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 C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#pragma GCC optimize("Ofast")
#include "bits/stdc++.h"
using namespace std;
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;
/*
            common mistakes
           1) long long vs int overflow
           2) read question carefully please!!!! it can cost you a lot
           3) unorderedmap is O(1) look up but worst case can be bad
           4) maps have higher constant so many lookups can cause TLE
           5) leaving debugs or cerrs in the program at times can cause TLE
*/
#define IOS { ios :: sync_with_stdio(false); cin.tie(0); cout.tie(0); }
#define umapii unordered_map <int,int>
#define umapll unordered_map<ll,ll>
#define mapll map<ll,ll>
#define mapii map<int,int>
#define ll long long
#define pi pair<int,int>
#define pll pair<ll,ll>
#define pb push_back
#define eb emplace_back
#define fi first
#define se second
#define vs vector<string>
#define vi vector<int>
#define vll vector<ll>
#define vpi vector<pi>
#define vpll vector<pll>
#define vvi vector<vi>
#define vvll vector<vll>
#define all(v) (v).begin(),(v).end()
#define FOR(i,a,b) for(ll i=a;i<=b;i++)
#define MP(a,b) make_pair(a,b)
#define debug(x)       cerr<< #x <<" : "<< x << endl;
#define debuga(A,N)    cerr<< #A <<" : [";for(int i = 0; i<N;i++) cerr<<A[i]<<" "; cerr<<"]\n";
#define debuga2(A,N,M) cerr<< #A << " : \n"; for(int i=0;i<N;i++){cerr<<"[";for(int j=0;j<M;++j) cerr<<A[i][j]<<" ";cerr<<"]\n";}
#define debugp(p)      cerr<< #p <<" : "<<"("<<(p).first<<","<<(p).second<<")\n";
#define debugv(v)      cerr<< #v <<" : "<<"[";for(int i = 0; i< (v).size(); i++) cerr<<v[i]<<" "; cerr<<"]\n";
#define debugv2(v)     cerr<< #v << " : \n"; for(int i=0;i<v.size();i++){cerr<<"[";for(int j=0;j<(v[0].size());++j) cerr<<v[i][j]<<" ";cerr<<"]\n";}
#define debugs(m)      cerr<< #m <<" : [ "; for(auto itr = m.begin(); itr!=m.end();itr++) cerr<<*itr<<" "; cerr<<"]\n";
#define debugm(m)      cerr<< #m <<" : [ "; for(auto itr = m.begin();itr!=m.end(); itr++) cerr<<"("<<itr->first<<","<<itr->second<<") ";cerr<<"]\n";

/*STL:
       (1) PRIORITY_QUEUE pq:
       -> priority_queue < int > pq; // max_heap
       -> priority_queue < int , vector < int > , greater < int > > pq; // min_heap 
       -> pq.top()   --> returns the maximum or minimum element
       -> pq.push()  --> push element into the queue
       -> pq.pop()   --> remove top most element
       -> pq.empty() --> checks whether queue is empty or not

       (2) set / multiset  ss; 
       --> ss.insert(key); (insertion)
       --> ss.erase(ss.find(key)) .. (deletion)
       --> ss.find(key) != ss.end() ..existence check
       --> ss.lower_bound(x) -> smallest value which is greater than or equals to x
       --> ss.upper_bound(x) --> first value strictly greater than x
     
       (3) common functions
       lower_bound(v.begin(),v.end(),x) - v.begin() --> smallest value greater than or equals to x
       upper_bound(v.begin(), v.end(),x)- v.begin() --> smallest value greater than x
       memset(dp , val , sizeof(dp)) -> sets the value of array dp to val
*/

template <typename T>
class Modular {
 public:
  using Type = typename decay<decltype(T::value)>::type;
 
  constexpr Modular() : value() {}
  template <typename U>
  Modular(const U& x) {
    value = normalize(x);
  }
 
  template <typename U>
  static Type normalize(const U& x) {
    Type v;
    if (-mod() <= x && x < mod()) v = static_cast<Type>(x);
    else v = static_cast<Type>(x % mod());
    if (v < 0) v += mod();
    return v;
  }
 
  const Type& operator()() const { return value; }
  template <typename U>
  explicit operator U() const { return static_cast<U>(value); }
  constexpr static Type mod() { return T::value; }
  Modular& operator+=(const Modular& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; }
  Modular& operator-=(const Modular& other) { if ((value -= other.value) < 0) value += mod(); return *this; }
  template <typename U> Modular& operator+=(const U& other) { return *this += Modular(other); }
  template <typename U> Modular& operator-=(const U& other) { return *this -= Modular(other); }
  Modular& operator++() { return *this += 1; }
  Modular& operator--() { return *this -= 1; }
  Modular operator++(int) { Modular result(*this); *this += 1; return result; }
  Modular operator--(int) { Modular result(*this); *this -= 1; return result; }
  Modular operator-() const { return Modular(-value); }
 
