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 (2.7.6) (judge verdict: Accepted)

#!/usr/bin/env python

from collections import deque
import itertools as it
import sys
import math

sys.setrecursionlimit(1000000)

N, K = map(int, raw_input().split())
DP = [[0] * (K + 1) for i in range(K + 1)]
bias = 0

for i in range(N):
    x, y, c = raw_input().split()
    x, y, c = int(x), int(y), (c == 'W')
    p = (x / K + y / K + c) % 2
    DP[y % K + 1][x % K + 1] += 1 - 2 * p
    bias += p

for i in range(K):
    for j in range(K + 1):
        DP[i + 1][j] += DP[i][j]

for i in range(K + 1):
    for j in range(K):
        DP[i][j + 1] += DP[i][j]

ans = 0

for i in range(K + 1):
    for j in range(K + 1):
        S = DP[i][K] + DP[K][j] - DP[i][j] * 2
        ans = max(ans, S, DP[K][K] - S)

print ans + bias