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 Rust (1.15.1) (judge verdict: Accepted)

fn main() {
    let (mut a, mut b): (isize, isize) = readln();
    let mut answer = [['.'; 100]; 100];
    a -= 1;
    b -= 1;
    println!("100 100");
    for y in 0..50 {
        for x in 0..100 {
            if y % 2 == 1 || x % 2 == 1 {
                answer[y][x] = '.';
            } else {
                if b > 0 {
                    answer[y][x] = '#';
                    b -= 1;
                } else {
                    answer[y][x] = '.';
                }
            }
        }
    }
    for y in 50..100 {
        for x in 0..100 {
            if y % 2 == 0 || x % 2 == 1 {
                answer[y][x] = '#';
            } else {
                if a > 0 {
                    answer[y][x] = '.';
                    a -= 1;
                } else {
                    answer[y][x] = '#';
                }
            }
        }
    }
    for y in 0..100 {
        for x in 0..100 {
            print!("{}", answer[y][x]);
        }
        println!("");
    }
}

#[allow(unused_imports)]
use std::cmp::{max, min};
#[allow(unused_imports)]
use std::collections::{HashMap, HashSet, VecDeque};

pub trait FromLn {
    fn fromln(s: &str) -> Self;
}
pub fn readln<T: FromLn>() -> T {
    let mut buf = String::new();
    let _ = ::std::io::stdin().read_line(&mut buf).unwrap();
    T::fromln(buf.trim())
}
pub fn readlns<T: FromLn>(n: usize) -> Vec<T> {
    let mut vs = vec![];
    for _ in 0..n {
        vs.push(readln());
    }
    vs
}
macro_rules! fromln_primitives {
    ($($t:ty),*) => { $(
        impl FromLn for $t {
            fn fromln(s: &str) -> $t {
                s.parse().unwrap()
            }
        }
    )* }
}
fromln_primitives!(
    String,
    bool,
    f32,
    f64,
    isize,
    i8,
    i16,
    i32,
    i64,
    usize,
    u8,
    u16,
    u32,
    u64
);
impl<T> FromLn for Vec<T>
where
    T: FromLn,
{
    fn fromln(s: &str) -> Vec<T> {
        s.split_whitespace().map(T::fromln).collect()
    }
}
impl FromLn for Vec<char> {
    fn fromln(s: &str) -> Vec<char> {
        s.chars().collect()
    }
}
macro_rules! fromln_tuple {
    ($($t:ident),*) => {
        impl<$($t),*> FromLn for ($($t),*) where $($t: FromLn),* {
            fn fromln(s: &str) -> ($($t),*) {
                let mut it = s.split_whitespace();
                let t = ($($t::fromln(it.next().unwrap())),*);
                assert_eq!(it.next(), None);
                t
            }
        }
    }
}
fromln_tuple!(A, B);
fromln_tuple!(A, B, C);
fromln_tuple!(A, B, C, D);
fromln_tuple!(A, B, C, D, E);
fromln_tuple!(A, B, C, D, E, F);
mod comp {
    pub mod math {
        pub mod comb {
            use super::mint::{Mint, MOD};
            pub struct Comb {
                fact: Vec<Mint>,
                factinv: Vec<Mint>,
            }
            impl Comb {}
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    let c = Comb::new(100);
                    assert_eq!(Mint::from(336), c.perm(8, 3));
                    assert_eq!(Mint::from(56), c.comb(8, 3));
                    assert_eq!(Mint::from(10), c.multi_comb(3, 3));
                }
                #[test]
                fn test_fact() {
                    let c = Comb::new(100);
                    let p = 8721234;
                    let mut f = Mint::from(1);
                    for i in 1..(p + 1) {
                        f *= i;
                    }
                    assert_eq!(f, c.fact(p));
                }
            }
        }
        pub mod mint {
            use std;
            pub const MOD: u32 = 1000000007;
            #[derive(Debug, Copy, Clone, Eq)]
            pub struct Mint {
                val: u32,
            }
            impl<T: Into<Mint>> std::ops::Add<T> for Mint {
                type Output = Mint;
                fn add(self, other: T) -> Mint {
                    let nval = self.val + other.into().val;
                    Mint {
                        val: if nval >= MOD { nval - MOD } else { nval },
                    }
                }
            }
            impl<T: Into<Mint>> std::ops::Sub<T> for Mint {
                type Output = Mint;
                fn sub(self, other: T) -> Mint {
                    let nval = self.val + MOD - other.into().val;
                    Mint {
                        val: if nval >= MOD { nval - MOD } else { nval },
                    }
                }
            }
            impl<T: Into<Mint>> std::ops::Mul<T> for Mint {
                type Output = Mint;
                fn mul(self, other: T) -> Mint {
                    let nval = self.val as u64 * other.into().val as u64;
