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: Accepted)

#include <iostream>
#include <iomanip>
#include <vector>
#include <set>
#include <string>
#include <queue>
#include <algorithm>
#include <map>
#include <cmath>
#include <numeric>
#include <list>
#include <stack>
#include <cstdio>
#include <cstdlib>
#include <cstring>

#define rep(i, n) for (int i = 0; i < n; i++)
#define rep1(i, n) for (int i = 1; i <= n; i++)

using namespace std;
typedef vector<int> vi;
typedef pair<int, int> pii;
typedef pair<long long, long long> pll;

const int INF = 1 << 30;
const long long INFL = 1LL << 62;
const int MOD = 1000000007;
const int MAX = 100000;
const int N = 2000;

int b[N][N],w[N][N],lb[N],rb[N],lw[N],rw[N];

int main() {
	int n,k;cin>>n>>k;
	rep(i,2*k) rep(j,2*k) {b[i][j]=0;w[i][j]=0;}
	rep(i,n) {
		int x,y;char c;cin>>x>>y>>c;
		if(c=='B') b[y%(2*k)][x%(2*k)]++;
		else w[y%(2*k)][x%(2*k)]++;
	}
	rep(i,2*k) {
		lb[i]=0;rb[i]=0;lw[i]=0;rw[i]=0;
		rep(j,k) {
			lb[i]+=b[i][j];
			rb[i]+=b[i][j+k];
			lw[i]+=w[i][j];
			rw[i]+=w[i][j+k];
		}
	}
	int ans=0,lb_sum,rb_sum,lw_sum,rw_sum;
	rep(j,2*k) {
		if(j>0) {
			rep(i,2*k) {
				lb[i]+=-b[i][j-1]+b[i][(j-1+k)%(2*k)];
				rb[i]+= b[i][j-1]-b[i][(j-1+k)%(2*k)];
				lw[i]+=-w[i][j-1]+w[i][(j-1+k)%(2*k)];
				rw[i]+= w[i][j-1]-w[i][(j-1+k)%(2*k)];
			}
		}
		lb_sum=0;rb_sum=0;lw_sum=0;rw_sum=0;
		rep(i,k) {
			lb_sum+=lb[i];
			rb_sum+=rb[i+k];
			lw_sum+=lw[i+k];
			rw_sum+=rw[i];
		}
		ans=max(ans,lb_sum+rb_sum+lw_sum+rw_sum);
		rep(i,2*k-1) {
			lb_sum+=-lb[i]+lb[(i+k)%(2*k)];
			rb_sum+= rb[i]-rb[(i+k)%(2*k)];
			lw_sum+= lw[i]-lw[(i+k)%(2*k)];
			rw_sum+=-rw[i]+rw[(i+k)%(2*k)];
			ans=max(ans,lb_sum+rb_sum+lw_sum+rw_sum);
		}
	}
	cout<<ans<<endl;
}