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 Kotlin (1.0.0) (judge verdict: Wrong Answer)

fun main() {
    val h = nextInt()
    val w = nextInt()
    val k = nextInt()
    val s = Array(h) {
        next().toCharArray()
    }
    var min = h + w
    for (i in 0 until (Math.pow(2.0, (h - 1).toDouble()).toInt())) {
        val cutList = mutableListOf<Cut>()
        var current = Cut()
        cutList += current
        current.lines += s[0]
        var mod = 1
        for (j in 1 until h) {
            if (i and mod != 0) {
                current = Cut()
                cutList += current
            }
            current.lines += s[j]
            mod *= 2
        }
        var count = Integer.bitCount(i)
        var ok = true
        var needCut = false
        loop@ for (j in 0 until w) {
            for (cut in cutList) {
                if (cut.add(j) > k) {
                    ok = false
                    break@loop
                }
                if (cut.sum > k) {
                    needCut = true
                }
            }
            if (needCut) {
                count++
                for (cut in cutList) {
                    cut.sum = 0
                }
                needCut = false
            }
        }
        if (ok) {
            min = Math.min(min, count)
        }
    }
    println(min)
}

class Cut {
    val lines = mutableListOf<CharArray>()
    var sum = 0

    fun add(i: Int): Int {
        var res = 0
        for (line in lines) {
            res += line[i] - '0'
        }
        sum += res
        return res
    }
}

val scanner = java.util.Scanner(System.`in`)
fun next(): String = scanner.next()
fun nextInt() = scanner.nextInt()
fun nextLong() = scanner.nextLong()
fun nextLine(): String = scanner.nextLine()
fun nextList() = nextLine().split(" ")
fun nextIntList() = nextList().map(String::toInt)
fun nextIntArray(n: Int) = IntArray(n) { nextInt() }
fun nextLongList() = nextList().map(String::toLong)
fun printYesNo(isYes: Boolean) = println(if (isYes) "Yes" else "No")

fun main(args: Array<String>) = main()