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 Scala (2.11.7) (judge verdict: Time Limit Exceeded)

import scala.language.postfixOps

object Main {
  def main(args: Array[String]): Unit = {
    val Array(n, k) = io.StdIn.readLine().split(' ') map { _.toInt }

    val inputs = Stream.continually( io.StdIn.readLine().split(' ') ) take n
    val whites = inputs.filter( x => x(2) == "W" ) map { _.take(2) map { _.toInt } }
    val blacks = inputs.filter( x => x(2) == "B" ) map { _.take(2) map { _.toInt } }

    def eval_offset(x:Int, y:Int):Int =
      return (whites map { a: Array[Int] =>
        (Array((a(0) + x) / k, (a(1) + y) / k) sum) % 2
      } filter ({
        _ == 1
      }) length) +
        (blacks map { a: Array[Int] =>
        (Array((a(0) + x) / k, (a(1) + y) / k) sum) % 2
      } filter ({
        _ == 0
      }) length)

    var max = 0
    for (x <- 0 to k-1; y <- 0 to 2*k-1) {
      val a = eval_offset(x, y)
      if (a > max)
        max = a
    }

    println(max)
  }
}