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: Accepted)

#exec({'RUBY_THREAD_VM_STACK_SIZE'=>'100000000'},'/usr/bin/ruby', $0) if !ENV['RUBY_THREAD_VM_STACK_SIZE']

def inpf() a=gets.chomp.split(" ").map(&:to_f)end

def inps() a=gets.chomp.split(" ")end

def copy(a) Marshal.load(Marshal.dump(a)) end

def kaijo(n,r = 10**9+7)(n < 2)? 1 : (2..n).inject{|memo,u|memo=(memo*u)%r} end

def na(n,d=0) Array.new(n,d)end

def na2(n,m,d=0) Array.new(n){Array.new(m,d)}end

def na3(n,m,l,d=0) Array.new(n){Array.new(m){Array.new(l,d)}}end

def inp() a=gets.chomp.split(" ").map(&:to_i)end

def r_up(a, b) (a+b-1)/b end

def sum(a) a.inject(:+) end

def big(a,b) return (a>b)? a:b end

def small(a,b) return (a<b)? a:b end

h,w,k = inp
ans  = 1e20
s = []
h.times{s<<gets.chomp.split("").map(&:to_i)}
(0...(2**(h-1))).each do |bi|
  wc = [0] * h

  c = 0
  f = false
  is_break = false
  is_continue = false
  (0...w).each do |wi|
    wdi = 0
    (0...h).each do |hi|
      if(s[hi][wi] == 1)
        wc[wdi] += 1
        if(wc[wdi] > k)
          unless(f)

            is_break = true
            break
          else
            wc = [0] * h
            c += 1
            f = false
            is_continue = true
            break
          end
        end
      end
      if(bi[hi] == 1)
        wdi += 1
      end
    end
#    p wc
    if(is_continue)
      wdi = 0
      is_continue = false
      (0...h).each do |hi|
        if(s[hi][wi] == 1)
          wc[wdi] += 1
          if(wc[wdi] > k)
            unless(f)
              is_break = true
              break
            else
              wc = [0] * h
              c += 1
              break
            end
          end
        end
        if(bi[hi] == 1)
          wdi += 1
        end
      end
      f = true
      next
    end
    if(is_break)
      break
    end
    f = true
  end
  ans = small(ans,bi.to_s(2).count("1")+c) unless is_break
  bi.to_s(2)
end
p ans