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 Go (1.6) (judge verdict: Accepted)

package main

import (
	"bufio"
	"bytes"
	"fmt"
	"io"
	"os"
	"strconv"
)

// -----------------------------------------------------------------------------

// IO helper functions

// Returns next token from input.  It must be initialized by SetInput()
// before the first call.
var nextToken func() ([]byte, error)

// Holds a buffer for output.  It must be initialized by SetOutput().
// All IO fucntions (read*() and [e]print*()) should write to OutputWriter
// instead of this.
var OutputBuffer *bufio.Writer

// Holds an io.Writer.  It must be initialized by SetOutput()
var OutputWriter io.Writer

// Set IO functions for interactive input/output.
func SetInteractive(w io.Writer, r io.Reader) {
	SetUnbefferedInput(r)
	OutputBuffer = nil
	OutputWriter = w
}

// Setup OutputBuffer and OutputWriter.
func SetOutput(w io.Writer) {
	OutputBuffer = bufio.NewWriter(w)
	OutputWriter = OutputBuffer
}

// Flushes OutputBuffer
func Flush() {
	if OutputBuffer != nil {
		OutputBuffer.Flush()
	}
}

// Returns true if c is a white space
func IsSpace(c byte) bool {
	switch c {
	case '\t', '\n', '\v', '\f', '\r', ' ':
		return true
	}
	return false
}

// Setup nextToken with input buffer.
func SetInput(r io.Reader) {
	buf := new(bytes.Buffer)
	var b []byte

	var i int
	rest := func() ([]byte, error) {
		for i < len(b) && IsSpace(b[i]) {
			i++
		}
		j := i
		for i < len(b) && !IsSpace(b[i]) {
			i++
		}
		if i == j {
			return nil, io.ErrUnexpectedEOF
		}
		return b[j:i], nil
	}
	initial := func() ([]byte, error) {
		io.Copy(buf, r)
		b = buf.Bytes()
		nextToken = rest
		return rest()
	}
	nextToken = initial
}

// Setup nextToken without input buffer.
func SetUnbefferedInput(r io.Reader) {
	buf := bufio.NewReader(r)
	var b []byte

	var i int
	nextToken = func() ([]byte, error) {
		var err error
		if i == len(b) {
			b, err = buf.ReadBytes('\n')
			if err != nil {
				return nil, err
			}
			i = 0
			j := len(b) - 1
			for 0 <= j && IsSpace(b[j]) {
				j--
			}
			b = b[0 : j+1]
		}
		for i < len(b) && IsSpace(b[i]) {
			i++
		}
		j := i
		for i < len(b) && !IsSpace(b[i]) {
			i++
		}
		if i == j {
			return nil, io.ErrUnexpectedEOF
		}
		return b[j:i], nil
	}
}

// -----------------------------------------------------------------------------

// IO functions

// Reads next token and return it as []byte
func readb() []byte {
	b, err := nextToken()
	if err != nil {
		panic(err)
	}
	return b
}

// Reads next token and return it as string
func reads() string {
	return string(readb())
}

// Reads next token and return it as int64
func readll() int64 {
	i, err := strconv.ParseInt(reads(), 10, 64)
	if err != nil {
		panic(err.Error())
	}
	return i
}

// Reads next token and return it as int
func readi() int {
	return int(readll())
}

// Reads next token and return it as float64
func readf() float64 {
	f, err := strconv.ParseFloat(reads(), 64)
	if err != nil {
		panic(err.Error())
	}
	return f
}

// Write args to OutputWriter with the format f
func printf(f string, args ...interface{}) (int, error) {
	return fmt.Fprintf(OutputWriter, f, args...)
}

// Write args to OutputWriter without format
func println(args ...interface{}) (int, error) {
	return fmt.Fprintln(OutputWriter, args...)
}

// Write args to stderr with the format f
func eprintf(f string, args ...interface{}) (int, error) {
	return fmt.Fprintf(os.Stderr, f, args...)
}

// Write args to stderr without format
func eprintln(args ...interface{}) (int, error) {
	return fmt.Fprintln(os.Stderr, args...)
}

// -----------------------------------------------------------------------------

// Simple math functions

func min(a, b int) int {
	if a < b {
		return a
	}
	return b
}

func min64(a, b int64) int64 {
	if a < b {
		return a
	}
	return b
}

func max(a, b int) int {
	if a < b {
		return b
	}
	return a
}

func max64(a, b int64) int64 {
	if a < b {
		return b
	}
	return a
}

func abs(a int) int {
	if a < 0 {
		return -a
	}
	return a
}

func abs64(a int64) int64 {
	if a < 0 {
		return -a
	}
	return a
}

var counter int

func gcd(a, b int) int {
	if b == 0 {
		return a
	}
	counter++
	return gcd(b, a%b)
}

func gcd64(a, b int64) int64 {
	if b == 0 {
		return a
	}
	return gcd64(b, a%b)
}

// set up IO functions
func init() {
	// for non-interactive
	SetInput(os.Stdin)
	SetOutput(os.Stdout)

	// Enable below when interactive.  Its ok to leave above intact.
	// SetInteractive(os.Stdout, os.Stdin)
}

func main() {
	defer Flush()

	A := readi()
	B := readi()

	f := solve(A, B)
	println(len(f), len(f[0]))
	for i := range f {
		println(string(f[i]))
	}
}

func main0() {
	defer Flush()
	for A := 1; A <= 500; A++ {
		for B := 1; B <= 500; B++ {
			f := solve(A, B)
			if !check(f, A, B) {
				println("NG", A, B)
				for i := range f {
					println(string(f[i]))
				}
				return
			}
		}
	}
}

func solve(A, B int) [][]byte {
	H := 50
	W := 100
	f := make([][]byte, H)
	for i := range f {
		f[i] = make([]byte, W)
		for j := range f[i] {
			if j == 0 {
				f[i][j] = '#'
			} else if j == W-1 {
				f[i][j] = '.'
			} else if i%2 == 0 {
				f[i][j] = '.'
			} else {
				f[i][j] = '#'
			}
		}
	}

	i := 0
	j := 2
	for k := 1; k < A; k++ {
		f[i][j] = '#'
		j += 2
		if W-1 <= j {
			j = 2
			i += 2
		}
	}

	i = H - 1
	j = W - 3
	for k := 1; k < B; k++ {
		f[i][j] = '.'
		j -= 2
		if j < 1 {
			j = W - 3
			i -= 2
		}
	}

	return f
}

func check(f [][]byte, A, B int) bool {
	visited := make([][]bool, len(f))
	for i := range visited {
		visited[i] = make([]bool, len(f[i]))
	}

	dx := []int{0, 1, 0, -1}
	dy := []int{1, 0, -1, 0}

	var dfs func(c byte, i, j int)
	dfs = func(c byte, i, j int) {
		visited[i][j] = true
		for k := range dx {
			nx, ny := j+dx[k], i+dy[k]
			if nx < 0 || len(f[0]) <= nx || ny < 0 || len(f) <= ny || visited[ny][nx] || f[ny][nx] != c {
				continue
			}
			dfs(c, ny, nx)
		}
	}

	w, b := 0, 0
	for i := range f {
		for j := range f[i] {
			if visited[i][j] {
				continue
			}
			if f[i][j] == '.' {
				w++
				dfs('.', i, j)
			} else {
				b++
				dfs('#', i, j)
			}
		}
	}
	return w == A && b == B
}