Step by Step solution about How to resolve the algorithm Mind boggling card trick step by step in the Haskell programming language

The provided Haskell code simulates a card game with a shuffled deck of 52 cards, where the objective is to distribute the cards into "red" and "black" piles while ensuring that the number of red cards in the red pile is equal to the number of black cards in the black pile. A step-by-step explanation of the code:

1. Importing Necessary Modules: The code imports the System.Random module to generate random numbers and the Data.List and Data.Monoid modules for utility functions and data structures.

2. main Function: The entry point of the program is the main function, which performs the following tasks:

   • Calls knuthShuffle to randomly shuffle a deck of numbers from 1 to 52, representing the cards in the deck.
   • Converts the shuffled list of numbers into a string of characters, where 'R' represents red cards and 'B' represents black cards, using the rb function.
   • Calls threeStacks to distribute the shuffled cards into three piles: the red pile (rs_), the black pile (bs_), and a discard pile (discards).
   • Generates a random number nSwap between 1 and the minimum length of the red and black piles to determine how many cards should be swapped between the piles.
   • Calls exchange to swap nSwap cards between the red and black piles, resulting in the final rs and bs lists.
   • Checks the validity of the distribution by counting the number of red cards in rs (rrs) and black cards in bs (bbs). It prints out various messages, including the discards, the number of cards swapped, the contents of the red and black piles, and whether the number of red cards in the red pile equals the number of black cards in the black pile.

3. rb Function: This function takes an integer representing a card and maps it to 'R' (red) if the number is even and 'B' (black) if it is odd.

4. threeStacks Function: This function takes a string of characters representing the shuffled deck and distributes them into three piles: rs (red pile), bs (black pile), and ds (discards). It recursively iterates through the input string, adding cards to the red or black pile based on their color and putting any remaining cards in the discards pile.

5. exchange Function: This function takes an integer n and two lists, xs and ys, and swaps the first n elements between the two lists and returns them as the (xs', ys') pair.

6. knuthShuffle Function: This function takes a list and generates a randomly shuffled version of the list using the Knuth shuffle algorithm. It iterates through the input list, zips it with a list of indexes, and applies the swapElems function to each pair to randomly swap elements until the entire list is shuffled.

7. randoms Function: This function generates a list of random integers between 0 and x-1, where x is the length of the input list.

8. swapElems Function: This function takes a pair of indexes (i, j) and a list xs and swaps the elements at indexes i and j in the list.

9. replaceAt Function: This function takes an index i, a value c, and a list l and replaces the element at index i in the list with the value c, returning a new list.

import System.Random (randomRIO)
import Data.List (partition)
import Data.Monoid ((<>))
 
main :: IO [Int]
main = do
 
  -- DEALT
  ns <- knuthShuffle [1 .. 52]
  let (rs_, bs_, discards) = threeStacks (rb <$> ns)
 
  -- SWAPPED
  nSwap <- randomRIO (1, min (length rs_) (length bs_))
  let (rs, bs) = exchange nSwap rs_ bs_
 
  -- CHECKED
  let rrs = filter ('R' ==) rs
  let bbs = filter ('B' ==) bs
  putStrLn $
    unlines
      [ "Discarded: " <> discards
      , "Swapped: " <> show nSwap
      , "Red pile: " <> rs
      , "Black pile: " <> bs
      , rrs <> " = Red cards in the red pile"
      , bbs <> " = Black cards in the black pile"
      , show $ length rrs == length bbs
      ]
  return ns
 
-- RED vs BLACK ----------------------------------------
rb :: Int -> Char
rb n
  | even n = 'R'
  | otherwise = 'B'
 
-- THREE STACKS ----------------------------------------
threeStacks :: String -> (String, String, String)
threeStacks = go ([], [], [])
  where
    go tpl [] = tpl
    go (rs, bs, ds) [x] = (rs, bs, x : ds)
    go (rs, bs, ds) (x:y:xs)
      | 'R' == x = go (y : rs, bs, x : ds) xs
      | otherwise = go (rs, y : bs, x : ds) xs
 
exchange :: Int -> [a] -> [a] -> ([a], [a])
exchange n xs ys =
  let [xs_, ys_] = splitAt n <$> [xs, ys]
  in (fst ys_ <> snd xs_, fst xs_ <> snd ys_)
 
-- SHUFFLE -----------------------------------------------
-- (See Knuth Shuffle task)
knuthShuffle :: [a] -> IO [a]
knuthShuffle xs = (foldr swapElems xs . zip [1 ..]) <$> randoms (length xs)
 
randoms :: Int -> IO [Int]
randoms x = traverse (randomRIO . (,) 0) [1 .. (pred x)]
 
swapElems :: (Int, Int) -> [a] -> [a]
swapElems (i, j) xs
  | i == j = xs
  | otherwise = replaceAt j (xs !! i) $ replaceAt i (xs !! j) xs
 
replaceAt :: Int -> a -> [a] -> [a]
replaceAt i c l =
  let (a, b) = splitAt i l
  in a ++ c : drop 1 b