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 Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.Arrays;
import java.util.StringTokenizer;

public class Main {
    public static void main(String[] args) throws Throwable {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));

        StringTokenizer st = new StringTokenizer(br.readLine());
        int N = Integer.parseInt(st.nextToken());
        int K = Integer.parseInt(st.nextToken());
        int[] x = new int[N];
        int[] y = new int[N];
        char[] c = new char[N];
        for (int i = 0; i < N; ++i) {
            st = new StringTokenizer(br.readLine());
            x[i] = Integer.parseInt(st.nextToken());
            y[i] = Integer.parseInt(st.nextToken());
            c[i] = st.nextToken().charAt(0);
        }

        System.out.println(solve(x, y, c, K));

        br.close();
    }

    static int solve(int[] x, int[] y, char[] c, int K) {
        int result = 0;
        int[] deltas = new int[2 * K + 1];
        for (int minX = 0; minX < 2 * K; ++minX) {
            Arrays.fill(deltas, 0);

            for (int i = 0; i < x.length; ++i) {
                int blackMinY;
                int blackMaxY;
                if ((x[i] - minX + 2 * K) % (2 * K) < K) {
                    blackMinY = 0;
                    blackMaxY = K - 1;
                } else {
                    blackMinY = K;
                    blackMaxY = 2 * K - 1;
                }

                int unitY = y[i] % (2 * K);
                if (c[i] == 'B') {
                    if (unitY >= blackMinY && unitY <= blackMaxY) {
                        ++deltas[0];

                        if (blackMinY == 0) {
                            --deltas[unitY + 1];
                            ++deltas[unitY + 1 + K];
                        } else {
                            --deltas[unitY - K + 1];
                            ++deltas[unitY + 1];
                        }
                    } else {
                        if (blackMinY == 0) {
                            ++deltas[unitY - K + 1];
                            --deltas[unitY + 1];
                        } else {
                            ++deltas[unitY + 1];
                            --deltas[unitY + 1 + K];
                        }
                    }
                } else {
                    if (unitY >= blackMinY && unitY <= blackMaxY) {
                        if (blackMinY == 0) {
                            ++deltas[unitY + 1];
                            --deltas[unitY + 1 + K];
                        } else {
                            ++deltas[unitY - K + 1];
                            --deltas[unitY + 1];
                        }
                    } else {
                        ++deltas[0];

                        if (blackMinY == 0) {
                            --deltas[unitY - K + 1];
                            ++deltas[unitY + 1];
                        } else {
                            --deltas[unitY + 1];
                            ++deltas[unitY + 1 + K];
                        }
                    }
                }
            }

            int matchNum = 0;
            for (int i = 0; i < deltas.length; ++i) {
                matchNum += deltas[i];
                result = Math.max(result, matchNum);
            }
        }

        return result;
    }
}