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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

import sys
import numpy as np
def input(): return sys.stdin.readline().strip()
n, k = [int(item) for item in input().split()]
xyc = [[item for item in input().split()] for _ in range(n)]
 
a = [[0] * 2*k for _ in range(2*k)]
for x, y, c in xyc:
    x = int(x)
    y = int(y)
    if c == "W":
        a[x%(2*k)][y%(2*k)] +=  1
    else: 
        a[(x+k)%(2*k)][(y)%(2*k)] += 1

cumsum = np.cumsum(np.cumsum(a, axis=0), axis=1)
 
ans = 0
for i in range(k):
    for j in range(k):
        ret = 0
        ik = i+k
        jk = j+k
        hm = 2*k-1
        ret += (cumsum[ik][hm] - cumsum[i][hm] + 2 * cumsum[i][jk] + cumsum[hm][j+k] - cumsum[hm][j] + 2 * cumsum[ik][j] - 2 * cumsum[ik][jk] - 2 * cumsum[i][j])
        ans = max(ans, ret, n-ret)
print(ans)