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: $$10101010000111010101...$$

$$Q:$$ 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.

Note by Apratim Ghosh
6 months, 2 weeks ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

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

- 5 months, 3 weeks ago

For n=2, surely this is 6?

- 6 months, 2 weeks ago