Step by Step solution about How to resolve the algorithm 4-rings or 4-squares puzzle step by step in the Ruby programming language

The provided Ruby code is an implementation of the famous Four Squares puzzle, which asks for ways to express a given number as a sum of four perfect squares. It primarily consists of nested f and four_squares functions that efficiently find and present the unique or non-unique solutions.

1. f Function:

   • This is a lambda function that takes seven integer arguments: a, b, c, d, e, f, and g.
   • It checks if the sum of the first, second, third, and fourth arguments ([a+b, b+c+d, d+e+f, f+g]) contains only one unique value.
   • It returns true if the size of the uniq array is 1, and false otherwise. This condition ensures that f only evaluates to true when the sums are all identical.

2. four_squares Function:

   • It is the primary function for finding the solutions to the puzzle.
   • It takes three arguments: low, high, unique, and show (optional).
   • low and high represent the lower and upper bounds for the search, while unique determines whether to find unique or non-unique solutions.
   • show controls whether to print the actual solutions.
   • It initializes uniq with the appropriate string based on the value of the unique argument, either "unique" or "non-unique".
   • It creates a range from low to high with [*low..high].
   • If unique is set to true, it uses permutation(7) to generate all possible permutations of 7 elements from the range, and filters them using the select method with the f function as the condition.
   • If unique is false, it uses repeated_permutation(7) instead, which allows for repeated elements in the permutations.
   • The result is assigned to the solutions variable.
   • If show is true, it prints the headings for the columns ("a" to "g") and iterates over each solutions array, printing its elements as rows.
   • Finally, it prints the total number of solutions found and the type (unique or non-unique) within the specified range.

3. Usage:

   • The code calls four_squares twice with different input ranges: [1,7] and [3,9]. In both cases, it finds and prints the unique solutions.
   • It then calls four_squares a third time with [0,9] and false for unique, which finds and prints all non-unique solutions within that range.

def four_squares(low, high, unique=true, show=unique)
  f = -> (a,b,c,d,e,f,g) {[a+b, b+c+d, d+e+f, f+g].uniq.size == 1}
  if unique
    uniq = "unique"
    solutions = [*low..high].permutation(7).select{|ary| f.call(*ary)}
  else
    uniq = "non-unique"
    solutions = [*low..high].repeated_permutation(7).select{|ary| f.call(*ary)}
  end
  if show
    puts " " + [*"a".."g"].join("  ")
    solutions.each{|ary| p ary}
  end
  puts "#{solutions.size} #{uniq} solutions in #{low} to #{high}"
  puts
end

[[1,7], [3,9]].each do |low, high|
  four_squares(low, high)
end
four_squares(0, 9, false)