I recently had a problem in my mind and am having some trouble proving my solution, have a look.

Imagine I make a string of numbers with only 1's and 0's. ex: $10101010000111010101...$

$Q:$ How many numbers (1's and 0's) would I have to write (at least) to guarantee a repetition of any $n$-string number. Ex: Let $n=2$, generate a random sequence of 1's and 0's: $100110$. Notice that the first 2 digits are "$10$", so is the 5th and 6th "$10$" a repetition!

For $n=2$, I have proved a string of length $>5$ must have at least one repetition. For $n=2$ we have answer $5$. Similarly, for $n=3$, we found the answer to be $10$, the string length cannot exceed $10$ without repeating a $3$-string number. I couldn't find a number for $n=4$ but I have shown that for any $n$ the string length does not exceed $2^n+n-1$ but I suspect $2^n+n-1$ might be the general formula (If you substitute $n=2$ and $n=3$ you will find the results match), but I haven't been able to prove this for all $n$.

PS: I think the solution might be related to graph theory.

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TopNewestFor n=2, surely this is 6?

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This is a well-studied problem. Maybe try OEIS first next time: http://oeis.org/A052944

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