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_macros)]
macro_rules! parse_line {
    ( $t:ty ) => (
        {
            let mut line = String::new();
            ::std::io::stdin().read_line(&mut line).unwrap();
            let mut iter = line.split_whitespace();
            iter.next().unwrap().parse::<$t>().unwrap()
        }
    );

    ( $( $t:ty), +) => (
        {
            let mut line = String::new();
            ::std::io::stdin().read_line(&mut line).unwrap();
            let mut iter = line.split_whitespace();
            ( $(iter.next().unwrap().parse::<$t>().unwrap()),* )
        }
    );
}

#[allow(unused_macros)]
macro_rules! read_line {
    () => (
        {
            let mut line = String::new();
            ::std::io::stdin().read_line(&mut line).unwrap();
            line.pop();
            line
        }
    )
}

#[allow(unused_macros)]
macro_rules! parse_vec {
    ( $t:ty ) => (
        {
            let mut line = String::new();
            ::std::io::stdin().read_line(&mut line).unwrap();
            let iter = line.split_whitespace();
            iter.map(|v| v.parse::<$t>().unwrap()).collect::<Vec<_>>()
        }
    )
}

fn imos(v: &mut Vec<Vec<i32>>, lb: (usize, usize), rt: (usize, usize)) {
    v[lb.0][lb.1] += 1;
    v[lb.0][rt.1] -= 1;
    v[rt.0][lb.1] -= 1;
    v[rt.0][rt.1] += 1;
}

fn solve(k: usize, cells: Vec<(usize, usize, char)>) -> usize {

    let cs = cells.iter().map(|&(x, y, c)| {
        if c == 'B' {
            (x%(2*k),y%(2*k))
        } else {
            (x%(2*k), (y+k)%(2*k))
        }
    });

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

    for (x,y) in cs {
        if (x<k && y<k) || (k<=x && k<=y) {
            let x = x%k; let y = y%k;

            imos(&mut v, (0, 0), (x+1, y+1)); // left bottom
            imos(&mut v, (x+k+1, 0), (2*k, y+1)); // right bottom
            imos(&mut v, (0, y+k+1), (x+1, 2*k)); // left top
            imos(&mut v, (x+k+1, y+k+1), (2*k, 2*k)); // right top
            imos(&mut v, (x+1, y+1), (x+k+1, y+k+1)); // center
        } else {
            let x = x%k; let y = y%k;

            imos(&mut v, (x+1, 0), (x+k+1, y+1)); // bottom
            imos(&mut v, (0, y+1), (x+1, y+k+1)); // left
            imos(&mut v, (x+k+1, y+1), (2*k, y+k+1)); // right
            imos(&mut v, (x+1, y+k+1), (x+k+1, 2*k)); // top
        }
    }

    for y in 0..(2*k+1) {
        for x in 1..(2*k+1) {
            let x = x as usize; let y = y as usize;
            v[x][y] += v[x-1][y];
        }
    }

    for x in 0..(2*k+1) {
        for y in 1..(2*k+1) {
            let x = x as usize; let y = y as usize;
            v[x][y] += v[x][y-1];
        }
    }

    let mut max = 0;
    for x in 0..(2*k) {
        for y in 0..(2*k) {
            let x = x as usize; let y = y as usize;
            if max < v[x][y] {
                max = v[x][y];
            }
        }
    }

    max as usize
}

fn main() {
    let (n, k) = parse_line!(usize, usize);

    println!("{}", solve(k, (0..n).map(|_| {
        let (x,y,c) = parse_line!(usize, usize, String);
        (x,y,c.chars().next().unwrap())
    }).collect()));
}

#[test]
fn solve_test() {
    assert_eq!(solve(3, vec![(0, 1, 'W'),
                             (1, 2, 'W'),
                             (5, 3, 'B'),
                             (5, 4, 'B')]), 4);

    assert_eq!(solve(1000, vec![(0, 0, 'B'),
                                (0, 1, 'W')]), 2);

    assert_eq!(solve(2, vec![(1, 2, 'B'),
                             (2, 1, 'W'),
                             (2, 2, 'B'),
                             (1, 0, 'B'),
                             (0, 6, 'W'),
                             (4, 5, 'W'),
    ]), 4);

}