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: Time Limit Exceeded)

using System;

class MainClass
{
    public static void Main(string[] args)
    {
        string[] s1 = Console.ReadLine().Split(' ');

        int N = int.Parse(s1[0]);
        long K = long.Parse(s1[1]);
       
        long[] x = new long[N];
        long[] y = new long[N];
        string[] c = new string[N];

        for (int i = 0; i < N;i++)
        {
            string[] s = Console.ReadLine().Split(' ');

            x[i] = long.Parse(s[0]);
            y[i] = long.Parse(s[1]);
            c[i] = s[2];
        }

        int max = 0;

        for (int width = 0; width <= K;width++)
        {
            for (int height = 0; height <= K; height++)
            {
                int sum = 0;

                for (int i = 0; i < N; i++)
                {
                    if(c[i] == "W")
                    {
                        if (((x[i] + width) / K + (y[i] + height) / K) % 2 == 0)
                        {
                            sum++;
                        }

                    }
                    else
                    {
                        if (((x[i] + width) / K + (y[i] + height) / K) % 2 == 1)
                        {
                            sum++;
                        }
                    }

                    if (sum + N - i <= max && sum >=max) break;

                }

                if (sum <= N / 2) sum = N - sum;
                if (sum >= max) max = sum;

                if (max >= N)
                {
                    Console.WriteLine(max);
                    return;
                }

            } 
        }

        Console.WriteLine(max);
        return;

    }
}