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 C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using static System.Convert;
using static System.Math;
using static Constants;
using static Extensions;
using static MathExtensions;

public class Program
{
    public static void Solve()
    {
        var white = I;
        var black = I;
        var grid = new Grid2D<char>(100, 100);
        foreach (var p in Point2D.InnerPoints(grid.Size))
            grid[p] = '.';

        var row = true;
        var h = 3;
        var w = 3;
        var whiteCnt = 1;

        while (whiteCnt < white)
        {
            if (row)
            {
                var j = grid.Width - 1;
                while (whiteCnt < white && j >= 0)
                {
                    while (j >= 0 && grid[h, j] == '.')
                    {
                        grid[h, j] = '#';
                        j--;
                    }
                    j--;
                    whiteCnt++;
                }
                h += 3;
            }
            else
            {
                var i = grid.Height - 1;
                while (whiteCnt < white && i >= 0)
                {
                    while (i >= 0 && grid[i, w] == '.')
                    {
                        grid[i, w] = '#';
                        i--;
                    }
                    i--;
                    whiteCnt++;
                }
                w += 3;
            }
            row = !row;
        }

        h = 3;
        w--;
        var initW = w;
        var blackCnt = white == 1 ? 0 : 1;

        while (blackCnt < black)
        {
            if (w > grid.Width)
            {
                h += 3;
                w = initW;
                continue;
            }

            if (black - blackCnt == 1)
            {
                foreach (var j in Range(w, grid.Width))
                {
                    grid[h - 1, j] = '#';
                    grid[h, j] = '.';
                }
                blackCnt++;
                break;
            }

            if (grid[h, w] == '#')
            {
                grid[h - 1, w] = '#';
                grid[h, w] = '.';
            }
            else
            {
                grid[h - 1, w] = grid[h, w + 1] = '#';
            }
            blackCnt += 2;
            w += 2;
        }

        Console.WriteLine($"{grid.Height} {grid.Width}");

        foreach (var i in Range(grid.Height))
        {
            foreach (var j in Range(grid.Width))
                Console.Write(grid[i, j]);
            Console.WriteLine();
        }
    }

    #region Scanners
    static TextScanner _ts;

    static char C => char.Parse(_ts.Next());
    static string S => _ts.Next();
    static int I => int.Parse(_ts.Next());
    static long L => long.Parse(_ts.Next());
    static BigInteger B => BigInteger.Parse(_ts.Next());
    static double D => double.Parse(_ts.Next());

    static char[] Cs(int count) => Repeat(count, () => C).ToArray();
    static string[] Ss(int count) => Repeat(count, () => S).ToArray();
    static int[] Is(int count) => Repeat(count, () => I).ToArray();
    static long[] Ls(int count) => Repeat(count, () => L).ToArray();
    static BigInteger[] Bs(int count) => Repeat(count, () => B).ToArray();
    static double[] Ds(int count) => Repeat(count, () => D).ToArray();
    #endregion

    [DebuggerStepThrough]
    public static void Main()
    {
        var sw = new StreamWriter(Console.OpenStandardOutput());
        sw.NewLine = "\n";
#if DEBUG
        sw.AutoFlush = true;
#else
        sw.AutoFlush = false;
#endif
        Console.SetOut(sw);
        _ts = new TextScanner(Console.In);
        Solve();
        Console.Out.Flush();
    }
}

public struct Point2D : IEquatable<Point2D>
{
    public Point2D(int y, int x)
    {
        this.Y = y;
        this.X = x;
    }

    public int Y { get; }

    public int X { get; }

    public static readonly Point2D Origin
        = new Point2D(0, 0);

    public static readonly IReadOnlyCollection<Point2D> NeumannNeighborhoodUnits
        = new[] { new Point2D(0, 1), new Point2D(1, 0), new Point2D(0, -1), new Point2D(-1, 0) };

    public static readonly IReadOnlyCollection<Point2D> MooreNeighborhoodUnits
        = new[] { new Point2D(-1, -1), new Point2D(-1, 0), new Point2D(-1, 1), new Point2D(0, -1), new Point2D(0, 1), new Point2D(1, -1), new Point2D(1, 0), new Point2D(1, 1) };

    public static IEnumerable<Point2D> InnerPoints(Point2D max)
        => InnerPoints(Origin, max);

    public static IEnumerable<Point2D> InnerPoints(Point2D min, Point2D max)
    {
        if (min.X > max.X || min.Y > max.Y) throw new ArgumentException();
        for (var i = min.Y; i < max.Y; i++)
            for (var j = min.X; j < max.X; j++)
                yield return new Point2D(i, j);
    }

