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: Accepted)

n,k = gets.split.map(&:to_i)
arr = Array.new(k+1){Array.new(k+1,0)}
arr2 = Array.new(k+1){Array.new(k+1,0)}
col = Array.new(k+1,0)
col2 = Array.new(k+1,0)
n.times do
  x,y,c = gets.split
  x,y,c = x.to_i,y.to_i,c.chomp
  t = (c == "W") ? 0 : 1
  xm = x % k
  ym = y % k
  arr[xm][ym] += 1
  arr2[xm][ym] += 1 if ((x/k + y/k) % 2) == t
  # p [x,y,c,t,((x/k + y/k) % 2),((x/k + y/k) % 2) == t]
end
k.times do |i|
  k.times do |j|
    col[i] += arr[j][i]
    col2[i] += arr2[j][i]
  end
end

k.times do |i|
  k.times do |j|
    arr[i][j] += arr[i][j-1]
    arr2[i][j] += arr2[i][j-1]
  end
end

max = min = ni = num = col2.inject(:+)

(k+1).times do |i|
  num = ni - col2[i-1] + (col[i-1] - col2[i-1])
  ni = num
  max = num if num > max
  min = num if num < min
  # p [i,num]
  k.times do |j|
    num -= arr2[j][k-1] - arr2[j][i-1]
    num += arr2[j][i-1]
    num += arr[j][k-1] - arr[j][i-1] - (arr2[j][k-1] - arr2[j][i-1])
    num -= arr[j][i-1] - arr2[j][i-1]
    max = num if num > max
    min = num if num < min
    # p [i,j,num]
  end
end

puts max > (n-min) ? max : n-min