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 C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <sstream>
#include <algorithm>
#include <functional>
#include <iterator>
#include <bitset>
#include <string>
#include <list>
#include <vector>
#include <stack>
#include <queue>
#include <set>
#include <map>

typedef   signed long long int slli;
typedef unsigned long long int ulli;
const slli MAX_SLLI = 0x7FFFFFFFFFFFFFFF;
const slli MIN_SLLI = 0x8000000000000000;
const ulli MAX_ULLI = 0xFFFFFFFFFFFFFFFF;
const ulli MIN_ULLI = 0x0000000000000000;

#define N_TIMES(i, n)     for (ulli i = 0;     i <  n; ++i)
#define N_TIMES_REV(i, n) for (slli i = n - 1; i >= 0; --i)

#ifdef __DEBUG__
template<typename T>
std::ostream& operator<<(std::ostream &os, const std::list<T> &list)
{
  const std::string delim = ", ";
  typename std::list<T>::const_iterator itr = list.begin();
  while (itr != list.end()) {
    os << *itr;
    ++itr;
    if(itr != list.end()) os << delim;
  }
  return os;
}

template<typename T>
std::ostream& operator<<(std::ostream &os, const std::vector<T> &v)
{
  const std::string delim = ", ";
  for (unsigned n = 0; n < v.size(); ++n) {
    os << v[n];
    if((n + 1) < v.size()) os << delim;
  }
  return os;
}
#endif /* __DEBUG__ */

inline unsigned count(
  const std::vector< std::vector<unsigned> > &sum,
  unsigned left, unsigned right, unsigned top, unsigned bottom 
)
{
  return sum[bottom][right]
       + sum[top][left]
       - sum[bottom][left]
       - sum[top][right];
}

int main(int argc, char **argv)
{
  unsigned N, K;
  std::cin >> N >> K;
  
  unsigned M = 2 * K;
  
  std::vector< std::vector<unsigned> > table(M);
  for (unsigned m = 0; m < M; ++m) {
    table[m] = std::vector<unsigned>(M, 0);
  }
  
  for (unsigned n = 0; n < N; ++n) {
    unsigned x, y;
    char color;
    std::cin >> x >> y >> color;
    
    ++table[(y + (color == 'W' ? K : 0)) % M][x % M];
  }
  
  std::vector< std::vector<unsigned> > sum(M + 1);
  for (unsigned m = 0; m < M + 1; ++m) {
    sum[m] = std::vector<unsigned>(M + 1, 0);
  }
  
  for (unsigned y = 0; y < M; ++y) {
    for (unsigned x = 0; x < M; ++x) {
      sum[y + 1][x + 1] = table[y][x]
                        + sum[y + 1][x] 
                        + sum[y][x + 1] 
                        - sum[y][x];
    }
  }
  
  unsigned max = 0;
  
  for (unsigned py = 0; py < K; ++py) {
    for (unsigned px = 0; px < K; ++px) {
      unsigned S = 0;
      
      /*
      S += count(sum, 0, px, py, py + K);
      S += count(sum, px, px + K, 0, py);
      S += count(sum, px, px + K, py + K, M);
      S += count(sum, px + K, M, py, py + K);
      */
      
      S += count(sum, 0, px, 0, py);
      S += count(sum, 0, px, py + (K - 1), M - 1);
      S += count(sum, px, px + (K - 1), py, py + (K - 1));
      S += count(sum, px + (K - 1), M - 1, 0, py);
      S += count(sum, px + (K - 1), M - 1, py + (K - 1), M - 1);
      
      max = std::max(max, std::max(S, N - S));
    }
  }
  
  std::cout << max << std::endl;
  
  return 0;
}