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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using static System.Console;
using System.Text;
using System.IO;
using static System.Math;
using System.Numerics;

namespace AtCoder
{
    public class Program
    {
        public static void Main(string[] args)
        {
            //SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
            Solve();
            //Out.Flush();
            Read();
        }

        static void Solve()
        {
            var (N, K) = Slice2<long>();
            var KKB = new long[K, K];
            var KKW = new long[K, K];
            for (var i = 0; i < N; ++i)
            {
                var xyc = ReadLine().Split();
                var x = long.Parse(xyc[0]);
                var y = long.Parse(xyc[1]);
                var c = xyc[2][0] == 'B';
                var rev = false;
                if ((x / K % 2) != 0) rev = !rev;
                if ((y / K % 2) != 0) rev = !rev;
                x = x % K;
                y = y % K;
                c = rev ? !c : c;
                if (c)
                    ++KKB[x, y];
                else
                    ++KKW[x, y];
            }
            var ruiB = new long[K + 1, K + 1];
            var ruiW = new long[K + 1, K + 1];
            for (var i = 1; i <= K; ++i)
                for (var j = 1; j <= K; ++j)
                {
                    ruiB[i, j] = KKB[i - 1, j - 1] + ruiB[i, j - 1] + ruiB[i - 1, j] - ruiB[i - 1, j - 1];
                    ruiW[i, j] = KKW[i - 1, j - 1] + ruiW[i, j - 1] + ruiW[i - 1, j] - ruiW[i - 1, j - 1];
                }
            var ans = 0L;
            for (var i = 0; i <= K; ++i)
                for (var j = 0; j <= K; ++j)
                {
                    var num = 0L;
                    num += ruiB[K, K] - ruiB[i, K] - ruiB[K, j] + 2 * ruiB[i, j];
                    num += ruiW[i, K] + ruiW[K, j] - 2 * ruiW[i, j];
                    ans = Max(ans, Max(num, N - num));
                }
            WriteLine(ans);
        }
        static long Mod = 998244353;
        static public long[] Sarray() { return ReadLine().Split().Select(long.Parse).ToArray(); }
        static public List<long> Slist() { return ReadLine().Split().Select(long.Parse).ToList(); }
        static public (T1 a, T2 b) Slice<T1, T2>()
        {
            var t = ReadLine().Split();
            return (
            (T1)Convert.ChangeType(t[0], typeof(T1)),
            (T2)Convert.ChangeType(t[1], typeof(T2)));
        }
        static public (T a, T b) Slice2<T>()
        {
            var t = ReadLine().Split();
            return (
            (T)Convert.ChangeType(t[0], typeof(T)),
            (T)Convert.ChangeType(t[1], typeof(T)));
        }
        static public (T a, T b, T c) Slice3<T>()
        {
            var t = ReadLine().Split();
            return (
            (T)Convert.ChangeType(t[0], typeof(T)),
            (T)Convert.ChangeType(t[1], typeof(T)),
            (T)Convert.ChangeType(t[2], typeof(T)));
        }
    }
}