Find a simple arithmetic expression that evaluates to 24 using the numbers 1, 3, 4, 6 exactly once.

Number Puzzle

import           Data.Ratio
import qualified Data.Set   as S

data Op  = Add | Sub | Mul | Div
  deriving Eq
data Exp a = Bin Op (Exp a) (Exp a)
           | Num a
  deriving Eq

instance Show Op where
  show Add = "+"
  show Sub = "-"
  show Mul = "*"
  show Div = "/"

instance Show a => Show (Exp a) where
  show (Num x) = show x
  show (Bin op e1 e2) = "(" ++ show e1 ++ show op  ++ show e2 ++ ")" where

numbers = [1.0,3.0,4.0,6.0]
ops = [Add, Sub, Mul, Div]

value (Num x) = x
value (Bin op e1 e2) = (value e1) `op'` (value e2) where
  op' = case op of
    Add -> (+)
    Sub -> (-)
    Mul -> (*)
    Div -> (/)

base (Num x) = S.singleton x
base (Bin _ e1 e2) = base e1 `S.union` base e2

different e1 e2 = S.size (base e1 `S.intersection` base e2) == 0

size (Num _) = 1
size (Bin _ e1 e2) = size e1 + size e2

generate es = es ++ [Bin op e1 e2 | e1 <- es, e2 <- es, op <- ops, e1 `differen
t` e2]

allExpressions = generate . generate . generate . map Num $ numbers

allValidExpressions = filter (\e -> size e == 4) allExpressions

solutions = filter (\e -> value e == 24.0) allValidExpressions