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# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Runtime.CompilerServices;
using System.Text;
using System.Threading.Tasks;

namespace AtCorder
{
	public class Program
	{
		static void Main(string[] args)
		{
			var cin = new Scanner2();
			int h = cin.Int();
			int w = cin.Int();
			int k = cin.Int();

			var s = new string[h];
			for (int i = 0; i < h; i++) {
				s[i] = cin.Next();
			}

			
			int hh = h - 1;
			int ww = w - 1;
			long ans = h * w;
			for (int i = 0; i < 1 << hh; i++) {
				var hs = new List<int>();
				hs.Add(0);
				for (int j = 0; j < hh; j++) {
					if (((1 << j) & i) != 0) {
						hs.Add(j + 1);
					}
				}

				hs.Add(h);
				var cnt = new int[hs.Count - 1];
				long temp = hs.Count - 2;
				int width = 1;
				for (int l = 0; l < w; l++) {
					for (int j = 0; j < hs.Count - 1; j++) {
						int st = hs[j];
						int en = hs[j + 1];
						for (int kk = st; kk < en; kk++) {
							if (s[kk][l] == '1') {
								cnt[j]++;
							}
						}
					}

					bool ok = true;
					for (int j = 0; j < hs.Count - 1; j++) {
						if (cnt[j] > k) {
							ok = false;
							break;
						}
					}

					if (ok == false) {
						if (width == 1) {
							break;
						} else {
							l--;
							width = 1;
							++temp;
							for (int j = 0; j < hs.Count - 1; j++) {
								cnt[j] = 0;
							}
						}
					} else {
						++width;
					}
				}

				ans = Math.Min(ans, temp);

				/*
				temp = hs.Count - 2;
				width = 1;
				for (int l = w - 1; l >= 0; l--) {
					for (int j = 0; j < hs.Count - 1; j++) {
						int st = hs[j];
						int en = hs[j + 1];
						for (int kk = st; kk < en; kk++) {
							if (s[kk][l] == '1') {
								cnt[j]++;
							}
						}
					}

					bool ok = true;
					for (int j = 0; j < hs.Count - 1; j++) {
						if (cnt[j] > k) {
							ok = false;
							break;
						}
					}

					if (ok == false) {
						if (width == 1) {
							break;
						} else {
							l++;
							width = 1;
							++temp;
							for (int j = 0; j < hs.Count - 1; j++) {
								cnt[j] = 0;
							}
						}
					} else {
						++width;
					}
				}

				ans = Math.Min(ans, temp);
				*/
			}

			Console.WriteLine(ans);
		}

		public static int PopCount(ulong bits)
		{
			bits = ((bits & 0xaaaaaaaaaaaaaaaa) >> 1) + (bits & 0x5555555555555555);
			bits = ((bits & 0xcccccccccccccccc) >> 2) + (bits & 0x3333333333333333);
			bits = ((bits & 0xf0f0f0f0f0f0f0f0) >> 4) + (bits & 0x0f0f0f0f0f0f0f0f);
			bits = ((bits & 0xff00ff00ff00ff00) >> 8) + (bits & 0x00ff00ff00ff00ff);
			bits = ((bits & 0xffff0000ffff0000) >> 16) + (bits & 0x0000ffff0000ffff);
			bits = ((bits & 0xffffffff00000000) >> 32) + (bits & 0x00000000ffffffff);
			return (int)bits;
		}
	}

	public static class Helper
	{
		public static void UpdateMin32<T>(T array, int index, int value) where T : IList<int>
			=> array[index] = Math.Min(array[index], value);
		public static void UpdateMax32<T>(T array, int index, int value) where T : IList<int>
			=> array[index] = Math.Max(array[index], value);
		public static void UpdateMin64<T>(T array, int index, long value) where T : IList<long>
			=> array[index] = Math.Min(array[index], value);
		public static void UpdateMax64<T>(T array, int index, long value) where T : IList<long>
			=> array[index] = Math.Max(array[index], value);

		public static T[] Array<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToArray();
		public static List<T> List<T>(int n, Func<int, T> init)
			=> Enumerable.Range(0, n).Select(x => init(x)).ToList();
	}

	public class Scanner2
	{
		private readonly char[] delimiter_ = new char[] { ' ' };
		private readonly string filePath_;
		private readonly Func<string> reader_;
		private string[] buf_;
		private int index_;

		public Scanner2(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				reader_ = Console.ReadLine;
			} else {
				filePath_ = file;
				var fs = new StreamReader(file);
				reader_ = fs.ReadLine;
			}
			buf_ = new string[0];
			index_ = 0;
		}

		public string Next()
		{
			if (index_ < buf_.Length) {
				return buf_[index_++];
			}

			string st = reader_();
			while (st == "") {
				st = reader_();
			}

			buf_ = st.Split(delimiter_, StringSplitOptions.RemoveEmptyEntries);
			if (buf_.Length == 0) {
				return Next();
			}

			index_ = 0;
			return buf_[index_++];
		}

		public int Int() => int.Parse(Next());
		public long Long() => long.Parse(Next());
		public double Double() => double.Parse(Next());

		public int[] ArrayInt(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Int() + add;
			}
			return Array;
		}

		public long[] ArrayLong(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Long() + add;
			}
			return Array;
		}

		public double[] ArrayDouble(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Double() + add;
			}
			return Array;
		}

		public void Save(string text)
		{
			if (string.IsNullOrWhiteSpace(filePath_)) {
				return;
			}

			File.WriteAllText(filePath_ + "_output.txt", text);
		}
	}
}