User blog:Emlightened/Competition - Nov 2018

Hi everyone!

We started hosting a monthly competition on the forum, so (after some delays) I've brought it here too.

There are three questions, one relatively easy, one relatively medium, and one relatively hard. Answer whichever you want to - rankings for both individual questions and total scores will be given.

Each question is worth seven points. You get three points for the correct answer, and up to four points for showing your working/how you got the answer.

(1) Find the smallest \(n\) such that \(3\uparrow\uparrow (n+2) < f_3(n)\)

(2) What's the least \(n\) such that \(f_\infty(n)\) is greater than Graham's number? \(f_n\) is defined as:

\(C_n(0) = \{0,1\}\) \(C_n(k+1) = \{a+b, f_c(a) | c<n; a,b,c \in C_n(k)\}\) \(f_n(k) = \max(C_n(k))\)

(3) Let Worm(n) be the number of steps the Beklemishev worm consisting of a single ‘n’ takes to get to the empty string. Find natural numbers a,b,c with 0 < b < googol and 1 < c < googol such that:

\(f_{\omega a+b}(c)<Worm(3)<f_{\omega a+b}(c+1)\)

Clarifications on scoring and submission:

You get three (3) points for a correct answer, and up to four (4) points for an explanation. Points aren't affected by when you submit, and changing/modifying your answers is allowed at any time before the deadline with no penalty.

Points are awarded as:
 * (0) Little or no relevant working.
 * (1) Partial working or partially incorrect working.
 * (2) Working is complete and correct, but may not be detailed.
 * (3) Comprehensive/complete working, but not necessarily as detailed as a proof.
 * (4) Complete proofs.

If you don't want to bother with a proof (I wouldn't blame you), you can still get 6/7 with a good explanation.

Up to one mark is lost for errors in the explanation, unless it's a serious error (like, serious enough that the explanation doesn't work).

Submit answers to me at largenum.competition@gmail.com by 23:59 (UTC) on the 15th November. Credit for (2) goes to Simply Beautiful Art, and credit for (3) goes to Deedlit.

If you want to chat about the problems or to offer suggestions for improvement, feel free to comment below!