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 Lua (5.3.2) (judge verdict: Accepted)

local mmi, mma = math.min, math.max
local h, w = io.read("*n", "*n")
if h % 3 == 0 or w % 3 == 0 then print(0)
else
  local mincand = 0
  if 2 < h then
    mincand = w
  end
  if 2 < w then
    if mincand == 0 then mincand = h
    else mincand = mmi(h, mincand)
    end
  end
  if mincand == 0 then
    -- 2 x 2
    print(1)
  else
    local h_cut_pos = h // 3
    if 0 < h_cut_pos then
      local cut1 = h_cut_pos * w
      local rem_h = h - h_cut_pos
      local cut2 = rem_h * (w // 2)
      local cut3 = rem_h * w - cut2
      mincand = mmi(mincand, mma(cut3 - cut2, cut3 - cut1))
    end
    h_cut_pos = h_cut_pos + 1
    do
      local cut1 = h_cut_pos * w
      local rem_h = h - h_cut_pos
      local cut2 = rem_h * (w // 2)
      local cut3 = rem_h * w - cut2
      mincand = mmi(mincand, mma(cut3 - cut2, cut1 - cut2))
    end
    local w_cut_pos = w // 3
    if 0 < w_cut_pos then
      local cut1 = w_cut_pos * h
      local rem_w = w - w_cut_pos
      local cut2 = rem_w * (h // 2)
      local cut3 = rem_w * h - cut2
      mincand = mmi(mincand, mma(cut3 - cut2, cut3 - cut1))
    end
    w_cut_pos = w_cut_pos + 1
    do
      local cut1 = w_cut_pos * h
      local rem_w = w - w_cut_pos
      local cut2 = rem_w * (h // 2)
      local cut3 = rem_w * h - cut2
      mincand = mmi(mincand, mma(cut3 - cut2, cut1 - cut2))
    end
    print(mincand)
  end
end