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 <string>
       
#include <iostream>
       
 
       
// return the digit at position "pos"
       
// first digit after then decimal dot has pos = 1 (not zero !)

       
unsigned int getDigit(unsigned long long pos)
       
{
       
  // assume pos has one digit
       
  unsigned int       digits = 1;
       
  // then there are 9 other numbers
       
  unsigned long long range  = 9;
       
  // the smallest of them is 1
       
  unsigned long long first  = 1;
       
 
       
  // there are    9 numbers with 1 digit
       
  // there are   90 numbers with 2 digits
       
  // there are  900 numbers with 3 digits
       
  // there are 9000 numbers with 4 digits
       
  // ...
       
  // let's figure out the number of digits
       
 
       
  // skip numbers with too few digits
       
  unsigned long long skip = 0;
       
  while (skip + digits*range < pos)
       
  {
       
    skip += digits*range;
       
    // digits = 2 => range = 90 and
       
    // digits = 3 => range = 900
       
    // digits = 4 => range = 9000, etc.
       
    digits++;
       
    range *= 10;
       
    first *= 10;
       
  }
       
 
       
  // now that we know the number of digits
       
  // adjust "first" and "skip" such that the left-most/highest digit of pos and skip are identical
       
  // then continue with the next digit
       
  while (range > 9)
       
  {
       
    // could be replace by some modular arithmetic, but I'm too lazy for tough thinking ;-)
       
    while (skip + digits*range < pos)
       
    {
       
      skip  += digits*range;
       
      first += range;
       
    }
       
 
       
    // next lower digit
       
    range /= 10;
       
  }
       
 
       
  // right-most digit (basically same inner loop as above when range == 1)
       
  while (skip + digits < pos)
       
  {
       
    first++;
       
    skip += digits;
       
  }
       
 
       
  // skip all "skippable" digits
       
  pos -= skip;
       
  // strings are zero-based whereas input is one-based
       
  pos--;
       
 
       
  // create a string version of our number
       
  auto s = std::to_string(first);
       
  // extract digit and convert from ASCII to an integer
       
  return s[pos] - '0';
       
}
       
 
       
int main()
       
{
       
  unsigned int tests;
       
  std::cin >> tests;
       
  while (tests--)
       
  {
       
    unsigned int product = 1;
       
    // read 7 positions
       
    for (unsigned int i = 0; i < 7; i++)
       
    {
       
      unsigned long long pos;
       
      std::cin >> pos;
       
 
       
      // multiply all digits
       
      product *= getDigit(pos);
       
    }
       
 
       
    // print result
       
    std::cout << product << std::endl;
       
  }
       
 
       
  return 0;
       
}

This solution contains 12 empty lines, 28 comments and 2 preprocessor commands.