Problem:
Score : 500 points

Problem Statement

We have a chocolate bar partitioned into H horizontal rows and W vertical columns of squares.

The square (i, j) at the i-th row from the top and the j-th column from the left is dark if S_{i,j} is 0, and white if S_{i,j} is 1.

We will cut the bar some number of times to divide it into some number of blocks. In each cut, we cut the whole bar by a line running along some boundaries of squares from end to end of the bar.

How many times do we need to cut the bar so that every block after the cuts has K or less white squares?

Constraints

1 \leq H \leq 10

1 \leq W \leq 1000

1 \leq K \leq H \times W

S_{i,j} is 0 or 1.

Input

Input is given from Standard Input in the following format:

H W K
S_{1,1}S_{1,2}...S_{1,W}
:
S_{H,1}S_{H,2}...S_{H,W}

Output

Print the number of minimum times the bar needs to be cut so that every block after the cuts has K or less white squares.

Sample Input 1

3 5 4
11100
10001
00111

Sample Output 1

2

For example, cutting between the 1-st and 2-nd rows and between the 3-rd and 4-th columns - as shown in the figure to the left - works.

Note that we cannot cut the bar in the ways shown in the two figures to the right.

Sample Input 2

3 5 8
11100
10001
00111

Sample Output 2

0

No cut is needed.

Sample Input 3

4 10 4
1110010010
1000101110
0011101001
1101000111

Sample Output 3

3

Solution in Ruby (2.3.3) (judge verdict: Wrong Answer)

class Array
  def sum; inject(:+) end # v2.5
end
def min(a,b); a < b ? a : b end

h,w,k = gets.to_s.split.map(&:to_i)
a = Array.new(h){ gets.to_s.chomp.bytes }

ans = (h-1) * (w-1)
2**(h-1).times do |bit|
  
  div  = [0] * h
  t    = 0
  able = true
  
  (0...h-1).each do |i|
    t += 1 if bit[i] == 1
    div[i+1] = t
  end
  
  wc = [0] * (t+1)
  cut = bit.to_s(2).count('1')
  
  w.times do |j|
    over = false
    h.times do |i|
      wc[div[i]] += 1 if a[i][j] == 49
      if wc[div[i]] > k
        over = true
        break
      end
    end
    if over
      (t+1).times{|k| wc[k] = 0 }
      cut += 1
      h.times do |i|
        wc[div[i]] += 1 if a[i][j] == 49
        if wc[div[i]] > k
          able = false
          break
        end
      end
    end
    
    break if not able
  end
  
  ans = min(ans, cut) if able
end

p ans