                    Mint {
                        val: (nval % (MOD as u64)) as u32,
                    }
                }
            }
            impl<T: Into<Mint>> std::ops::Div<T> for Mint {
                type Output = Mint;
                fn div(self, other: T) -> Mint {
                    self * other.into().inv()
                }
            }
            impl Mint {
                pub fn inv(self) -> Mint {
                    let mut a = self.val as i32;
                    let mut b = MOD as i32;
                    let mut u = 1 as i32;
                    let mut v = 0 as i32;
                    while b != 0 {
                        let t = a / b;
                        a -= t * b;
                        std::mem::swap(&mut a, &mut b);
                        u -= t * v;
                        std::mem::swap(&mut u, &mut v);
                    }
                    Mint {
                        val: if u < 0 { u + MOD as i32 } else { u } as u32,
                    }
                }
            }
            impl PartialEq for Mint {
                fn eq(&self, other: &Mint) -> bool {
                    self.val == other.val
                }
            }
            impl std::fmt::Display for Mint {
                fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
                    self.val.fmt(f)
                }
            }
            macro_rules! impl_signed_mint(( $ ( $ t : ty ) * ) => (
                                      $ (
                                      impl From < $ t > for Mint {
                                      # [ inline ] fn from ( x : $ t ) -> Mint
                                      {
                                      let t = ( x as i64 ) % ( MOD as i64 ) ;
                                      if x >= 0 { Mint { val : t as u32 } }
                                      else {
                                      Mint { val : ( MOD as i64 + t ) as u32 }
                                      } } } ) * ));
            macro_rules! impl_unsigned_mint(( $ ( $ t : ty ) * ) => (
                                        $ (
                                        impl From < $ t > for Mint {
                                        # [ inline ] fn from ( x : $ t ) ->
                                        Mint {
                                        let t = x as u64 % MOD as u64 ; Mint {
                                        val : t as u32 } } } ) * ));
            impl_signed_mint! (i8 i16 i32 i64 isize);
            impl_unsigned_mint! (u8 u16 u32 u64 usize);
            impl<T: Into<Mint>> std::ops::AddAssign<T> for Mint {
                fn add_assign(&mut self, other: T) {
                    *self = *self + other.into();
                }
            }
            impl<T: Into<Mint>> std::ops::SubAssign<T> for Mint {
                fn sub_assign(&mut self, other: T) {
                    *self = *self - other.into();
                }
            }
            impl<T: Into<Mint>> std::ops::MulAssign<T> for Mint {
                fn mul_assign(&mut self, other: T) {
                    *self = *self * other.into();
                }
            }
            impl<T: Into<Mint>> std::ops::DivAssign<T> for Mint {
                fn div_assign(&mut self, other: T) {
                    *self = *self / other.into();
                }
            }
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    let a: Mint = Mint::from(3);
                    let b: Mint = Mint::from(1000000000);
                    assert_eq!(Mint::from(3000000000u64 % MOD as u64), a * b);
                }
            }
        }
        pub mod prime {
            #[derive(Copy, Clone, PartialEq, PartialOrd, Ord, Eq, Debug)]
            struct U128 {
                hi: u64,
                lo: u64,
            }
        }
    }
    pub mod data_structure {
        pub mod bit {
            pub struct BIT {
                tree: Vec<i64>,
            }
            pub struct BITRange {
                bit0: BIT,
                bit1: BIT,
            }
            impl BIT {}
            impl BITRange {}
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    let mut bit = BIT::new(10);
                    bit.add(2, 1);
                    bit.add(3, 3);
                    bit.add(5, 10);
                    bit.add(0, -4);
                    bit.add(9, -5);
                    assert_eq!(5, bit.get(0, 10));
                    assert_eq!(4, bit.get(2, 4));
                    assert_eq!(1, bit.get(2, 3));
                    assert_eq!(0, bit.get(2, 2));
                    assert_eq!(0, bit.get(2, 0));
