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

namespace D
{
	internal class Program
	{
		public static void Main(string[] args)
		{
			var line = Console.ReadLine().Split().Select(int.Parse).ToArray();
			var n = line[0];
			var k = line[1];
			var xn = new List<int>(n);
			var yn = new List<int>(n);
			var cn = new List<bool>(n); // 'W'ならtrue
			var xMods = new bool[k]; // kで割った余りがiのx座標が存在するか
			var yMods = new bool[k]; // kで割った余りがiのy座標が存在するか
			for (var i = 0; i < n; i++)
			{
				var line2 = Console.ReadLine().Split().ToArray();
				var x = int.Parse(line2[0]);
				var y = int.Parse(line2[1]);
				xn.Add(x);
				yn.Add(y);
				xMods[x % k] = true;
				yMods[y % k] = true;
				cn.Add(line2[2][0] == 'W');
			}

			var result = 0;
			for (var leftBottomX = 0; leftBottomX < k; leftBottomX++)
			{
				if (!xMods[leftBottomX]) { continue; }

				for (var leftBottomY = 0; leftBottomY < k; leftBottomY++)
				{
					if (!yMods[leftBottomY]) { continue; }

					var groupNum = 0;
					for (var i = 0; i < n; i++)
					{
						var tmp = (int) (Math.Floor((xn[i] - leftBottomX) / (double) k)) % 2 +
						          (int) (Math.Floor((yn[i] - leftBottomY) / (double) k)) % 2;
						if ((tmp % 2 == 0) ^ cn[i]) { groupNum++; }
					}

					if (groupNum * 2 < n) { groupNum = n - groupNum; }

					if (result < groupNum) { result = groupNum; }

					if (result == n)
					{
						Console.WriteLine(result);
						return;
					}
				}
			}

			Console.WriteLine(result);
		}
	}
}