Step by Step solution about How to resolve the algorithm Esthetic numbers step by step in the Haskell programming language

The provided Haskell code implements functions for generating and working with esthetic numbers in various bases. An esthetic number is a number where the absolute difference between consecutive digits is always 1.

1. Esthetic Numbers:

• The program defines the isEsthetic function, which checks if a given number (represented as a list of digits) in a specified base is esthetic. It does this by calculating the differences between consecutive digits and checking if all differences are equal to 1.

2. Monadic Solution (esthetics_m):

• The esthetics_m function uses the monadic approach to generate esthetic numbers. It generates a list of differences for various lengths (using the replicateM function) and combines them with a starting digit to create potential esthetic numbers. However, this approach can be inefficient, especially for larger bases.

3. Iterative Solution (esthetics):

• The esthetics function is a more efficient iterative approach that generates esthetic numbers. It uses a loop (implemented using iterate and step) to incrementally build up esthetic numbers by considering new digits that maintain the esthetic property.

4. Base Conversion:

• The code includes functions for converting numbers between different bases:
  • fromBase b takes a number in base 10 and converts it to the specified base b.
  • toBase b converts a number from base b to base 10.

5. Display Functions:

• showInBase b takes a number in base 10 and converts it to the specified base b, then displays it as a string of digits.

Overall: This code provides various functions for working with esthetic numbers, including generating them in different bases and verifying if a given number is esthetic. The iterative approach (esthetics) is more efficient for generating large numbers of esthetic numbers.

import Data.List (unfoldr, genericIndex)
import Control.Monad (replicateM, foldM, mzero)

-- a predicate for esthetic numbers
isEsthetic b = all ((== 1) . abs) . differences . toBase b
  where
    differences lst = zipWith (-) lst (tail lst)

-- Monadic solution, inefficient for small bases.
esthetics_m b =
  do differences <- (\n -> replicateM n [-1, 1]) <$> [0..]
     firstDigit <- [1..b-1]
     differences >>= fromBase b <$> scanl (+) firstDigit

-- Much more efficient iterative solution (translation from Python).
-- Uses simple list as an ersatz queue.
esthetics b = tail $ fst <$> iterate step (undefined, q)
  where
    q = [(d, d) | d <- [1..b-1]]
    step (_, queue) =
      let (num, lsd) = head queue
          new_lsds = [d | d <- [lsd-1, lsd+1], d < b, d >= 0]
      in (num, tail queue ++ [(num*b + d, d) | d <- new_lsds])

-- representation of numbers as digits
fromBase b = foldM f 0
  where f r d | d < 0 || d >= b = mzero
              | otherwise = pure (r*b + d)

toBase b = reverse . unfoldr f
  where
    f 0 = Nothing
    f n = let (q, r) = divMod n b in Just (r, q)

showInBase b = foldMap (pure . digit) . toBase b
  where digit = genericIndex (['0'..'9'] <> ['a'..'z'])