                    assert_eq!(-4, bit.get(0, 1));
                    assert_eq!(5, bit.get(5, 10));
                    let mut bit = BITRange::new(10);
                    bit.add(1, 3, 1);
                    bit.add(2, 5, 2);
                    assert_eq!(1, bit.get(1, 2));
                    assert_eq!(4, bit.get(1, 3));
                    assert_eq!(6, bit.get(1, 4));
                    assert_eq!(6, bit.get(0, 4));
                }
            }
        }
        pub mod union_find {
            pub struct UnionFind {
                data: Vec<i32>,
            }
            impl UnionFind {}
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    let mut uf = UnionFind::new(3);
                    assert_eq!(1, uf.size(0));
                    assert_eq!(1, uf.size(1));
                    assert_eq!(1, uf.size(2));
                    assert_eq!(0, uf.root(0));
                    assert_eq!(1, uf.root(1));
                    assert_eq!(2, uf.root(2));
                    assert!(!uf.same(1, 2));
                    uf.unite(1, 2);
                    assert_ne!(uf.root(0), uf.root(2));
                    assert_eq!(uf.root(1), uf.root(2));
                    assert_eq!(1, uf.size(0));
                    assert_eq!(2, uf.size(1));
                    assert_eq!(2, uf.size(2));
                    assert!(uf.same(1, 2));
                    uf.unite(0, 1);
                    assert_eq!(uf.root(0), uf.root(2));
                    assert_eq!(uf.root(1), uf.root(2));
                    assert_eq!(3, uf.size(0));
                    assert_eq!(3, uf.size(1));
                    assert_eq!(3, uf.size(2));
                    assert!(uf.same(1, 2));
                    assert!(uf.same(0, 2));
                }
            }
        }
    }
    pub mod util {
        pub mod permutation {
            pub trait Permutation {
                fn next_permutation(&mut self) -> bool;
                fn prev_permutation(&mut self) -> bool;
            }
            impl<T> Permutation for [T]
            where
                T: Ord,
            {
                fn next_permutation(&mut self) -> bool {
                    if self.len() <= 1 {
                        return false;
                    }
                    let mut i = self.len() - 1;
                    while i > 0 && self[i - 1] >= self[i] {
                        i -= 1;
                    }
                    if i == 0 {
                        return false;
                    }
                    let mut j = self.len() - 1;
                    while self[j] <= self[i - 1] {
                        j -= 1;
                    }
                    self.swap(j, i - 1);
                    self[i..].reverse();
                    true
                }
                fn prev_permutation(&mut self) -> bool {
                    if self.len() <= 1 {
                        return false;
                    }
                    let mut i = self.len() - 1;
                    while i > 0 && self[i - 1] <= self[i] {
                        i -= 1;
                    }
                    if i == 0 {
                        return false;
                    }
                    self[i..].reverse();
                    let mut j = self.len() - 1;
                    while self[j - 1] < self[i - 1] {
                        j -= 1;
                    }
                    self.swap(i - 1, j);
                    true
                }
            }
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    let mut data = [0, 0, 1, 1, 2];
                    data.next_permutation();
                    assert_eq!(&data, &[0, 0, 1, 2, 1]);
                    data.next_permutation();
                    assert_eq!(&data, &[0, 0, 2, 1, 1]);
                    data.next_permutation();
                    assert_eq!(&data, &[0, 1, 0, 1, 2]);
                    data.next_permutation();
                    assert_eq!(&data, &[0, 1, 0, 2, 1]);
                    data.next_permutation();
                    assert_eq!(&data, &[0, 1, 1, 0, 2]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 1, 0, 2, 1]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 1, 0, 1, 2]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 0, 2, 1, 1]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 0, 1, 2, 1]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 0, 1, 1, 2]);
                    assert!(!data.prev_permutation());
                }
            }
        }
        pub mod rand {
            use std::{marker, mem};
            use std::io::{self, Read};
            use std::fs::File;
            use std::char;
            #[derive(Debug, Clone, Copy)]
