0-1 String Problem

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...10101010000111010101...

Q:Q: How many numbers (1's and 0's) would I have to write (at least) to guarantee a repetition of any nn-string number. Ex: Let n=2n=2, generate a random sequence of 1's and 0's: 100110100110. Notice that the first 2 digits are "1010", so is the 5th and 6th "1010" a repetition!

For n=2n=2, I have proved a string of length >5>5 must have at least one repetition. For n=2n=2 we have answer 55. Similarly, for n=3n=3, we found the answer to be 1010, the string length cannot exceed 1010 without repeating a 33-string number. I couldn't find a number for n=4n=4 but I have shown that for any nn the string length does not exceed 2n+n1 2^n+n-1 but I suspect 2n+n1 2^n+n-1 might be the general formula (If you substitute n=2n=2 and n=3n=3 you will find the results match), but I haven't been able to prove this for all nn.

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

Note by Apratim Ghosh
3 years, 4 months ago

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For n=2, surely this is 6?

Stephen Mellor - 3 years, 4 months ago

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

Richard Xu - 3 years, 4 months ago

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