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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.IOException;
import java.io.InputStream;
import java.util.Arrays;
import java.util.NoSuchElementException;
import java.util.function.IntBinaryOperator;
import java.util.function.IntUnaryOperator;
import java.util.function.LongUnaryOperator;

public class Main {
    public static void main(String[] args) throws Exception {
        FastScanner fsc = new FastScanner();
        int n = fsc.nextInt();
        int k = fsc.nextInt();
        int[][] f = new int[2 * k][2 * k];
        for (int i = 0; i < n; i++) {
            int x = fsc.nextInt();
            int y = fsc.nextInt();
            char c = fsc.next().charAt(0);
            if (c == 'W') {
                f[x % (2 * k)][y % (2 * k)]++;
            } else {
                f[(x + k) % (2 * k)][y % (2 * k)]++;
            }
        }
        int[][] s = ArrayUtil.cumsum(f);
        int max = 0;
        for (int i = 0; i < k; i++) {
            for (int j = 0; j < k; j++) {
                max = IntUtil.max(max, u(s, k, i, j), n - u(s, k, i, j));
            }
        }
        System.out.println(max);
    }

    public static int t(int[][] s, int x1, int y1, int x2, int y2) {
        return s[x2][y2] - s[x1][y2] - s[x2][y1] + s[x1][y1];
    }

    public static int u(int[][] s, int k, int i, int j) {
        return t(s, 0, 0, i, j) + t(s, i, j, i + k, j + k) + t(s, 0, j + k, i, 2 * k) + t(s, i + k, 0, 2 * k, j)
                + t(s, i + k, j + k, 2 * k, 2 * k);
    }
}

class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    private boolean hasNextByte() {
        if (ptr < buflen) {
            return true;
        } else {
            ptr = 0;
            try {
                buflen = in.read(buffer);
            } catch (IOException e) {
                e.printStackTrace();
            }
            if (buflen <= 0) {
                return false;
            }
        }
        return true;
    }

    private int readByte() {
        if (hasNextByte())
            return buffer[ptr++];
        else
            return -1;
    }

    private static boolean isPrintableChar(int c) {
        return 33 <= c && c <= 126;
    }

    public boolean hasNext() {
        while (hasNextByte() && !isPrintableChar(buffer[ptr]))
            ptr++;
        return hasNextByte();
    }

    public String next() {
        if (!hasNext())
            throw new NoSuchElementException();
        StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (isPrintableChar(b)) {
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }

    public long nextLong() {
        if (!hasNext())
            throw new NoSuchElementException();
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {
            minus = true;
            b = readByte();
        }
        if (b < '0' || '9' < b) {
            throw new NumberFormatException();
        }
        while (true) {
            if ('0' <= b && b <= '9') {
                n *= 10;
                n += b - '0';
            } else if (b == -1 || !isPrintableChar(b)) {
                return minus ? -n : n;
            } else {
                throw new NumberFormatException();
            }
            b = readByte();
        }
    }

    public int nextInt() {
        long nl = nextLong();
        if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE)
            throw new NumberFormatException();
        return (int) nl;
    }

    public double nextDouble() {
        return Double.parseDouble(next());
    }
}

class IntUtil {
    public static int min(int... a) {
        int ret = Const.IINF;
        for (int c : a)
            ret = Math.min(ret, c);
        return ret;
    }

    public static int max(int... a) {
        int ret = -Const.IINF;
        for (int c : a)
            ret = Math.max(ret, c);
        return ret;
    }

    /**
     * Caluculate the ceil of a/b. Returns the smallest integer greater than or
     * equal to a/b while 'a/b' rounds fractional parts to ZERO.
     * 
     * @param a
     * @param b
     * @return the smallest integer greater than or equal to a/b
     */
    public static int cld(int a, int b) {
        if (a > 0 && b > 0)
            return (a + b - 1) / b;
        else if (a < 0 && b < 0)
            return (a + b + 1) / b;
        else
            return a / b;
    }

    /**
     * Caluculate the floor of a/b. Returns the largest integer less than or equal
     * to a/b while 'a/b' rounds fractional parts to ZERO.
     * 
     * @param a
     * @param b
     * @return the largest integer less than or equal to a/b
     */
    public static int fld(int a, int b) {
        if (a <= 0 && b > 0)
            return (a - b + 1) / b;
        else if (a > 0 && b <= 0)
            return (a - b - 1) / b;
        else
            return a / b;
    }

    public static int fold(IntBinaryOperator func, int a, int... b) {
        int ret = a;
        for (int c : b)
            ret = func.applyAsInt(ret, c);
        return ret;
    }
}

class Const {
    public static final long MOD7 = 1_000_000_007;
    public static final long MOD9 = 1_000_000_009;
    public static long MOD = MOD7;

    public static final long LINF = 1_000_000_000_000_000_000l;
    public static final int IINF = 1_000_000_000;

    public static void setMod(long mod) {
        MOD = mod;
    }
}

class ArrayUtil {
    public static int[] map(IntUnaryOperator op, int[] a) {
        int[] b = new int[a.length];
        for (int i = 0; i < a.length; i++)
            b[i] = op.applyAsInt(a[i]);
        return b;
    }

