String of 0 and 1's question There are sixteen different ways of writing four-digit strings using 1s and 0s. Three
of these strings are 1010, 0100 and 1001. These three can be found as substrings
of 101001. There is a string of nineteen 1s and 0s which contains all sixteen strings
of length 4 exactly once. If this string starts with 1111, the last four digits are?
Interesting Question from the australian mathematics compeititon, not sure of an efficient approach.
 A: Consider the term after the initial string 1111. The only possibilities are 1111, 1110, or nothing. But since 1111 has been used, and this is the start of the sequence, the next term must be 1110.
Consider the term after 0111. The only possibilities are 1111, 1110, or nothing. But since 1111 and 1110 have been used, thus there is no next term. This means that, if a sequence exists, then the sequence must end with 0111.

Notes:

*

*For completeness, we should show that at least one such string exists, which it does.

*I don't consider the above a full complete solution. I'm guessing that the referenced AMC is MCQ, so this is sufficient to select the answer.

*Of course, there is no preventing the possibility that another string abcd can have no term after it, hence note 1.

A: This is similar to a system of probability that I have developed for numerical systems and their vlaues based around the computational logic of a numerical triangle.
If you take the string 1111 below.
                                   0
                                  0 0
                                 0 0 0
                                1 1 1 1

You see I have turned it into a computational system based around a numerical triangle based on the value that divides the two values below it.
Now, everytime you generate a new sequence of this four value string where each quantity has a value of 0 or 1, the numerical sequence of the triangle above it changes.
You can discover the formulae for calculations based on the sequence above for what the string sequence is.
However I am yet to solve the problem of what the formuale for the calculations are, as such a concept of probability can be used to calculate the value and location of a string with infinite quanity and infinite value and therefore an infinite triangle.
I hope you like my idea.
