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: Wrong Answer)

use std::io::{BufRead, Write};
use std::io;
use std::cmp;

fn range_sum(table: &Vec<Vec<usize>>, start: (usize, usize), end: (usize, usize)) -> usize {
    assert!(end.0 >= start.0);
    assert!(end.1 >= start.1);
    table[end.0][end.1] + table[start.0][start.1] - table[start.0][end.1] - table[end.0][start.1]
}

fn solve<R: BufRead, W: Write>(mut input: R, mut output: W) {
    let mut buf = String::new();
    input.read_line(&mut buf).unwrap();
    let (n, k): (usize, usize) = {
        let mut split = buf.split_whitespace();
        (split.next().unwrap().parse().unwrap(), split.next().unwrap().parse().unwrap())
    };
    let mut white = vec![vec![0usize; k * 2]; k * 2];
    let mut black = white.clone();
    let mut white_sums = vec![vec![0usize; k * 2 + 1]; k * 2 + 1];
    let mut black_sums = white_sums.clone();
    for _ in 0..n {
        buf.clear();
        input.read_line(&mut buf).unwrap();
        let (x, y, c): (usize, usize, bool) = {
            let mut split = buf.split_whitespace();
            (split.next().unwrap().parse().unwrap(), split.next().unwrap().parse().unwrap(), split.next().unwrap() == "B")
        };
        if c {
            black[x % (k * 2)][y % (k * 2)] += 1;
        } else {
            white[x % (k * 2)][y % (k * 2)] += 1;
        }
    }
    for x in 0..k * 2 {
        for y in 0..k * 2 {
            white_sums[x + 1][y + 1] = white_sums[x + 1][y] + white_sums[x][y + 1] - white_sums[x][y] + white[x][y];
            black_sums[x + 1][y + 1] = black_sums[x + 1][y] + black_sums[x][y + 1] - black_sums[x][y] + black[x][y];
        }
    }
    let result = (0..k).map(|x| (0..k).map(|y| {
        let white = range_sum(&white_sums, (0, 0), (x, y))
            + range_sum(&white_sums, (x, y), (x + k, y + k))
            + range_sum(&white_sums, (x + k, 0), (k * 2, y))
            + range_sum(&white_sums, (0, y + k), (x, k * 2))
            + range_sum(&white_sums, (x + k, y + k), (k * 2, k * 2));
        let black = range_sum(&black_sums, (x, 0), (x + k, y))
            + range_sum(&black_sums, (0, y), (x, y + k))
            + range_sum(&black_sums, (x + k, y), (k * 2, y + k))
            + range_sum(&black_sums, (x, y + k), (y + k, k * 2));
        cmp::max(white + black, n - white - black)
    }).max().unwrap()).max().unwrap();
    writeln!(output, "{}", result).unwrap();
}

fn main() {
    let stdin = io::stdin();
    let stdout = io::stdout();
    solve(stdin.lock(), stdout.lock());
}

#[cfg(test)]
mod tests {
    fn test(input: &str, output: &str) {
        let mut v: Vec<u8> = Vec::new();
        ::solve(input.as_bytes(), &mut v);
        assert_eq!(String::from_utf8(v).unwrap(), output);
    }

    #[test]
    fn case1() {
        test(
            "4 3
0 1 W
1 2 W
5 3 B
5 4 B
",
            "4\n",
        );
    }

    #[test]
    fn case2() {
        test(
            "2 1000
0 0 B
0 1 W
",
            "2\n",
        );
    }

    #[test]
    fn case3() {
        test(
            "6 2
1 2 B
2 1 W
2 2 B
1 0 B
0 6 W
4 5 W
",
            "4\n",
        );
    }
}