    public static int ManhattanDistance(Point2D source, Point2D target)
        => System.Math.Abs(source.X - target.X) + System.Math.Abs(source.Y - target.Y);

    public static int ChebyshevDistance(Point2D source, Point2D target)
        => System.Math.Max(System.Math.Abs(source.X - target.X), System.Math.Abs(source.Y - target.Y));

    public static Point2D operator +(Point2D left, Point2D right)
        => new Point2D(left.Y + right.Y, left.X + right.X);

    public static Point2D operator -(Point2D left, Point2D right)
        => new Point2D(left.Y - right.Y, left.X - right.X);

    public static bool operator ==(Point2D left, Point2D right)
        => left.Equals(right);

    public static bool operator !=(Point2D left, Point2D right)
        => !left.Equals(right);

    public bool IsIn(Point2D max)
        => this.IsIn(Origin, max);

    public bool IsIn(Point2D min, Point2D max)
    {
        if (min.X > max.X || min.Y > max.Y) throw new ArgumentException();
        return min.Y <= this.Y && this.Y < max.Y && min.X <= this.X && this.X < max.X;
    }

    public bool Equals(Point2D other)
        => this.Y == other.Y && this.X == other.X;

    public override bool Equals(object obj)
        => obj is Point2D ? this.Equals((Point2D)obj) : false;

    public override int GetHashCode()
        => (this.X << 16) ^ this.Y;

    public override string ToString()
        => $"{this.X} {this.Y}";
}

public partial class Grid2D<T>
{
    private T[] _grid;

    public Grid2D(int height, int width) : this(new Point2D(height, width)) { }

    public Grid2D(Point2D size)
    {
        _grid = new T[size.Y * size.X];
        this.Height = size.Y;
        this.Width = size.X;
        this.Size = size;
    }

    public Grid2D(T[][] grid)
    {
        this.Height = grid.Length;
        this.Width = grid.Max(r => r.Length);
        this.Size = new Point2D(this.Height, this.Width);
        _grid = new T[this.Height * this.Width];
        for (var i = 0; i < grid.Length; i++)
            for (var j = 0; j < grid[i].Length; j++)
                _grid[i * this.Width + j] = grid[i][j];
    }

    public int Height { get; }

    public int Width { get; }

    public Point2D Size { get; }

    public T this[int i, int j]
    {
        get { return _grid[i * this.Width + j]; }
        set { _grid[i * this.Width + j] = value; }
    }

    public T this[Point2D point]
    {
        get { return this[point.Y, point.X]; }
        set { this[point.Y, point.X] = value; }
    }
}

[DebuggerStepThrough]
public class TextScanner
{
    private readonly TextReader _tr;

    public TextScanner(TextReader tr)
    {
        _tr = tr;
    }

    public string Next()
    {
        var sb = new StringBuilder();
        int i;
        do
        {
            i = _tr.Read();
            if (i == -1) throw new EndOfStreamException();
        }
        while (char.IsWhiteSpace((char)i));
        while (i != -1 && !char.IsWhiteSpace((char)i))
        {
            sb.Append((char)i);
            i = _tr.Read();
        }
        return sb.ToString();
    }
}

public static class Constants
{
    public const int AnswerToLifeTheUniverseAndEverything = 42;
    public const string Yes = "Yes";
    public const string No = "No";
    public const string YES = "YES";
    public const string NO = "NO";
    public const string Possible = "Possible";
    public const string Impossible = "Impossible";
    public const string POSSIBLE = "POSSIBLE";
    public const string IMPOSSIBLE = "IMPOSSIBLE";
}

[DebuggerStepThrough]
public static class Extensions
{
    public static void Answer(object value)
    {
        Console.WriteLine(value);
        Exit(0);
    }

    public static void Assert(bool condition)
    {
        if (!condition) throw new Exception("Assertion failed");
    }

    public static string AsString(this IEnumerable<char> source) => new string(source.ToArray());

    public static Dictionary<T, int> Bucket<T>(this IEnumerable<T> source) where T : IEquatable<T>
    {
        var dict = new Dictionary<T, int>();
        foreach (var item in source) if (dict.ContainsKey(item)) dict[item]++; else dict[item] = 1;
        return dict;
    }

    public static int[] Bucket<T>(this IEnumerable<T> source, int maxValue, Func<T, int> selector)
    {
        var arr = new int[maxValue + 1];
        foreach (var item in source) arr[selector(item)]++;
        return arr;
    }

    public static IComparer<T> CreateDescendingComparer<T>()
        where T : IComparable<T>
        => Comparer<T>.Create((x, y) => y.CompareTo(x));

