My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <iostream>
       
 
       
int main()
       
{
       
  // maximum digit
       
  unsigned int maxDigit = 9;
       
  std::cin >> maxDigit;
       
 
       
  // restrict variable "a"
       
  bool even = (maxDigit + 1) % 2 == 0;
       
  auto maxA = even ? (maxDigit - 1) / 2 : maxDigit;
       
 
       
  unsigned int result   = 0;
       
  for (unsigned int a = 0; a <= maxA; a++) // limit a to 0..4 => for every solution found there is a "inverted" version where x = 9 - x and x = a,b,c,d,...,p
       
    for (unsigned int b = 0; b <= maxDigit; b++)
       
      for (unsigned int c = 0; c <= maxDigit; c++)
       
        for (unsigned int d = 0; d <= maxDigit; d++)
       
        {
       
          // all rows, columns, diagonals must share the same sum
       
          auto sum = a + b + c + d;
       
 
       
          for (unsigned int e = b; e <= maxDigit; e++) // !!! start at b instead of zero
       
            for (unsigned int f = 0; f <= maxDigit; f++)
       
              for (unsigned int g = 0; g <= maxDigit; g++)
       
              {
       
                // sum of second row must be identical to first row
       
                auto h = sum - e - f - g;
       
                if (h > maxDigit)
       
                  continue;
       
 
       
                for (unsigned int i = 0; i <= maxDigit; i++)
       
                {
       
                  // sum of first column must be identical to first row
       
                  auto m = sum - a - e - i;
       
                  if (m > maxDigit)
       
                    continue;
       
 
       
                  // sum of anti-diagonal must be identical to first row
       
                  auto j = sum - d - g - m;
       
                  if (j > maxDigit)
       
                    continue;
       
 
       
                  // sum of second column must be identical to first row
       
                  auto n = sum - b - f - j;
       
                  if (n > maxDigit)
       
                    continue;
       
 
       
                  for (unsigned int k = 0; k <= maxDigit; k++)
       
                  {
       
                    // sum of third column must be identical to first row
       
                    auto o = sum - c - g - k;
       
                    if (o > maxDigit)
       
                      continue;
       
 
       
                    // sum of third row must be identical to first row
       
                    auto l = sum - i - j - k;
       
                    if (l > maxDigit)
       
                      continue;
       
 
       
                    // sum of fourth row must be identical to first row
       
                    auto p = sum - m - n - o;
       
                    if (p > maxDigit)
       
                      continue;
       
 
       
                    // check diagonal, too
       
                    if (sum != a + f + k + p)
       
                      continue;
       
 
       
                    // yes, found a solution
       
                    result++;
       
                    // mirror grid along diagonal, too
       
                    if (b < e)
       
                      result++;
       
                  }
       
                }
       
              }
       
        }
       
 
       
  // add "inverted" solutions
       
  if (even)
       
    result *= 2;
       
 
       
  std::cout << result << std::endl;
       
  return 0;
       
}

This solution contains 14 empty lines, 14 comments and 1 preprocessor command.