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: Wrong Answer)

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

class Solution
{
    static int GetSum(int[][] table, int x0, int y0, int x1, int y1)
    {
        var maxX = Math.Max(x0, x1);
        var maxY = Math.Max(y0, y1);
        var minX = Math.Min(x0, x1);
        var minY = Math.Min(y0, y1);

        int sum = table[maxY][maxX];
        if (minX > 0)
        {
            sum -= table[maxY][minX];
        }
        if (minY > 0)
        {
            sum -= table[minY][maxX];
        }
        if (minX > 0 && minY > 0)
        {
            sum += table[minY][minX];
        }

        return sum;
    }

    static void Main()
    {
        var vals = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
        int n = vals[0];
        int k = vals[1];
        var blackCount = Enumerable.Range(0, k).Select(_ => new int[k]).ToArray();
        var whiteCount = Enumerable.Range(0, k).Select(_ => new int[k]).ToArray();
        for (int i = 0; i < n; i++)
        {
            var v = Console.ReadLine().Split(' ').ToArray();
            var x = int.Parse(v[0]) % (2 * k);
            var y = int.Parse(v[1]) % (2 * k);
            var color = v[2][0];
            var isBlack = color == 'B';

            if (x >= k && y >= k)
            {
                x -= k;
                y -= k;
            }
            else if (x >= k)
            {
                x -= k;
                isBlack ^= true;
            }
            else if (y >= k)
            {
                y -= k;
                isBlack ^= true;
            }

            if (isBlack)
            {
                Console.Error.WriteLine($"({x},{y}) black");
                blackCount[y][x]++;
            }
            else
            {
                Console.Error.WriteLine($"({x},{y}) white");
                whiteCount[y][x]++;
            }
        }

        // 右へ加算
        for (int y = 0; y < k; y++)
        {
            for (int x = 1; x < k; x++)
            {
                blackCount[y][x] += blackCount[y][x - 1];
                whiteCount[y][x] += whiteCount[y][x - 1];
            }
        }

        // 下へ加算
        for (int x = 0; x < k; x++)
        {
            for (int y = 1; y < k; y++)
            {
                blackCount[y][x] += blackCount[y - 1][x];
                whiteCount[y][x] += whiteCount[y - 1][x];
            }
        }

        int max = 0;
        int min = int.MaxValue;
        for (int x = 0; x < k; x++)
        {
            for (int y = 0; y < k; y++)
            {
                int satisfied = 0;
                // 左上から(x-1,y-1)まで黒
                if (x > 0 && y > 0)
                {
                    satisfied += GetSum(blackCount, 0, 0, x - 1, y - 1);
                }
                // (x,y)から右下まで黒
                satisfied += GetSum(blackCount, x, y, k - 1, k - 1);
                // (x,y-1)から右上まで白
                if (y > 0)
                {
                    satisfied += GetSum(whiteCount, x, y - 1, k - 1, 0);
                }
                // 左下から(x-1,y)まで白
                if (x > 0)
                {
                    satisfied += GetSum(whiteCount, 0, k - 1, x - 1, y);
                }

                max = Math.Max(max, satisfied);
                min = Math.Min(min, satisfied);
            }
        }
        Console.Error.WriteLine($"max {max}");
        Console.Error.WriteLine($"min {min}");

        max = Math.Max(max, n - min);
        Console.WriteLine(max);
    }
}