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 (1.15.1) (judge verdict: Accepted)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use io::*;
use std::*;

fn search(
    H: i64,
    W: i64,
    K: usize,
    S: &Vec<Vec<u8>>,
    pos: i64,
    sumss: Vec<Vec<usize>>,
) -> Option<i64> {
    let mut min = None;
    if pos == H {
        let mut acc = vec![0; sumss.len()];
        let mut split = 0;
        'outer: for y in 0..W as usize {
            for x in 0..sumss.len() {
                let sums = &sumss[x];
                acc[x] += sums[y];
                if acc[x] > K {
                    split += 1;
                    for x in 0..sumss.len() {
                        acc[x] = sumss[x][y];
                        if acc[x] > K {
                            return None;
                        }
                    }
                    continue 'outer;
                }
            }
        }
        return Some(split);
    }
    for i in 1..=H - pos {
        let mut sumss = sumss.clone();
        let sums: Vec<usize> = (0..W)
            .map(|y| {
                (pos..pos + i)
                    .filter(|&x| S[x as usize][y as usize] == b'1')
                    .count()
            })
            .collect();
        sumss.push(sums);
        if let Some(result) = search(H, W, K, S, pos + i, sumss) {
            let target = result + 1;
            if let Some(m) = min {
                if target < m {
                    min = Some(target);
                }
            } else {
                min = Some(target);
            }
        }
    }
    min
}

// Generated by 1.1.6 https://github.com/kyuridenamida/atcoder-tools
fn main() {
    let con = read_string();
    let mut scanner = Scanner::new(&con);
    let H: i64 = scanner.next();
    let W: i64 = scanner.next();
    let K: usize = scanner.next();
    let mut S = Vec::new();
    for i in 0..H {
        let buf: String = scanner.next();
        S.push(buf.into_bytes());
    }
    println!("{}", search(H, W, K, &S, 0, Vec::new()).unwrap() - 1);
}

pub mod io {
    use std;
    use std::str::FromStr;

    pub struct Scanner<'a> {
        iter: std::str::SplitWhitespace<'a>,
    }

    impl<'a> Scanner<'a> {
        pub fn new(s: &'a str) -> Scanner<'a> {
            Scanner {
                iter: s.split_whitespace(),
            }
        }

        pub fn next<T: FromStr>(&mut self) -> T {
            let s = self.iter.next().unwrap();
            if let Ok(v) = s.parse::<T>() {
                v
            } else {
                panic!("Parse error")
            }
        }

        pub fn next_vec_len<T: FromStr>(&mut self) -> Vec<T> {
            let n: usize = self.next();
            self.next_vec(n)
        }

        pub fn next_vec<T: FromStr>(&mut self, n: usize) -> Vec<T> {
            (0..n).map(|_| self.next()).collect()
        }
    }

    pub fn read_string() -> String {
        use std::io::Read;

        let mut s = String::new();
        std::io::stdin().read_to_string(&mut s).unwrap();
        s
    }
}
*/

fn main() {
    let exe = "/tmp/bin8E2C7C4A";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";