# Puzzle-Solving

Contents

## MIT IAP Mystery Hunt 2016 "Huntception" (aka "Muttstery Hunt")

### Identify, SORT, Index, Solve

Link to puzzle: MMH 2016 "Huntception" : Sleeping Beauty identify, SORT, index, solve

"Just how huge are things from Thumbelina's perspective? The answer may be well beyond the powers of Google."

There is a list of 29 clues, each consisting of a photo and a name, formula, expression, or description of a very large number rendered in MathJax. At the bottom is a blank labeled "Lexicographic rank of this permutation:".

In the first phase, solvers identify the photos. A combination of brain-storming and Google Image Search quickly reveals that all represent a word or name that sounds like a pun on large number names like million, trillion, and "zillion" (Chuquet names) such as Trillian, Pavillion, Napoleon, etc.

The next stage is to "sort". In this puzzle the sorting phase is both the most difficult and the most unique aspect of the puzzle. Most of the numbers listed are fairly difficult to sort, because they are so large that it takes a lot of effort (or past experience) to even understand the notation and the properties of the mathematical functions used. But they are all finite positive integers, and sufficiently well-defined that a definitive ordering is possible:

2↑^{10100}2

Number of particles in the known Universe

43^{454749413}

6

3→2→3→3

etc.

To figure out the proper ordering of these numbers, the puzzler should consult:

- the Googology wiki
- My own Large Numbers page (and a few entries on the Numbers page as well)
- My Hypercalc for the bigger power towers, and a high-precision numeric calculator for two of the smaller ones

The ordering I ended up with is shown at the beginning of the Perl program that I posted on Pastebin.

Once the "sorted" ordering is known, the puzzler is asked to
identify the "lexicographic rank" of the (unsorted) permutation of the
clues as originally presented in the puzzle. Lexicographic rank is
described
here
reasonably well. For a list of 29 items, there are 29! =
8841761993739701954543616000000 ~ 8.84×10^{30} permutations, a
31-digit number. The lexicographic rank will be anything from 1 (from
properly sorted) to 29! (if reverse-sorted largest-to-smallest).

As it so happens (and this is fairly easy to determine) the first
item given is in fact the smallest number: 2↑↑↑↑...↑2 is 4,
no matter how many ↑'s there are. The second item given ("Number of
particles in the known Universe") is the 3^{rd} smallest. One can see
fairly quickly that the lexicographic rank will be a 29-digit number
starting with 13895... The precise value will depend on getting the
sorting right in every case.

Since there are 29 digits and 29 clues, it seems likely that the 29
digits can be used to "index" or extract letters from the clue words
(Trillian, Pavillion, Napoleon, etc.). There might be some additional
twist, like taking the next letter each time (a rotating Caesar
cypher), indexing letters from the clue words in their original
pre-sorted order, etc. I tried the normal things, like just taking the
1^{st} letter of "Trillian", the 3^{rd} letter of "Aphelion", the
8^{th} letter of "Reptilian", etc. but found nothing. I had made two
errors: computing A(1,5) instead of A(5,1), and having my list sorted
in the wrong direction (largest first).

One generally looks at the flavour text, which is the little quote in italics at the top of this article, for hints. In this case everything seems to have an explanation: "Thumbelina" ties in with the fairy tale theme already seen in the Hunt artwork so far, and provides a smallness perspective from which "huge things" can be appreciated. "Beyond the powers of Google" has a double meaning: it is a pun on "(exponential) powers of googol", and a reference to the common puzzler's use of Google search queries as a powerful tool for solving e.g. crossword clues. There seems to be no additional clue there.

## For devjoe

From: mrob27 2016-01-23 04:08 pm (UTC)

Thank you for clarifying, I'll remember Fedora when I set up a new multi-boot system. I only checked my own package managers, then saw this site claiming "for historical interest only [...] [last updated] 2007-07-16" and decided I'd just add an extra stage to the puzzle, which turned out to work just fine. I didn't have to hack OUTPUT, but gfortran with -std=legacy won't pass an integer to IBENJ, so I changed e.g. "IBENJ(7)" to "IBENJ(7.0)" and made line 00000234 say "IBENJ(REAL(PUL(4)))".

The important insight is that I got myself worked up beyond what was needed, thinking I could comment out some lines to make another working program that would provide more needed clue(s) (having worked on the beautiful Something For Every{one|body}), and that my team is more interested in having fun than getting points, which is perfectly okay. I kind of like that g77 isn't really needed, and it appears I could have manually translated it into another language if I really wanted to.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2016 Feb 13. s.27