Problem:
Score : 500 points

Problem Statement

AtCoDeer is thinking of painting an infinite two-dimensional grid in a checked pattern of side K.
Here, a checked pattern of side K is a pattern where each square is painted black or white so that each connected component of each color is a K × K square.
Below is an example of a checked pattern of side 3:

AtCoDeer has N desires.
The i-th desire is represented by x_i, y_i and c_i.
If c_i is B, it means that he wants to paint the square (x_i,y_i) black; if c_i is W, he wants to paint the square (x_i,y_i) white.
At most how many desires can he satisfy at the same time?

Constraints

1 ≤ N ≤ 10^5

1 ≤ K ≤ 1000

0 ≤ x_i ≤ 10^9

0 ≤ y_i ≤ 10^9

If i ≠ j, then (x_i,y_i) ≠ (x_j,y_j).

c_i is B or W.

N, K, x_i and y_i are integers.

Input

Input is given from Standard Input in the following format:

N K
x_1 y_1 c_1
x_2 y_2 c_2
:
x_N y_N c_N

Output

Print the maximum number of desires that can be satisfied at the same time.

Sample Input 1

4 3
0 1 W
1 2 W
5 3 B
5 4 B

Sample Output 1

4

He can satisfy all his desires by painting as shown in the example above.

Sample Input 2

2 1000
0 0 B
0 1 W

Sample Output 2

2

Sample Input 3

6 2
1 2 B
2 1 W
2 2 B
1 0 B
0 6 W
4 5 W

Sample Output 3

4

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[macro_use]
mod input_mcr {
    // ref: tanakh <https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8>
    // diff: using Parser
    #[macro_export(local_inner_macros)]
    macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut parser = Parser::from_str($s);
        input_inner!{parser, $($r)*}
    };
    (parser = $parser:ident, $($r:tt)*) => {
        input_inner!{$parser, $($r)*}
    };
    (new_stdin_parser = $parser:ident, $($r:tt)*) => {
        let stdin = std::io::stdin();
        let reader = std::io::BufReader::new(stdin.lock());
        let mut $parser = Parser::new(reader);
        input_inner!{$parser, $($r)*}
    };
    ($($r:tt)*) => {
        input!{new_stdin_parser = parser, $($r)*}
    };
}

    #[macro_export(local_inner_macros)]
    macro_rules! input_inner {
    ($parser:ident) => {};
    ($parser:ident, ) => {};
    ($parser:ident, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($parser, $t);
        input_inner!{$parser $($r)*}
    };
}

    #[macro_export(local_inner_macros)]
    macro_rules! read_value {
    ($parser:ident, ( $($t:tt),* )) => {
        ( $(read_value!($parser, $t)),* )
    };
    ($parser:ident, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($parser, $t)).collect::<Vec<_>>()
    };
    ($parser:ident, chars) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()
    };
    ($parser:ident, char_) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()[0]
    };
    ($parser:ident, usize1) => {
        read_value!($parser, usize) - 1
    };
    ($parser:ident, i64_) => {
        $parser.fast_i64()
    };
    ($parser:ident, usize_) => {
        $parser.fast_i64() as usize
    };
    ($parser:ident, usize1_) => {
        ($parser.fast_i64() - 1) as usize
    };
    ($parser:ident, $t:ty) => {
        $parser.next::<$t>().expect("Parse error")
    };
}

    use std::io;
    use std::io::BufRead;
    use std::str;

    // ref: tatsuya6502 <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e>
    // ref: wariuni <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e#comment-7040a5ae96305e884eb9>
    // diff: using std::io::BufRead::fill_buf()
    pub struct Parser<R> {
        pub reader: R,
        buf: Vec<u8>,
        pos: usize,
    }

    impl Parser<io::Empty> {
        pub fn from_str(s: &str) -> Parser<io::Empty> {
            Parser {
                reader: io::empty(),
                buf: s.as_bytes().to_vec(),
                pos: 0,
            }
        }
    }