    /**
     * s[i] is the sum of the range [0, i) in a.
     * 
     * @param a
     * @return cumulative sum array of a.
     */
    public static int[] cumsum(int[] a) {
        int n = a.length;
        int[] s = new int[n + 1];
        for (int i = 1; i <= n; i++)
            s[i] = s[i - 1] + a[i - 1];
        return s;
    }

    public static int[][] cumsum(int[][] a) {
        int n = a.length;
        int m = a[0].length;
        int[][] s = new int[n + 1][m + 1];
        for (int i = 1; i <= n; i++)
            for (int j = 1; j <= m; j++)
                s[i][j] = s[i - 1][j] + s[i][j - 1] - s[i - 1][j - 1] + a[i - 1][j - 1];
        return s;
    }

    public static void reverse(int[] a, int begin, int end) {
        for (int i = begin; i < begin + (end - begin) / 2; i++)
            swap(a, i, begin + end - i - 1);
    }

    public static void reverse(int[] a) {
        reverse(a, 0, a.length);
    }

    public static void revSort(int[] a) {
        Arrays.sort(a);
        reverse(a);
    }

    public static int[] nextPermutation(int[] a) {
        int[] ret = a.clone();
        int n = ret.length;
        for (int i = n - 1; i >= 1; i--) {
            if (ret[i] > ret[i - 1]) {
                int fail = n, pass = i;
                while (fail - pass > 1) {
                    int mid = pass + (fail - pass) / 2;
                    if (ret[mid] > ret[i - 1])
                        pass = mid;
                    else
                        fail = mid;
                }
                swap(ret, pass, i - 1);
                reverse(ret, i, n);
                return ret;
            }
        }
        return null;
    }

    public static int[] predPermutation(int[] a) {
        int[] ret = a.clone();
        int n = ret.length;
        for (int i = n - 1; i >= 1; i--) {
            if (ret[i] < ret[i - 1]) {
                int fail = n, pass = i;
                while (fail - pass > 1) {
                    int mid = pass + (fail - pass) / 2;
                    if (ret[mid] < ret[i - 1])
                        pass = mid;
                    else
                        fail = mid;
                }
                swap(ret, pass, i - 1);
                reverse(ret, i, n);
                return ret;
            }
        }
        return null;
    }

    public static void swap(int[] a, int u, int v) {
        int tmp = a[u];
        a[u] = a[v];
        a[v] = tmp;
    }

    public static int compare(int[] a, int[] b) {
        for (int i = 0; i < a.length; i++) {
            if (i >= b.length) {
                return -1;
            } else if (a[i] > b[i]) {
                return 1;
            } else if (a[i] < b[i]) {
                return -1;
            }
        }
        if (a.length < b.length) {
            return 1;
        } else {
            return 0;
        }
    }

    public static boolean equals(int[] a, int[] b) {
        return compare(a, b) == 0;
    }

    /*
     * IntArray _________________________ LongArray
     */
    public static long[] map(LongUnaryOperator op, long[] a) {
        long[] b = new long[a.length];
        for (int i = 0; i < a.length; i++)
            b[i] = op.applyAsLong(a[i]);
        return b;
    }

    /**
     * s[i] is the sum of the range [0, i) in a.
     * 
     * @param a
     * @return cumulative sum array of a.
     */
    public static long[] cumsum(long[] a) {
        int n = a.length;
        long[] s = new long[n + 1];
        for (int i = 1; i <= n; i++)
            s[i] = s[i - 1] + a[i - 1];
        return s;
    }

    public static long[][] cumsum(long[][] a) {
        int n = a.length;
        int m = a[0].length;
        long[][] s = new long[n + 1][m + 1];
        for (int i = 1; i <= n; i++)
            for (int j = 1; j <= m; j++)
                s[i][j] = s[i - 1][j] + s[i][j - 1] - s[i - 1][j - 1] + a[i - 1][j - 1];
        return s;
    }

    public static void reverse(long[] a, int begin, int end) {
        for (int i = begin; i < begin + (end - begin) / 2; i++)
            swap(a, i, begin + end - i - 1);
    }

    public static void reverse(long[] a) {
        reverse(a, 0, a.length);
    }

    public static void revSort(long[] a) {
        Arrays.sort(a);
        reverse(a);
    }

    public static void compress(long[] a) {
        int n = a.length;
        long[] sorted = a.clone();
        Arrays.sort(sorted);
        for (int i = 0; i < n; i++) {
            int l = 0, r = n;
            while (r - l > 1) {
                int m = l + (r - l) / 2;
                if (sorted[m] <= a[i])
                    l = m;
                else
                    r = m;
            }
            a[i] = l;
        }
    }

    public static void swap(long[] a, int u, int v) {
        long tmp = a[u];
        a[u] = a[v];
        a[v] = tmp;
    }

    public static int compare(long[] a, long[] b) {
        for (int i = 0; i < a.length; i++) {
            if (i >= b.length) {
                return -1;
            } else if (a[i] > b[i]) {
                return 1;
            } else if (a[i] < b[i]) {
                return -1;
            }
        }
        if (a.length < b.length) {
            return 1;
        } else {
            return 0;
        }
    }

    public static boolean equals(long[] a, long[] b) {
        return compare(a, b) == 0;
    }
}