Problem:
Score : 500 points

Problem Statement

You are given two integers A and B.

Print a grid where each square is painted white or black that satisfies the following conditions, in the format specified in Output section:

Let the size of the grid be h \times w (h vertical, w horizontal). Both h and w are at most 100.

The set of the squares painted white is divided into exactly A connected components.

The set of the squares painted black is divided into exactly B connected components.

It can be proved that there always exist one or more solutions under the conditions specified in Constraints section.
If there are multiple solutions, any of them may be printed.

Notes

Two squares painted white, c_1 and c_2, are called connected when the square c_2 can be reached from the square c_1 passing only white squares by repeatedly moving up, down, left or right to an adjacent square.

A set of squares painted white, S, forms a connected component when the following conditions are met:

Any two squares in S are connected.

No pair of a square painted white that is not included in S and a square included in S is connected.

A connected component of squares painted black is defined similarly.

Constraints

1 \leq A \leq 500

1 \leq B \leq 500

Input

Input is given from Standard Input in the following format:

A B

Output

Output should be in the following format:

In the first line, print integers h and w representing the size of the grid you constructed, with a space in between.

Then, print h more lines. The i-th (1 \leq i \leq h) of these lines should contain a string s_i as follows:

If the square at the i-th row and j-th column (1 \leq j \leq w) in the grid is painted white, the j-th character in s_i should be ..

If the square at the i-th row and j-th column (1 \leq j \leq w) in the grid is painted black, the j-th character in s_i should be #.

Sample Input 1

2 3

Sample Output 1

3 3
##.
..#
#.#

This output corresponds to the grid below:

Sample Input 2

7 8

Sample Output 2

3 5
#.#.#
.#.#.
#.#.#

Sample Input 3

1 1

Sample Output 3

4 2
..
#.
##
##

Sample Input 4

3 14

Sample Output 4

8 18
..................
..................
....##.......####.
....#.#.....#.....
...#...#....#.....
..#.###.#...#.....
.#.......#..#.....
#.........#..####.

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.io.*;
import java.util.*;
import java.util.concurrent.*;

public class Main {
    void solve() {
        int a = in.nextInt();
        int b = in.nextInt();
        char[][] map = new char[100][100];
        char print;
        if (a <= b) {
            fill(map, '.');
            a--;
            print = '#';
            int tmp = a;
            a = b;
            b = tmp;
        } else {
            fill(map, '#');
            b--;
            print = '.';
        }
        
        draw(map, a, print);
        drawLeftover(map, b, print);
        out.print(100 + " ");
        out.print(100 + " ");
        out.println();
        print(map);
    }

    private void print(char[][] map) {
        for (int i = 0; i < map.length; i++) {
            for (int j = 0; j < map[0].length; j++) {
                out.print(map[i][j]);
            }
            out.println();
        }
    }

    private void drawLeftover(char[][] map, int b, char print) {
        int start = 0;
        for (int i = 1; b > 0; i+=2) {
            int firstNotJoined = -1;
            int lastNotJoined = -1;
            for (int j = start; j < map.length && b > 0; j+=2) {
                map[i][j] = print;
                b--;
                if (!joined(map, i, j, print)) {
                    if (firstNotJoined == -1) firstNotJoined = j;
                    lastNotJoined = j;
                }
            }
            join(map, i, firstNotJoined, lastNotJoined, print);
            start = start == 1 ? 0 : 1;
        }
    }

    private void join(char[][] map, int i, int firstNotJoined, int lastNotJoined, char print) {
        if (firstNotJoined == -1) return;
        if (firstNotJoined == map.length - 1) {
            map[i][firstNotJoined - 1] = print;
        } else {
            map[i][firstNotJoined + 1] = print;
        }
        if (firstNotJoined == lastNotJoined) return;
        int iFill = i + 1;
        for (int k = firstNotJoined; k <= lastNotJoined; k++) {
            map[iFill][k] = print;
        }
    }

//    private void join(char[][] map, int i, int j, char print) {
//        if (j == map.length - 1) {
//            map[i][j - 1] = print;
//        } else {
//            map[i][j+1] = print;
//        }
//        int iFill = i + 1;
//        int upto = j % 2 == 0 ? map.length - 1 : map.length;
//        for (int k = j; k < upto; k++) {
//            map[iFill][k] = print;
//        }
//    }

    private boolean joined(char[][] map, int i, int j, char print) {
        return map[i+1][j] == print || (j > 0 && map[i][j-1] == print);
    }

    private void draw(char[][] map, int a, char print) {
        int start = 1;
        for (int i = 0; a > 0; i+=2) {
            for (int j = start; j < map.length && a > 0; j+=2) {
                map[i][j] = print;
                a--;
            }
            start = start == 1 ? 0 : 1;
        }
    }

    private void fill(char[][] map, char c) {
        for (int i = 0; i < map.length; i++) {
            for (int j = 0; j < map[0].length; j++) {
                map[i][j] = c;
            }
        }
    }


    Scanner in;
    PrintWriter out;

    void run() {
        try {
            in = new Scanner(new File("A.in"));
            out = new PrintWriter(new File("A.out"));

            solve();

            out.close();
        } catch (FileNotFoundException e) {
            e.printStackTrace();
        }
    }

    void runIO() {

        in = new Scanner(System.in);
        out = new PrintWriter(System.out);

        solve();

        out.close();
    }

    class Scanner {
        BufferedReader br;
        StringTokenizer st;

        public Scanner(File f) {
            try {
                br = new BufferedReader(new FileReader(f));
            } catch (FileNotFoundException e) {
                e.printStackTrace();
            }
        }

        public Scanner(InputStream f) {
            br = new BufferedReader(new InputStreamReader(f));
        }

        String next() {
            while (st == null || !st.hasMoreTokens()) {
                String s = null;
                try {
                    s = br.readLine();
                } catch (IOException e) {
                    e.printStackTrace();
                }
                if (s == null)
                    return null;
                st = new StringTokenizer(s);
            }
            return st.nextToken();
        }

        private boolean isSpaceChar(int c) {
            return !(c >= 33 && c <= 126);
        }

        private int skip() {
            int b;
            while ((b = read()) != -1 && isSpaceChar(b)) ;
            return b;
        }

        private int read() {
            int res = -1;
            try {
                res = br.read();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return res;
        }

        char[] nextCharArray(int size) {
            char[] buf = new char[size];
            int b = skip(), p = 0;
            while (p < size && !(isSpaceChar(b))) {
                buf[p++] = (char) b;
                b = read();
            }
            return size == p ? buf : Arrays.copyOf(buf, p);
        }

        char[][] nextCharMap(int n, int m) {
            char[][] map = new char[n][];
            for (int i = 0; i < n; i++) {
                map[i] = nextCharArray(m);
            }
            return map;
        }

        boolean hasMoreTokens() {
            while (st == null || !st.hasMoreTokens()) {
                String s = null;
                try {
                    s = br.readLine();
                } catch (IOException e) {
                    e.printStackTrace();
                }
                if (s == null)
                    return false;
                st = new StringTokenizer(s);
            }
            return true;
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        long nextLong() {
            return Long.parseLong(next());
        }

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

    public static void main(String[] args) {
        new Main().runIO();
    }
}