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: Accepted)

import java.io.{BufferedReader, IOException, InputStream, InputStreamReader}
import java.util.StringTokenizer

import scala.util.Sorting

object Main {
  def solve(): Unit = {
    val sc = new InputReader(System.in)
    val N = sc.nextInt()
    val K = sc.nextInt()
    val m = 2 * K
    val c = Array.ofDim[Int](m, m)

    (0 until N) foreach { _ =>
      val x = sc.nextInt()
      val y = sc.nextInt()
      sc.nextChar() match {
        case 'W' => c(x % m)(y % m) += 1
        case 'B' => c((x % m + K) % m)(y % m) += 1
      }
    }

    val sums = Array.ofDim[Int](m, m)    // (x, y)から(x+K, y)までのsum
    (0 until m) foreach { y =>
      sums(0)(y) = ((0 until K) map (x => c(x)(y))).sum
    }
    for {
      x <- 1 until m
      y <- 0 until m
    } {
      sums(x)(y) = sums(x - 1)(y) + c((x + K - 1) % m)(y) - c(x - 1)(y)
    }

    def w1(x: Int, y: Int) = sums(x % m)(y % m)
    def w2(x: Int, y: Int) = sums((x + K) % m)((y + K) % m)

    var ans = 0
    (0 until m) foreach { x =>
      var sum1 = ((0 until K) map (y => w1(x, y))).sum
      var sum2 = ((0 until K) map (y => w2(x, y))).sum
      ans = math.max(ans, sum1 + sum2)
      (1 until m) foreach { y =>
        sum1 -= w1(x, y - 1)
        sum1 += w1(x, y + K - 1)
        sum2 -= w2(x, y - 1)
        sum2 += w2(x, y + K - 1)
        ans = math.max(ans, sum1 + sum2)
      }
    }
    println(ans)
  }

  def main(args: Array[String]): Unit = {
    solve()
  }

  class InputReader(val stream: InputStream) {
    val reader = new BufferedReader(new InputStreamReader(stream), 32768)
    var tokenizer: StringTokenizer = null

    def next: String = {
      while (tokenizer == null || !tokenizer.hasMoreTokens)
        tokenizer = new StringTokenizer(reader.readLine)
      tokenizer.nextToken
    }

    def nextInt(): Int = next.toInt
    def nextChar(): Char = next.charAt(0)
  }
}