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 Go (1.6) (judge verdict: Accepted)

package main

import (
	"fmt"
)

func sum(g [][]int32) {
	n := len(g)
	for i := 0; i < n; i++ {
		for j := 0; j < n; j++ {
			if i > 0 {
				g[i][j] += g[i-1][j]
			}
			if j > 0 {
				g[i][j] += g[i][j-1]
			}
			if i > 0 && j > 0 {
				g[i][j] -= g[i-1][j-1]
			}
		}
	}
}

func check(g [][]int32, sx, sy int, k int) int32 {
	k2 := k * 2

	k -= 1
	k2 -= 1
	a := g[sx+k2][sy+k2] - g[sx+k2][sy+k] - g[sx+k][sy+k2] + g[sx+k][sy+k]
	b := g[sx+k][sy+k]
	if sx > 0 {
		b -= g[sx-1][sy+k]
	}
	if sy > 0 {
		b -= g[sx+k][sy-1]
	}
	if sx > 0 && sy > 0 {
		b += g[sx-1][sy-1]
	}
	return a + b
}

func main() {
	var N, K int
	fmt.Scanf("%d %d", &N, &K)
	K2 := K * 2

	w := make([][]int32, 4*K)
	for i := 0; i < len(w); i++ {
		w[i] = make([]int32, 4*K)
	}
	b := make([][]int32, 4*K)
	for i := 0; i < len(b); i++ {
		b[i] = make([]int32, 4*K)
	}

	for i := 0; i < N; i++ {
		x, y, s := 0, 0, ""
		fmt.Scanf("%d %d %s", &y, &x, &s)
		c := s[0]

		if c == 'B' {
			b[x%K2][y%K2] += 1
			b[x%K2+K2][y%K2] += 1
			b[x%K2+K2][y%K2+K2] += 1
			b[x%K2][y%K2+K2] += 1
		} else {
			w[x%K2][y%K2] += 1
			w[x%K2+K2][y%K2] += 1
			w[x%K2+K2][y%K2+K2] += 1
			w[x%K2][y%K2+K2] += 1
		}
	}

	//for _, t := range w {
	//	fmt.Println(t)
	//}

	sum(b)
	sum(w)

	//fmt.Println("----")
	//for _, t := range w {
	//	fmt.Println(t)
	//}

	ans := 0
	for i := 0; i < K; i++ {
		for j := 0; j < K; j++ {
			nw := check(w, i, j, K)
			nb := check(b, i+K, j, K)
			if int(nw+nb) > ans {
				ans = int(nw + nb)
			}
		}
	}

	for i := 0; i < K; i++ {
		for j := 0; j < K; j++ {
			nb := check(b, i, j, K)
			nw := check(w, i+K, j, K)
			if int(nw+nb) > ans {
				ans = int(nw + nb)
			}
		}
	}

	fmt.Println(ans)
}