  template <typename U = T>
  typename enable_if<is_same<typename Modular<U>::Type, int>::value, Modular>::type& operator*=(const Modular& rhs) {
#ifdef _WIN32
    uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);
    uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m;
    asm(
      "divl %4; \n\t"
      : "=a" (d), "=d" (m)
      : "d" (xh), "a" (xl), "r" (mod())
    );
    value = m;
#else
    value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));
#endif
    return *this;
  }
  template <typename U = T>
  typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type& operator*=(const Modular& rhs) {
    int64_t q = static_cast<int64_t>(static_cast<long double>(value) * rhs.value / mod());
    value = normalize(value * rhs.value - q * mod());
    return *this;
  }
  template <typename U = T>
  typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type& operator*=(const Modular& rhs) {
    value = normalize(value * rhs.value);
    return *this;
  }
 
  Modular& operator/=(const Modular& other) { return *this *= Modular(inverse(other.value, mod())); }
 
  template <typename U>
  friend const Modular<U>& abs(const Modular<U>& v) { return v; }
 
  template <typename U>
  friend bool operator==(const Modular<U>& lhs, const Modular<U>& rhs);
 
  template <typename U>
  friend bool operator<(const Modular<U>& lhs, const Modular<U>& rhs);
 
  template <typename U>
  friend std::istream& operator>>(std::istream& stream, Modular<U>& number);
 
 private:
  Type value;
};
 
template <typename T> bool operator==(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value == rhs.value; }
template <typename T, typename U> bool operator==(const Modular<T>& lhs, U rhs) { return lhs == Modular<T>(rhs); }
template <typename T, typename U> bool operator==(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) == rhs; }
 
template <typename T> bool operator!=(const Modular<T>& lhs, const Modular<T>& rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(const Modular<T>& lhs, U rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(U lhs, const Modular<T>& rhs) { return !(lhs == rhs); }
 
template <typename T> bool operator<(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value < rhs.value; }
 
template <typename T> Modular<T> operator+(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }
 
template <typename T> Modular<T> operator-(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }
 
template <typename T> Modular<T> operator*(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }
 
template <typename T> Modular<T> operator/(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }

template<typename T, typename U>
Modular<T> power(const Modular<T>& a, const U& b) {
  assert(b >= 0);
  Modular<T> x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1) res *= x;
    x *= x;
    p >>= 1;
  }
  return res;
}
template <typename T>
bool IsZero(const Modular<T>& number) {
  return number() == 0;
}
template <typename T>
string to_string(const Modular<T>& number) {
  return to_string(number());
}
template <typename T>
std::ostream& operator<<(std::ostream& stream, const Modular<T>& number) {
  return stream << number();
}
template <typename T>
std::istream& operator>>(std::istream& stream, Modular<T>& number) {
  typename common_type<typename Modular<T>::Type, int64_t>::type x;
  stream >> x;
  number.value = Modular<T>::normalize(x);
  return stream;
}
constexpr int md = 1000000007;
using Mint = Modular<std::integral_constant<decay<decltype(md)>::type, md>>;

ll modexp(ll x, ll ex) {
    ll ans = 1ll;
    while (ex > 0) {
        if (ex & 1ll) ans = (ans * x) % md;
        ex >>= 1ll;
        x = (x * x) % md;
    }
    return ans;
}

const int inf = 1e9 + 7;

int main(){
    IOS;
    ll h,w,k;
    cin>>h>>w>>k;
    vs s(h);
    FOR(i,0,h-1){
      cin>>s[i];
    }
    ll ans=LLONG_MAX;
    FOR(mask,0,1<<(h-1)){
      ll c=0;
      // debug(mask);
      int nv=__builtin_popcountll(mask);
      vll v(nv+1,0);
      bool flag,flag2;
      FOR(i,0,w-1){
        vll sm(nv+1,0);
        int jj=0;
        FOR(j,0,h-1){
          v[jj] +=(s[j][i]=='1');
          sm[jj]+=(s[j][i]=='1');
          if((mask&(1ll<<j)) ) ++jj;
        }
        flag=false;
        flag2=false; 
        FOR(i,0,nv){
          if(v[i]>k)flag=true;
          if(sm[i]>k)flag2=true;
        }
        if(flag2)break;
        if(flag){
          ++c;
          v=sm;
        }
      }
      if(flag2)break;
      ans=min(ans,c+nv);
    }
    cout<<ans<<'\n';
}