lagrange_multiples.rb

def lagrange_four_squares(n)
  # Initialize the squares array with zeros
  squares = [0, 0, 0, 0]

  # First check if n is a perfect square
  if Math.sqrt(n) % 1 == 0
    squares[0] = Math.sqrt(n).to_i
    return squares.join('² + ') + '²'
  end

  # Check if n can be expressed as the sum of two squares
  (1..Math.sqrt(n).to_i).each do |i|
    if Math.sqrt(n - i*i) % 1 == 0
      squares[0] = i
      squares[1] = Math.sqrt(n - i*i).to_i
      return squares.join('² + ') + '²'
    end
  end

  # Check if n can be expressed as the sum of three squares
  (1..Math.sqrt(n).to_i).each do |i|
    (1..Math.sqrt(n - i*i).to_i).each do |j|
      if Math.sqrt(n - i*i - j*j) % 1 == 0
        squares[0] = i
        squares[1] = j
        squares[2] = Math.sqrt(n - i*i - j*j).to_i
        return squares.join('² + ') + '²'
      end
    end
  end

  # If n is not expressible as the sum of two or three squares,
  # use the four-square theorem to find the four squares
  (1..Math.sqrt(n).to_i).each do |i|
    (1..Math.sqrt(n - i*i).to_i).each do |j|
      (1..Math.sqrt(n - i*i - j*j).to_i).each do |k|
        if Math.sqrt(n - i*i - j*j - k*k) % 1 == 0
          squares[0] = i
          squares[1] = j
          squares[2] = k
          squares[3] = Math.sqrt(n - i*i - j*j - k*k).to_i
          return squares.join('² + ') + '²'
        end
      end
    end
  end

  # If no combination is found, return an empty string
  ''
end

# Example usage:
number = 123
puts "The number #{number} can be expressed as the sum of four squares: " + lagrange_four_squares(number)