f*n))) (1,1)
 fix (\f n -> if n <= 0 then 1 else n * f (n-1))

 -- interspersing with newlines
 -- ["hello","world"] -> "hello world"
 unlines
 intersperse '\n'

 -- sorting by a custom function
 -- length -> ["abc", "ab", "a"] -> ["a", "ab", "abc"]
 sortBy length
 map snd . sortBy fst . map (length &&& id) 
 
 
 -- zweierpotenzen
 iterate (*2) 1
 unfoldr (\z -> Just (z,2*z)) 1

 -- simulating lisp's cond
 case () of () | 1 > 2     -> True
               | 3 < 4     -> False
               | otherwise -> True

 -- add indices to list for later use
 -- [3,3,3] -> [(0,3),(1,3),(2,3)]
 zip [0..]
 

 -- fibonacci series
 unfoldr (\(f1,f2) -> Just (f1,(f2,f1+f2))) (0,1)
 fibs = 0:1:zipWith (+) fibs (tail fibs)
 fib = 0:scanl (+) 1 fib

 -- unjust'ify list of Maybe's
 -- [Just 4, Nothing, Just 3] -> [4,3]
 catMaybes

 -- find substring
 -- "ell" -> "hello" -> True
 substr a b = any (a `elem`) $ map inits (tails b)

 -- apply a list of functions to an argument
 -- even -> odd -> 4 -> [True,False]
 map ($4) [even,odd]
 sequence [even,odd] 4
 
 
 -- apply a function to two other function the same argument
 -- (lifting to the function monad (->))
 -- even 4 && odd 4 -> False
 liftM2 (&&) even odd 4
 liftM2 (>>) putStrLn return "hello"    -- putStrLn "hello" >> return "hello"

 -- match a constructor
 -- this is better than applying all the arguments, because this way the data type can be changed without touching the code (ideally).
 case a of Just{} -> True
           _      -> False

 -- prime numbers
 -- example of a memoising caf (??)
 primes = sieve [2..] where
          sieve (p:x) = p : sieve [ n | n <- x, n `mod` p > 0 ]

 unfoldr  sieve [2..] where 
          sieve (p:x) = Just(p,   [ n | n <- x, n `mod` p > 0 ])

 -- forward function concatenation with the function arrow
 (*3) >>> (+1) $ 2

 -- perform functions in/on a monad
 fmap (+2) (Just 2)
 liftM2 (+) (Just 4) (Just 2)

 -- ???
 (fmap . (!!)) ['a'..'z'] [0,25]       -- "az"
 (id >>= (+) >>= (+) >>= (+)) 3        -- (3+3)+(3+3) = 12
 (join . liftM2) (*) (+3) 5            -- 64

 -- all combinations of letters
 (inits . repeat) ['a'..'z'] >>= sequence

 {- 
 TODO, ideas:
   either
   maybe
   group
   fun with monad, monadPlus
   fun with arrows (second, first, &&&, ***)
   liftM, ap
   list monad vs comprehensions
 
 LINKS:
   bananas, envelopes, ...   (generic traversal)
   why functional fp matters (lazy search, ...)
 -}