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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

N, K = gets.split(' ').map(&:to_i)
grid = Array.new(2 * K + 1){[0] * (2 * K + 1)}
N.times do 
  x, y, c = gets.split(' ')
  if c == 'B'
    grid[x.to_i % (2 * K) + 1][y.to_i % (2 * K) + 1] += 1
  else
    grid[(x.to_i + K) % (2 * K) + 1][y.to_i % (2 * K) + 1] += 1
  end
end

1.upto(2 * K) do |i|
  1.upto(2 * K) do |j|
    if j >= 1
      grid[i][j] += grid[i][j - 1]
    end
  end
end
1.upto(2 * K) do |i|
  1.upto(2 * K) do |j|
    if i >= 1
      grid[i][j] += grid[i - 1][j]
    end
  end
end

ans = 0
1.upto(K) do |i|
  1.upto(K) do |j|
    score = 0
    score += grid[i + K - 1][j + K - 1] - grid[i - 1][j + K - 1] - grid[i + K - 1][j - 1] + grid[i - 1][j - 1]
    score += grid[2 * K][2 * K] - grid[i + K - 1][2 * K] - grid[2 * K][j + K - 1] + grid[i + K - 1][j + K - 1]
    score += grid[i - 1][2 * K] - grid[i - 1][j + K - 1]
    score += grid[2 * K][j - 1] - grid[i + K - 1][j - 1]
    score += grid[i - 1][j - 1]
    ans = [ans, score, N - score].max
  end
end

puts ans