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)

#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
mod util {
    use std::io::stdin;
    use std::str::FromStr;
    use std::fmt::Debug;
    #[allow(dead_code)]
    pub fn line() -> String {
        let mut line: String = String::new();
        stdin().read_line(&mut line).unwrap();
        line.trim().to_string()
    }
    #[allow(dead_code)]
    pub fn gets<T: FromStr>() -> Vec<T>
    where
        <T as FromStr>::Err: Debug,
    {
        let mut line: String = String::new();
        stdin().read_line(&mut line).unwrap();
        line.split_whitespace()
            .map(|t| t.parse().unwrap())
            .collect()
    }
}
#[allow(unused_macros)]
macro_rules ! get { ( $ t : ty ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; line . trim ( ) . parse ::<$ t > ( ) . unwrap ( ) } } ; ( $ ( $ t : ty ) ,* ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; let mut iter = line . split_whitespace ( ) ; ( $ ( iter . next ( ) . unwrap ( ) . parse ::<$ t > ( ) . unwrap ( ) , ) * ) } } ; ( $ t : ty ; $ n : expr ) => { ( 0 ..$ n ) . map ( | _ | get ! ( $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ ( $ t : ty ) ,*; $ n : expr ) => { ( 0 ..$ n ) . map ( | _ | get ! ( $ ( $ t ) ,* ) ) . collect ::< Vec < _ >> ( ) } ; ( $ t : ty ;; ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; line . split_whitespace ( ) . map ( | t | t . parse ::<$ t > ( ) . unwrap ( ) ) . collect ::< Vec < _ >> ( ) } } ; }
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { println ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }

fn main() {
    let (n, k) = get!(usize, usize);
    let xyc = get!(u64, u64, String; n);
    let mut xy: Vec<(usize, usize)> = xyc.into_iter()
        .map(|(x, y, c)| if c == "W" { (x, y + k as u64) } else { (x, y) })
        .map(|(x, y)| ((x % (2 * k as u64)) as usize, (y % (2 * k) as u64) as usize))
        .collect();

    let mut b = Vec::new();
    let mut w = Vec::new();

    for t in &mut xy {
        match (t.0 < k, t.1 < k) {
            (true, true) => b.push(t.clone()),
            (true, false) => w.push((t.0, t.1 - k)),
            (false, true) => w.push((t.0 - k, t.1)),
            (false, false) => b.push((t.0 - k, t.1 - k)),
        }
    }

    let mut imos = vec![vec![0i64; 2 * k + 1]; 2 * k + 1];

    for &(x, y) in &b {
        let xx = [(0, x), (x + 1, x + k), (x + k + 1, 2 * k - 1)];
        let yy = [(0, y), (y + 1, y + k), (y + k + 1, 2 * k - 1)];

        for (i, &(l, r)) in xx.iter().enumerate() {
            for (j, &(b, u)) in yy.iter().enumerate() {
                if (i + j) % 2 == 0 {
                    imos[b][l] += 1;
                    imos[u + 1][r + 1] += 1;
                    imos[b][r + 1] -= 1;
                    imos[u + 1][l] -= 1;
                }
            }
        }
    }

    for &(x, y) in &w {
        let xx = [(0, x), (x + 1, x + k), (x + k + 1, 2 * k - 1)];
        let yy = [(0, y), (y + 1, y + k), (y + k + 1, 2 * k - 1)];

        for (i, &(l, r)) in xx.iter().enumerate() {
            for (j, &(b, u)) in yy.iter().enumerate() {
                if (i + j) % 2 == 1 {
                    imos[b][l] += 1;
                    imos[u + 1][r + 1] += 1;
                    imos[b][r + 1] -= 1;
                    imos[u + 1][l] -= 1;
                }
            }
        }
    }

    for i in 0..2 * k + 1 {
        for j in 1..2 * k + 1 {
            imos[i][j] += imos[i][j - 1];
        }
    }

    for i in 0..2 * k + 1 {
        for j in 1..2 * k + 1 {
            imos[j][i] += imos[j - 1][i];
        }
    }

    let ans = imos.into_iter()
        .map(|v| v.iter().max().cloned().unwrap())
        .max()
        .unwrap();
    println!("{}", ans);
}