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)

def make_2dgrid(g)
  h = g.size
  s = Array.new(h + 1) { Array.new(h + 1, 0) }
  h.times do |i|
    h.times do |j|
      s[i + 1][j + 1] = s[i][j + 1] + s[i + 1][j] - s[i][j] + g[i][j]
    end
  end
  s
end

def query(s, x1, x2, y1, y2)
  s[x2][y2] - s[x1][y2] - s[x2][y1] + s[x1][y1]
end

n, k = gets.split.map(&:to_i)

k2 = 2 * k
gB = Array.new(k2) { Array.new(k2, 0) }
n.times do
  x, y, ci = gets.split
  x = x.to_i % k2
  y = (y.to_i + (ci == "W" ? k : 0)) % k2
  gB[x][y] += 1
end
ans = 0

sB = make_2dgrid(gB)
k.times do |i|
  k.times do |j|
    # score = query(sB, 0, i, 0, j)
    score = sB[i][j] - sB[0][j] - sB[i][0] + sB[0][0]
    # score += query(sB, i + k, k2, 0, j)
    score += sB[k2][j] - sB[i + k][j] - sB[k2][0] + sB[i + k][0]
    # score += query(sB, i, i + k, j, j + k)
    score += sB[i + k][j + k] - sB[i][j + k] - sB[i + k][j] + sB[i][j]
    # score += query(sB, 0, i, j + k, k2)
    score += sB[i][k2] - sB[0][k2] - sB[i][j + k] + sB[0][j + k]
    # score += query(sB, i + k, k2, j + k, k2)
    score += sB[k2][k2] - sB[i + k][k2] - sB[k2][j + k] + sB[i + k][j + k]
    ans = score if score > ans
    score = n - score
    ans = score if score > ans
  end
end
p ans