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: Wrong Answer)

# -*- coding: utf-8 -*-
import heapq
import math
import sys

def hanten(c):
    if c == 'B':
        return 'W'
    else:
        return 'B'

tmp = input()
N, K = [int(a) for a in tmp.split()]
requirements = [[{'B': 0, 'W': 0} for x in range(K)] for y in range(K)]
for n in range(N):
    tmp = input()
    x, y, c = [a for a in tmp.split()]
    x, y = int(x), int(y)
    x %= (K*2)
    y %= (K*2)
    x_pole = x // K
    y_pole = y // K
    assert x_pole <= 1 and y_pole <= 1
    if x_pole + y_pole == 1:
        c = hanten(c)
    requirements[x % K][y % K][c] += 1

Colors = ['B', 'W']
max_ret = 0
all_num = {'B': 0, 'W': 0}
for req1 in requirements:
    for req2 in req1:
        all_num['B'] += req2['B']
all_num['W'] += N - all_num['B']
for now_c in Colors:
    num = all_num[now_c]
    max_ret = max([max_ret, num])
    bx, ex = 0, K-1
    by, ey = 0, K-1
    while bx <= ex and by <= ey:
        # left ->
        ltor = 0
        for y in range(by, ey+1):
            ltor -= requirements[bx][y][now_c]
            ltor += requirements[bx][y][hanten(now_c)]
        # right <-
        rtol = 0
        for y in range(by, ey+1):
            rtol -= requirements[ex][y][now_c]
            rtol += requirements[ex][y][hanten(now_c)]
        # top shita
        ttob = 0
        for x in range(bx, ex+1):
            ttob -= requirements[x][by][now_c]
            ttob += requirements[x][by][hanten(now_c)]
        # bottom ue
        btot = 0
        for x in range(bx, ex+1):
            btot -= requirements[x][ey][now_c]
            btot += requirements[x][ey][hanten(now_c)]
        results = [ltor, rtol, ttob, btot]
        max_effective = max(results)
        if max_effective == ltor:
            bx += 1
        elif max_effective == rtol:
            ex -= 1
        elif max_effective == ttob:
            by += 1
        else:
            ey -= 1
        num += max_effective
        max_ret = max([max_ret, num])
print(max_ret)