    public static IEnumerable<int> CumSum(this IEnumerable<int> source)
    {
        var sum = 0;
        foreach (var item in source) yield return sum += item;
    }

    public static IEnumerable<long> CumSum(this IEnumerable<long> source)
    {
        var sum = 0L;
        foreach (var item in source) yield return sum += item;
    }

    public static void Exit(int exitCode)
    {
        Console.Out.Flush();
        Environment.Exit(exitCode);
    }

    public static void ForEach<T>(this IEnumerable<T> source, Action<T> action)
    {
        foreach (var item in source) action(item);
    }

    public static void ForEach<T, _>(this IEnumerable<T> source, Func<T, _> func)
    {
        foreach (var item in source) func(item);
    }

    public static void ForEach<T>(this IEnumerable<T> source, Action<T, int> action)
    {
        var i = 0;
        foreach (var item in source) action(item, i++);
    }

    public static void ForEach<T, _>(this IEnumerable<T> source, Func<T, int, _> func)
    {
        var i = 0;
        foreach (var item in source) func(item, i++);
    }

    // [l, r)
    public static bool IsIn<T>(this T value, T l, T r)
        where T : IComparable<T>
    {
        if (l.CompareTo(r) > 0) throw new ArgumentException();
        return l.CompareTo(value) <= 0 && value.CompareTo(r) < 0;
    }

    public static IEnumerable<int> Range(int start, int end, int step = 1)
    {
        for (var i = start; i < end; i += step) yield return i;
    }

    public static IEnumerable<int> Range(int end) => Range(0, end);

    public static void Repeat(int count, Action action)
    {
        for (var i = 0; i < count; i++) action();
    }

    public static void Repeat(int count, Action<int> action)
    {
        for (var i = 0; i < count; i++) action(i);
    }

    public static IEnumerable<T> Repeat<T>(int count, Func<T> func)
    {
        for (var i = 0; i < count; i++) yield return func();
    }

    public static IEnumerable<T> Repeat<T>(int count, Func<int, T> func)
    {
        for (var i = 0; i < count; i++) yield return func(i);
    }

    public static IEnumerable<int> RangeReverse(int start, int end, int step = 1)
    {
        for (var i = end - 1; i >= start; i -= step) yield return i;
    }

    public static IEnumerable<int> RangeReverse(int end) => RangeReverse(0, end);

    public static void Swap<T>(ref T x, ref T y)
    {
        var tmp = x; x = y; y = tmp;
    }
}

[DebuggerStepThrough]
public static class MathExtensions
{
    public static int DivCeil(int left, int right)
        => left / right + (left % right == 0 ? 0 : 1);

    public static long DivCeil(long left, long right)
        => left / right + (left % right == 0L ? 0L : 1L);

    public static int Gcd(int left, int right)
    {
        int r;
        while ((r = left % right) != 0) { left = right; right = r; }
        return right;
    }

    public static long Gcd(long left, long right)
    {
        long r;
        while ((r = left % right) != 0L) { left = right; right = r; }
        return right;
    }

    public static BigInteger Gcd(BigInteger left, BigInteger right)
        => BigInteger.GreatestCommonDivisor(left, right);

    public static int HighestOneBit(int x)
    {
        x |= x >> 01;
        x |= x >> 02;
        x |= x >> 04;
        x |= x >> 08;
        x |= x >> 16;
        return x - (x >> 1);
    }

    public static long HighestOneBit(long x)
    {
        x |= x >> 01;
        x |= x >> 02;
        x |= x >> 04;
        x |= x >> 08;
        x |= x >> 16;
        x |= x >> 32;
        return x - (x >> 1);
    }

    public static int Lcm(int left, int right) => left / Gcd(left, right) * right;

    public static long Lcm(long left, long right) => left / Gcd(left, right) * right;

    public static BigInteger Lcm(BigInteger left, BigInteger right)
        => left / Gcd(left, right) * right;

    public static int Pow(int value, int exponent)
    {
        var r = 1;
        while (exponent > 0)
        {
            if ((exponent & 1) == 1) r *= value;
            value *= value;
            exponent >>= 1;
        }
        return r;
    }

    public static long Pow(long value, int exponent)
    {
        var r = 1L;
        while (exponent > 0)
        {
            if ((exponent & 1) == 1) r *= value;
            value *= value;
            exponent >>= 1;
        }
        return r;
    }

    public static long Fact(int value)
    {
        var r = 1L;
        for (var i = 2; i <= value; i++) r *= i;
        return r;
    }
}