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 Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused, non_snake_case, dead_code, non_upper_case_globals)]
use {
    proconio::{marker::*, *},
    std::{cmp::*, collections::*, mem::*},
};
/**
 * snuke code
**/
fn main() {
    input! {
            h:usize,
            w:usize,
            k:usize,
        s:[Chars;h],
    }
    dbg!(&s);
    let mut ans = INF;
    // 横に切る方法を全探索
    for div in 0..(1usize << (h - 1)) {
        let mut g: usize = 0;
        let mut id = vec![0usize; h + 1];
        for i in 0..h {
            id[i] = g;
            if (((div >> i) & 1) == 1) {
                g += 1;
            }
        }
        // dbg!(div);
        g += 1;
        let mut c = vec![vec![0usize; w]; g];
        // groupにmapping
        for i in 0..h {
            for j in 0..w {
                c[id[i]][j] += (s[i][j] as u8 - b'0') as usize;
            }
        }

        // dbg!(&c);
        let mut num: usize = g - 1;
        // 各グループで何個のホワイトチョコがあるか
        let mut now = vec![0usize; g];
        for j in 0..w {
            // dbg!(j);
            if !add(&mut now, j, g, k, &c) {
                num += 1;
                now = vec![0usize; g];
                if !add(&mut now, j, g, k, &c) {
                    num = INF;
                    break;
                }
            }
            // dbg!(&now);
            // dbg!(num);
        }
        dbg!(div, num);
        ans = min(ans, num);
    }
    println!("{}", ans);
}
// j列目をnowに加算
fn add(now: &mut Vec<usize>, j: usize, g: usize, k: usize, c: &Vec<Vec<usize>>) -> bool {
    for i in 0..g {
        now[i] += c[i][j];
    }
    for i in 0..g {
        if now[i] > k {
            return false;
        }
    }
    return true;
}
const INF: usize = 1 << 62;

pub fn charconv(c: char, base: char) -> usize {
    return (c as u8 - base as u8) as usize;
}