            pub struct Xoroshiro128Rng {
                state: [u64; 2],
            }
            impl Xoroshiro128Rng {}
            impl Rng for Xoroshiro128Rng {
                fn next_u32(&mut self) -> u32 {
                    self.next_u64() as u32
                }
                fn next_u64(&mut self) -> u64 {
                    let s0: u64 = self.state[0];
                    let mut s1: u64 = self.state[1];
                    let result: u64 = s0.wrapping_add(s1);
                    s1 ^= s0;
                    self.state[0] = s0.rotate_left(55) ^ s1 ^ (s1 << 14);
                    self.state[1] = s1.rotate_left(36);
                    result
                }
            }
            pub trait Rand: Sized {
                fn rand<R: Rng>(rng: &mut R) -> Self;
            }
            pub trait Rng {
                fn next_u32(&mut self) -> u32;
                fn next_u64(&mut self) -> u64 {
                    let x = self.next_u32() as u64;
                    let y = self.next_u32() as u64;
                    (y << 32) | x
                }
                fn next_f32(&mut self) -> f32 {
                    const UPPER_MASK: u32 = 1065353216;
                    const LOWER_MASK: u32 = 8388607;
                    let tmp = UPPER_MASK | (self.next_u32() & LOWER_MASK);
                    let result: f32 = unsafe { mem::transmute(tmp) };
                    result - 1.0
                }
                fn next_f64(&mut self) -> f64 {
                    const UPPER_MASK: u64 = 4607182418800017408;
                    const LOWER_MASK: u64 = 4503599627370495;
                    let tmp = UPPER_MASK | (self.next_u64() & LOWER_MASK);
                    let result: f64 = unsafe { mem::transmute(tmp) };
                    result - 1.0
                }
                #[inline(always)]
                fn gen<T: Rand>(&mut self) -> T
                where
                    Self: Sized,
                {
                    Rand::rand(self)
                }
                fn gen_iter<'a, T: Rand>(&'a mut self) -> Generator<'a, T, Self>
                where
                    Self: Sized,
                {
                    Generator {
                        rng: self,
                        _marker: marker::PhantomData,
                    }
                }
                fn shuffle<T>(&mut self, values: &mut [T])
                where
                    Self: Sized,
                {
                    let mut i = values.len();
                    while i >= 2 {
                        i -= 1;
                        values.swap(i, (self.next_u64() as usize) % i);
                    }
                }
            }
            #[derive(Debug)]
            pub struct Generator<'a, T, R: 'a> {
                rng: &'a mut R,
                _marker: marker::PhantomData<fn() -> T>,
            }
            impl<'a, T: Rand, R: Rng> Iterator for Generator<'a, T, R> {
                type Item = T;
                fn next(&mut self) -> Option<T> {
                    Some(self.rng.gen())
                }
            }
            pub struct OsRng {
                reader: File,
            }
            impl OsRng {}
            impl Rng for OsRng {
                fn next_u32(&mut self) -> u32 {
                    let mut dest = [0u8; 4];
                    self.reader.read_exact(&mut dest).unwrap();
                    unsafe { mem::transmute::<[u8; 4], u32>(dest) }
                }
            }
            impl Rand for isize {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> isize {
                    if mem::size_of::<isize>() == 4 {
                        rng.gen::<i32>() as isize
                    } else {
                        rng.gen::<i64>() as isize
                    }
                }
            }
            impl Rand for i8 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> i8 {
                    rng.next_u32() as i8
                }
            }
            impl Rand for i16 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> i16 {
                    rng.next_u32() as i16
                }
            }
            impl Rand for i32 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> i32 {
                    rng.next_u32() as i32
                }
            }
            impl Rand for i64 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> i64 {
                    rng.next_u64() as i64
                }
            }
            #[cfg(feature = "i128_support")]
            impl Rand for i128 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> i128 {
                    rng.gen::<u128>() as i128
