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
9 months, 1 week ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. 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 1

paragraph 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:

Top Newest

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

Richard Xu - 8 months, 2 weeks ago

Log in to reply

For n=2, surely this is 6?

Stephen Mellor - 9 months, 1 week ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...