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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;

class D
{
	static void Main()
	{
		var h = Console.ReadLine().Split().Select(int.Parse).ToArray();
		var k = h[1];

		var bw = new int[k, k, 2];
		foreach (var x in new int[h[0]].Select(_ => Console.ReadLine().Split()))
		{
			var p = x.Take(2).Select(int.Parse).ToArray();
			bw[p[0] % k, p[1] % k, (p[0] / k + p[1] / k + (x[2] == "B" ? 0 : 1)) % 2]++;
		}

		var ti = new int[k, 2];
		for (int c = 0; c < 2; c++)
			for (int i = 0; i < k; i++)
				for (int j = 0; j < k; j++)
					ti[i, c] += bw[i, j, c];
		var tj = new int[k, 2];
		for (int c = 0; c < 2; c++)
			for (int j = 0; j < k; j++)
				for (int i = 0; i < k; i++)
					tj[j, c] += bw[i, j, c];
		var s = new int[2];
		for (int c = 0; c < 2; c++)
			for (int i = 0; i < k; i++)
				s[c] += ti[i, c];

		var M = 0;
		for (int j = 0; j < k; j++)
		{
			for (int i = 0; i < k; i++)
			{
				M = Math.Max(M, s[0]);
				M = Math.Max(M, s[1]);
				s[0] += ti[i, 1] - ti[i, 0];
				s[1] += ti[i, 0] - ti[i, 1];
			}

			for (int i = 0; i < k; i++)
			{
				ti[i, 0] += bw[i, j, 1] - bw[i, j, 0];
				ti[i, 1] += bw[i, j, 0] - bw[i, j, 1];
			}
			Array.Reverse(s);
			s[0] += tj[j, 1] - tj[j, 0];
			s[1] += tj[j, 0] - tj[j, 1];
		}
		Console.WriteLine(M);
	}
}