                }
            }
            impl Rand for usize {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> usize {
                    if mem::size_of::<usize>() == 4 {
                        rng.gen::<u32>() as usize
                    } else {
                        rng.gen::<u64>() as usize
                    }
                }
            }
            impl Rand for u8 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> u8 {
                    rng.next_u32() as u8
                }
            }
            impl Rand for u16 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> u16 {
                    rng.next_u32() as u16
                }
            }
            impl Rand for u32 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> u32 {
                    rng.next_u32()
                }
            }
            impl Rand for u64 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> u64 {
                    rng.next_u64()
                }
            }
            #[cfg(feature = "i128_support")]
            impl Rand for u128 {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> u128 {
                    ((rng.next_u64() as u128) << 64) | (rng.next_u64() as u128)
                }
            }
            impl Rand for f64 {
                fn rand<R: Rng>(rng: &mut R) -> f64 {
                    rng.next_f64()
                }
            }
            impl Rand for f32 {
                fn rand<R: Rng>(rng: &mut R) -> f32 {
                    rng.next_f32()
                }
            }
            impl Rand for char {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> char {
                    const CHAR_MASK: u32 = 2097151;
                    loop {
                        match char::from_u32(rng.next_u32() & CHAR_MASK) {
                            Some(c) => return c,
                            None => {}
                        }
                    }
                }
            }
            impl Rand for bool {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> bool {
                    rng.gen::<u8>() & 1 == 1
                }
            }
            macro_rules! tuple_impl(( $ ( $ tyvar : ident ) , * ) => {
                                impl < $ ( $ tyvar : Rand ) , * > Rand for (
                                $ ( $ tyvar ) , * , ) {
                                # [ inline ] fn rand < R : Rng > (
                                _rng : & mut R ) -> ( $ ( $ tyvar ) , * , ) {
                                ( $ ( _rng . gen :: < $ tyvar > (  ) ) , * , )
                                } } });
            impl Rand for () {
                #[inline]
                fn rand<R: Rng>(_: &mut R) -> () {
                    ()
                }
            }
            tuple_impl!(A);
            tuple_impl!(A, B);
            tuple_impl!(A, B, C);
            tuple_impl!(A, B, C, D);
            tuple_impl!(A, B, C, D, E);
            tuple_impl!(A, B, C, D, E, F);
            tuple_impl!(A, B, C, D, E, F, G);
            tuple_impl!(A, B, C, D, E, F, G, H);
            tuple_impl!(A, B, C, D, E, F, G, H, I);
            tuple_impl!(A, B, C, D, E, F, G, H, I, J);
            tuple_impl!(A, B, C, D, E, F, G, H, I, J, K);
            tuple_impl!(A, B, C, D, E, F, G, H, I, J, K, L);
            macro_rules! array_impl({
                                $ n : expr , $ t : ident , $ ( $ ts : ident ,
                                ) * } => {
                                array_impl ! { ( $ n - 1 ) , $ ( $ ts , ) * }
                                impl < T > Rand for [ T ; $ n ] where T : Rand
                                {
                                # [ inline ] fn rand < R : Rng > (
                                _rng : & mut R ) -> [ T ; $ n ] {
                                [
                                _rng . gen :: < $ t > (  ) , $ (
                                _rng . gen :: < $ ts > (  ) ) , * ] } } } ; {
                                $ n : expr , } => {
                                impl < T > Rand for [ T ; $ n ] {
                                fn rand < R : Rng > ( _rng : & mut R ) -> [
                                T ; $ n ] { [  ] } } } ;);
            array_impl!(
                32,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
                T,
            );
            impl<T: Rand> Rand for Option<T> {
                #[inline]
                fn rand<R: Rng>(rng: &mut R) -> Option<T> {
                    if rng.gen() {
                        Some(rng.gen())
                    } else {
                        None
                    }
                }
            }
        }
    }
}