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 AtCoderBeginnerContest159.Questions;
using AtCoderBeginnerContest159.Extensions;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace AtCoderBeginnerContest159.Questions
{
    public class QuestionE : AtCoderQuestionBase
    {
        int h, w, k;
        bool[][] isWhite;

        // 復習
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var hwk = inputStream.ReadIntArray();
            h = hwk[0];
            w = hwk[1];
            k = hwk[2];

            isWhite = new bool[h][];
            for (int i = 0; i < h; i++)
            {
                isWhite[i] = inputStream.ReadLine().Select(c => c == '1').ToArray();
            }

            int min = int.MaxValue;
            for (int bitFlag = 0; bitFlag < 1 << (h - 1); bitFlag++)
            {
                var result = SolveFor(bitFlag);
                min = Math.Min(min, result);
            }

            yield return min;
        }

        private int SolveFor(int bitFlag)
        {
            int horizontaiDivisionCount = GetHorizontalDivisionCount(bitFlag);

            var rowGroups = new List<List<int>>();
            var group = new List<int>();
            rowGroups.Add(group);
            group.Add(0);

            for (int i = 1; i < h; i++)
            {
                if ((bitFlag & 0x01 << (i - 1)) != 0)
                {
                    group = new List<int>();
                    rowGroups.Add(group);
                }

                group.Add(i);
            }

            var splitCount = rowGroups.Count - 1;
            var whiteCounts = new int[rowGroups.Count];

            for (int i = 0; i < w; i++)
            {
                var currentWhiteCounts = rowGroups.Select(g => g.Count(r => isWhite[r][i])).ToArray();
                for (int j = 0; j < whiteCounts.Length; j++)
                {
                    if (currentWhiteCounts.Any(n => n > k))
                    {
                        // この割り方だとNGであることを返す
                        return int.MaxValue;
                    }

                    whiteCounts[j] += currentWhiteCounts[j];
                    if (whiteCounts.Any(n => n > k))
                    {
                        whiteCounts = currentWhiteCounts;
                        splitCount++;
                    }
                }
            }

            return splitCount;
        }

        private int GetHorizontalDivisionCount(int bitFlag)
        {
            int count = 0;
            while (bitFlag != 0)
            {
                count += bitFlag & 0x01;
                bitFlag >>= 1;
            }
            return count;
        }
    }
}


namespace AtCoderBeginnerContest159
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionE();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace AtCoderBeginnerContest159.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }

}

#endregion

#region Extensions

namespace AtCoderBeginnerContest159.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion