Problem:
Score : 400 points

Problem Statement

There is a bar of chocolate with a height of H blocks and a width of W blocks.
Snuke is dividing this bar into exactly three pieces.
He can only cut the bar along borders of blocks, and the shape of each piece must be a rectangle.

Snuke is trying to divide the bar as evenly as possible.
More specifically, he is trying to minimize S_{max} - S_{min}, where S_{max} is the area (the number of blocks contained) of the largest piece, and S_{min} is the area of the smallest piece.
Find the minimum possible value of S_{max} - S_{min}.

Constraints

2 ≤ H, W ≤ 10^5

Input

Input is given from Standard Input in the following format:

H W

Output

Print the minimum possible value of S_{max} - S_{min}.

Sample Input 1

3 5

Sample Output 1

0

In the division below, S_{max} - S_{min} = 5 - 5 = 0.

Sample Input 2

4 5

Sample Output 2

2

In the division below, S_{max} - S_{min} = 8 - 6 = 2.

Sample Input 3

5 5

Sample Output 3

4

In the division below, S_{max} - S_{min} = 10 - 6 = 4.

Sample Input 4

100000 2

Sample Output 4

1

Sample Input 5

100000 100000

Sample Output 5

50000

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

-- import Prelude hiding (abs)
import Control.Monad
import Control.Arrow ((&&&))
import Data.List (nub, sort, group, find)
import Data.Maybe (fromMaybe, fromJust)
import qualified Data.IntMap as M

type Counter = M.IntMap Int

main :: IO ()
main = do
    [h, w] <- (map read . words) <$> getLine
    print $ solve (h,w)

-- |
--
-- >>> solve (3,5)
-- 0
--
-- >>> solve (4,5)
-- 2
--
-- >>> solve (5,5)
-- 4
--
-- >>> solve (100000, 2)
-- 1
solve :: (Int,Int) -> Int
solve (h,w) = minimum $ map (calc (h,w)) $ yxs1 ++ yxs2
  where
    yxs1 = [(y,w) | y <- [1 .. (h-1)]]
    yxs2 = [(h,x) | x <- [1 .. (w-1)]]


calc :: (Int,Int) -> (Int,Int) -> Int
calc (h,w) (y,x) = maximum ss - minimum ss
  where ss = splitRect3 (h,w) (y,x)

splitRect3 :: (Int,Int) -> (Int,Int) -> [Int]
splitRect3 (h,w) (y,x)
  | h == y = y * x : splitRect2 (h, w - x)
  | w == x = y * x : splitRect2 (h - y, w)

splitRect2 :: (Int,Int) -> [Int]
splitRect2 (h,w)
  | dh > dw   = [h * w1, h * w2]
  | otherwise = [w * h1, w * h2]
  where
    h1 = h `div` 2
    h2 = h - h1
    w1 = w `div` 2
    w2 = w - w1
    dh = abs (h1 - h2) * w
    dw = abs (w1 - w2) * h