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 Julia (0.5.0) (judge verdict: Memory Limit Exceeded)

const lines=readlines(open("/dev/fd/0"))
input()=shift!(lines)
int(s::String)=parse(Int,s)
intSub(s::SubString{String})=parse(Int,s)
intLine()=map(intSub,split(input()))
function nextLine(k::Int)
    x,y,c=split(input())
    intSub(x)%2k+1,(intSub(y)+(c=="W")k)%2k+1
end
function main()
    n,k=intLine()
    g=Array{Int,1}[[0for j in 1:2k]for i in 1:2k]
    for i in 1:n
        x,y=nextLine(k)
        g[x][y]+=1
    end
    g=cumsum(map(cumsum,g::Array{Array{Int,1},1})::Array{Array{Int,1},1})
    m=0
    for i in 1:2k
        for j in 1:2k
            a=0
            for x in -2:2
                for y in -2:2
                    if (x+y)%2!=0;continue;end
                    v,w=i+x*k,j+y*k
                    if v>0<w;a+=g[min(2k,v)][min(2k,w)];end
                    if v>k&&w>0;a-=g[min(2k,v-k)][min(2k,w)];end
                    if w>k&&v>0;a-=g[min(2k,v)][min(2k,w-k)];end
                    if v>k<w;a+=g[min(2k,v-k)][min(2k,w-k)];end
                end
            end
            m=max(m,a)
        end
    end
    println(m)
end
main()