The self descriptive number

Suppose that the leftmost digit is 9. Then the rightmost digit must be 1. But then there are not enough digits left for 9 0s, a contradiction.

$\underline{9} \ \underbrace{\underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}}}_{\text{Not enough digits for 9 0s}} \ \underline{1}$

Now suppose that the leftmost digit is 8. Then the second to last digit must be 1. Then, the second from the left digit must be at least 1. But then there are not enough digits left for 8 0s, a contradiction.

$\underbrace{\underline{8} \ \underline{1} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{1} \underline{\hphantom{0}}}_{\text{Not enough digits for 8 0s}}$

Now suppose that the leftmost digit is 7. Then the third to last digit must be 1. Now we can resolve 2s and 1s by placing a 2 in the second digit and a 1 in the third digit. But then there are not enough digits left for 7 0s, a contradiction.

$\underbrace{\underline{7} \ \underline{2} \ \underline{1} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}} \ \underline{1} \ \underline{\hphantom{0}} \ \underline{\hphantom{0}}}_{\text{Not enough digits for 7 0s}}$

Now suppose that the leftmost digit is 6. Then the fourth to last digit must be 1. Now we can resolve 2s and 1s by placing a 2 in the second digit and a 1 in the third digit. And this works! There are exactly enough digits left to place the 6 0s.

$\underline{6} \ \underline{2} \ \underline{1} \ \underline{0} \ \underline{0} \ \underline{0} \ \underline{1} \ \underline{0} \ \underline{0} \ \underline{0}$

C# is a great medium for problems like these. Here's my solution:

using System.Collections.Generic; using System.Linq; using System.Text; using System.Threading.Tasks; using System.IO;

namespace SelfDescribingNumbers { class Program { static void Main(string[] args) { StreamReader mystreamreader = new StreamReader(args[0]); List<string> LinesFromFile = new List<string>(); string LinesFromTheFile = null;

        LinesFromTheFile = ReadLinesFromFile(mystreamreader, LinesFromFile, LinesFromTheFile);

        foreach (string myline in LinesFromFile)
        {
            string mytrimline = myline.Trim();
            int[] array_converted_line = new int[mytrimline.Length];
            array_converted_line = ConvertStringToArray(mytrimline);

            if (the_number_is_self_describing(array_converted_line))
            {
                Console.WriteLine('1');
            }
            else
            {
                Console.WriteLine('0');
            }
        }

        Console.ReadLine();
    }

    private static bool the_number_is_self_describing(int[] d)
    {
        int number_of_elements = d.Length;
        bool is_self_describing_number = true;

        if (number_of_elements == 1)
        {
            if (d[0] == 0)
                return false;
            else
                return true;
        }
        else
        {
            for (int i = 0; i < number_of_elements; i++)
            {
                if (card_of_digit_in_array(d, i) != d[i])
                {
                    return false;
                }
            }
        }

        return is_self_describing_number;
    }

    public static int card_of_digit_in_array(int[] thearray, int x)
    {
        int counter = 0;
        foreach (int i in thearray)
        {
            if (i == x)
            {
                counter++;
            }
        }

        return counter;
    }

    private static int[] ConvertStringToArray(string line)
    {
        int[] toreturn = line.Select(n => (int)Char.GetNumericValue(n)).ToArray();
        return toreturn;
    }

    private static string ReadLinesFromFile(StreamReader reader, List<string> LinesFromFile, string linefromfile)
    {
        using (reader)
        {
            linefromfile = reader.ReadLine();

            while (linefromfile != null)
            {
                LinesFromFile.Add(linefromfile);
                linefromfile = reader.ReadLine();
            }
        }
        return linefromfile;
    }
}

}

I think 6210001000 is a unique solution, you cannot have 6++ zeros because of self contradictory, if you reduce the zero then you must add a number, which is not possible because there is none that will counter them.