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: Wrong Answer)

import java.util.Scanner;

public class Main {

	public static void main(String[] args) {
		// TODO 自動生成されたメソッド・スタブ
		Scanner sc = new Scanner(System.in);

		int n = sc.nextInt();
		int k = sc.nextInt();

		int[][] white = new int[k][k];
		int[][] black = new int[k][k];

		int whiteTotal = 0;
		int blackTotal = 0;

		int[][] whiteLeftBelow = new int[k][k];
		int[][] blackLeftBelow = new int[k][k];
		int[][] whiteRightAbove = new int[k][k];
		int[][] blackRightAbove = new int[k][k];

		for(int i=0;i<n;i++) {
			int x = sc.nextInt();
			int y = sc.nextInt();
			boolean isWhite = sc.next().equals("W");
			int xmod = x%k;
			int ymod = y%k;
			if((x/k)%2==0 ^ (y/k)%2==0) {
				isWhite = !isWhite;
			}

			if(isWhite) {
				white[xmod][ymod]++;
				whiteTotal++;
			} else {
				black[xmod][ymod]++;
				blackTotal++;
			}
		}

		whiteLeftBelow[0][0]=white[0][0];
		blackLeftBelow[0][0]=black[0][0];
		for(int i=0;i<k;i++) {
			for(int j=0;j<k-1;j++) {
				if(i==0) {
					whiteLeftBelow[i][j+1]=whiteLeftBelow[i][j]+white[i][j+1];
					blackLeftBelow[i][j+1]=blackLeftBelow[i][j]+black[i][j+1];
				} else {
					whiteLeftBelow[i][j+1]=whiteLeftBelow[i][j]+whiteLeftBelow[i-1][j+1]-whiteLeftBelow[i-1][j]+white[i][j+1];
					blackLeftBelow[i][j+1]=blackLeftBelow[i][j]+blackLeftBelow[i-1][j+1]-blackLeftBelow[i-1][j]+black[i][j+1];
				}
			}
			if(i<k-1) {
				whiteLeftBelow[i+1][0]=whiteLeftBelow[i][0]+white[i+1][0];
				blackLeftBelow[i+1][0]=blackLeftBelow[i][0]+black[i+1][0];
			}
		}

		whiteRightAbove[k-1][k-1]=white[k-1][k-1];
		blackRightAbove[k-1][k-1]=black[k-1][k-1];
		for(int i=k-1;i>=0;i--) {
			for(int j=k-1;j>=1;j--) {
				if(i==k-1) {
					whiteRightAbove[i][j-1]=whiteRightAbove[i][j]+white[i][j-1];
					blackRightAbove[i][j-1]=blackRightAbove[i][j]+black[i][j-1];
				} else {
					whiteRightAbove[i][j-1]=whiteRightAbove[i][j]+whiteRightAbove[i+1][j-1]-whiteRightAbove[i+1][j]+white[i][j-1];
					blackRightAbove[i][j-1]=blackRightAbove[i][j]+blackRightAbove[i+1][j-1]-blackRightAbove[i+1][j]+black[i][j-1];
				}
			}
			if(i>0) {
				whiteRightAbove[i-1][0]=whiteRightAbove[i][0]+white[i-1][0];
				blackRightAbove[i-1][0]=blackRightAbove[i][0]+black[i-1][0];
			}
		}

		//count total hopes

		int ans = 0;
		for (int i=0;i<k-1;i++) {
			for(int j=0;j<k-1;j++) {
				int tmp = whiteLeftBelow[i][j] + whiteRightAbove[(i+1)%k][(j+1)%k] + blackTotal
						- blackLeftBelow[i][j] - blackRightAbove[(i+1)%k][(j+1)%k];
				if(ans < tmp) {
					ans = tmp;
				}

				tmp = blackLeftBelow[i][j]+blackRightAbove[(i+1)%k][(j+1)%k] + whiteTotal
						-  whiteLeftBelow[i][j] - whiteRightAbove[(i+1)%k][(j+1)%k];

				if(ans < tmp) {
					ans = tmp;
				}
			}
		}

		System.out.println(ans);

	}

}