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.IO;
using System.Collections.Generic;
using System.Linq;

namespace ARC089D
{
    public class Program
    {
        int N;
        int K;
        int K2;
        int[,] imos;

        void Add(int lx, int ty, int rx, int by)
        {
            lx %= K2;
            ty %= K2;
            rx %= K2;
            by %= K2;

            if (lx > rx)
            {
                Add(lx, ty, K2 - 1, by);
                Add(0, ty, rx, by);
                return;
            }
            if (ty > by)
            {
                Add(lx, ty, rx, K2 - 1);
                Add(lx, 0, rx, by);
                return;
            }

            imos[ty, lx]++;
            imos[by + 1, lx]--;
            imos[ty, rx + 1]--;
            imos[by + 1, rx + 1]++;
        }

        void Solve(StreamScanner ss, StreamWriter sw)
        {
            //---------------------------------
            N = ss.Next(int.Parse);
            K = ss.Next(int.Parse);
            K2 = K * 2;
            imos = new int[K2 + 1, K2 + 1];

            for (var i = 0; i < N; i++)
            {
                var X = ss.Next(int.Parse);
                var Y = ss.Next(int.Parse);
                var C = ss.Next(String);
                if (C == "B") X += K;

                Add(X, Y, X + K - 1, Y + K - 1);
                Add(X + K, Y + K, X + K2 - 1, Y + K2 - 1);
            }

            for (var i = 0; i <= 2 * K; i++)
            for (var j = 0; j <= 2 * K; j++)
            {
                if (i != 0) imos[i, j] += imos[i - 1, j];
                if (j != 0) imos[i, j] += imos[i, j - 1];
                if (i != 0 && j != 0) imos[i, j] -= imos[i - 1, j - 1];
            }

            sw.WriteLine(imos.Cast<int>().Max());
            //---------------------------------
        }

        static void Main()
        {
            var ss = new StreamScanner(new StreamReader(Console.OpenStandardInput()));
            var sw = new StreamWriter(Console.OpenStandardOutput()) {AutoFlush = false};
            new Program().Solve(ss, sw);
            sw.Flush();
        }

        static readonly Func<string, string> String = s => s;
    }

    public class StreamScanner
    {
        static readonly char[] Sep = {' '};
        readonly Queue<string> buffer = new Queue<string>();
        readonly TextReader textReader;

        public StreamScanner(TextReader textReader)
        {
            this.textReader = textReader;
        }

        public T Next<T>(Func<string, T> parser)
        {
            if (buffer.Count != 0) return parser(buffer.Dequeue());
            var nextStrings = textReader.ReadLine().Split(Sep, StringSplitOptions.RemoveEmptyEntries);
            foreach (var nextString in nextStrings) buffer.Enqueue(nextString);
            return Next(parser);
        }
    }
}