Step by Step solution about How to resolve the algorithm Latin Squares in reduced form step by step in the Haskell programming language

The given Haskell code defines functions to generate and print Latin squares of various orders. A Latin square is an n×n square filled with n distinct symbols, each of which occurs exactly once in each row and column of the square.

Here's a breakdown of the code:

1. Importing Modules:

   import Data.List (permutations, (\\))
   import Control.Monad (foldM, forM_)

   This line imports necessary modules from the Haskell standard library.

2. latinSquares Function:

   latinSquares :: Eq a => [a] -> [[[a]]]

   This function generates a list of Latin squares of order n, where n is the number of symbols in the set provided as the input. It returns a list of lists of lists, representing all possible Latin squares of order n.

   • If the input set is empty, it returns an empty list.
   • Otherwise, it generates a list of perm (tail of grouped permutations of the set) and applies the addRow function to each permutation, accumulating the results in squares.

3. addRow Function:

   addRow tbl rows = [ zipWith (:) row tbl
                    | row <- rows                      
                    , and $ different (tail row) (tail tbl) ]

   This function takes a current Latin square tbl, a list of rows rows, and generates new Latin squares by adding each row from rows to tbl. It checks that the resulting square is still a Latin square by ensuring that the tail of the new row (excluding the first element) is different from the tails of all rows in tbl.

4. different Function:

   different = zipWith $ (not .) . elem

   This function checks if two lists are different. It uses the zipWith function to combine the elements of two lists into pairs and then applies the (not .) . elem function to each pair. This function returns True if the second element is not present in the first element (and False otherwise).

5. groupedPermutations Function:

   groupedPermutations :: Eq a => [a] -> [[[a]]]

   This function generates a list of lists of permutations of the input set. Each sublist represents a possible order in which the symbols can appear in a Latin square.

6. printTable Function:

   printTable :: Show a => [[a]] -> IO ()

   This function is used to print a Latin square as a table.

7. task1 and task2: These are two tasks defined in the code. task1 generates and prints Latin squares of order 4. task2 calculates and prints the number of possible Latin squares for different orders.

Overall, this code provides a way to generate and analyze Latin squares of various orders.

import Data.List (permutations, (\\))
import Control.Monad (foldM, forM_)

latinSquares :: Eq a => [a] -> [[[a]]]
latinSquares [] = []
latinSquares set = map reverse <$> squares
  where
    squares = foldM addRow firstRow perm
    perm = tail (groupedPermutations set)
    firstRow = pure <$> set
    addRow tbl rows = [ zipWith (:) row tbl
                      | row <- rows                      
                      , and $ different (tail row) (tail tbl) ]
    different = zipWith $ (not .) . elem
       
groupedPermutations :: Eq a => [a] -> [[[a]]]
groupedPermutations lst = map (\x -> (x :) <$> permutations (lst \\ [x])) lst

printTable :: Show a => [[a]] -> IO () 
printTable tbl = putStrLn $ unlines $ unwords . map show <$> tbl


task1 = do 
  putStrLn "Latin squares of order 4:"
  mapM_ printTable $ latinSquares [1..4]

task2 = do 
  putStrLn "Sizes of latin squares sets for different orders:"
  forM_ [1..6] $ \n -> 
    let size = length $ latinSquares [1..n]
        total = fact n * fact (n-1) * size
        fact i = product [1..i]
    in printf "Order %v: %v*%v!*%v!=%v\n" n size n (n-1) total