    impl<R: BufRead> Parser<R> {
        pub fn new(reader: R) -> Parser<R> {
            Parser {
                reader: reader,
                buf: vec![],
                pos: 0,
            }
        }
        pub fn update_buf(&mut self) {
            self.buf.clear();
            self.pos = 0;
            loop {
                let (len, complete) = {
                    let buf2 = self.reader.fill_buf().unwrap();
                    self.buf.extend_from_slice(buf2);
                    let len = buf2.len();
                    (len, buf2[len - 1] <= 0x20)
                };
                self.reader.consume(len);
                if complete {
                    break;
                }
            }
        }
        pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
            loop {
                let mut begin = self.pos;
                while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                    begin += 1;
                }
                let mut end = begin;
                while end < self.buf.len() && (self.buf[end] > 0x20) {
                    end += 1;
                }
                if begin != self.buf.len() {
                    self.pos = end;
                    return unsafe { str::from_utf8_unchecked(&self.buf[begin..end]) }.parse::<T>();
                } else {
                    self.update_buf();
                }
            }
        }
        pub fn fast_i64(&mut self) -> i64 {
            loop {
                let mut begin = self.pos;
                while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                    begin += 1;
                }
                if begin == self.buf.len() {
                    self.update_buf();
                    continue;
                }
                let mut res = 0;
                let (is_positive, mut end) = match self.buf[begin] {
                    b'+' => (true, begin + 1),
                    b'-' => (false, begin + 1),
                    _ => (true, begin),
                };
                while end < self.buf.len() && (self.buf[end] > 0x20) {
                    res = res * 10 + (self.buf[end] as i64 - '0' as i64);
                    end += 1;
                }
                if begin != self.buf.len() {
                    self.pos = end;
                    return if is_positive { res } else { -res };
                } else {
                    self.update_buf();
                }
            }
        }
    }
}

mod cumulative2 {
    use std::ops::*;

    pub fn gen_cu2<T: Add<Output = T> + Sub<Output = T> + Copy>(
        zero: T,
        xss: &Vec<Vec<T>>,
    ) -> Vec<Vec<T>> {
        let mut res = vec![vec![zero; xss[0].len() + 1]; xss.len() + 1];
        for (i, ref xs) in xss.iter().enumerate() {
            for (k, &x) in xs.iter().enumerate() {
                res[i + 1][k + 1] = res[i + 1][k] + res[i][k + 1] - res[i][k] + x;
            }
        }
        res
    }

    pub fn range2<T: Add<Output = T> + Sub<Output = T> + Copy>(
        cu2: &Vec<Vec<T>>,
        l: usize,
        d: usize,
        r: usize,
        u: usize,
    ) -> T {
        cu2[u][r] + cu2[d][l] - cu2[u][l] - cu2[d][r]
    }
}

use input_mcr::*;
use std::cmp::*;

use cumulative2::*;
use std::collections::BTreeSet;

fn main() {
    input! {
        n: usize,
        m: usize,
        xs: [(usize,usize,char_); n],
    }
    let mut xss = vec![vec![0i64; m]; m];
    let mut yss = vec![vec![0i64; m]; m];
    for &(x, y, c) in &xs {
        // eprintln!("c = {}", c);
        let x2 = x % (2 * m);
        let y2 = y % (2 * m);
        let c = if (x2 / m + y2 / m) % 2 == 0 {
            if c == 'B' {
                -1
            } else {
                1
            }
        } else {
            if c == 'B' {
                1
            } else {
                -1
            }
        };
        if c == 1 {
            xss[y % m][x % m] += 1;
        } else {
            yss[y % m][x % m] += 1;
        }
    }
    let xss_cu2 = gen_cu2(0, &xss);
    let yss_cu2 = gen_cu2(0, &yss);
    let mut res = 0;
    for y in 0..m + 1 {
        for x in 0..m + 1 {
            let t = range2(&xss_cu2, 0, 0, x, y)
                + range2(&xss_cu2, x, y, m, m)
                + range2(&yss_cu2, x, 0, m, y)
                + range2(&yss_cu2, 0, y, x, m);
            //// eprintln!("x = {}, y = {}, t = {}", x, y, t);
            res = max(res, t);
            let u = range2(&yss_cu2, 0, 0, x, y)
                + range2(&yss_cu2, x, y, m, m)
                + range2(&xss_cu2, x, 0, m, y)
                + range2(&xss_cu2, 0, y, x, m);
            //// eprintln!("x = {}, y = {}, t = {}", x, y, t);
            res = max(res, u);
        }
    }
    //// eprintln!("{:?}", xss_cu2);
    println!("{}", res);
}