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)

h,w,k=gets.split.map(&:to_i)
s=[]
h.times do
  s << gets.chomp.chars
end
s=s.transpose
w.times do |i|
  s[i]=s[i].join
end
m=2000
[0,1].repeated_permutation(h-1) do |b|
  l=[]
  (h-1).times do |i|
    if b[i]==1
      l << i
    end
  end
  z=l.size
  a=z
  if z>m
    next
  end
  c=[0]*(z+1)
  w.times do |i|
    f=true
    (z+1).times do |j|
      if j==0
        if z==0
          cc=s[i].count("1")
        else
          cc=s[i][0..l[0]].count("1")
        end
      elsif j==z
        cc=s[i][l[z-1]+1..h-1].count("1")
      else
        cc=s[i][l[j-1]+1..l[j]].count("1")
      end
      if cc>k
        f=false
        break
      end
      c[j]+=cc
      if c[j]>k
        a+=1
        if a>=m
          f=false
          break
        end
        (z+1).times do |jj|
          if jj==0
            if z==0
              c[0]=s[i].count("1")
            else
              c[0]=s[i][0..l[0]].count("1")
            end
          elsif jj==z
            c[z]=s[i][l[z-1]+1..h-1].count("1")
          else
            c[j]=s[i][l[jj-1]+1..l[jj]].count("1")
          end
        end
        break
      end
      if !f
        break
      end
    end
    if !f
      break
    end
  end
  if a<m
    m=a
  end
end
puts m