enigmatic puzzle - PSE Assessment Exam

^{An entry in Fortnightly Topic Challenge #27: Treasure-Hunt...}

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Welcome to the PSE Assessment Exam!

This exam is designed to assess your aptitude, knowledge and experience as a high profile member of the Puzzling Stack Exchange community.
This is a timed test conducted in a closed room with a moderator.
Please wait to be called before entering.

As you nervously walk toward the room you've been directed to, that is the sign that greets you. You've heard about this exam, but never imagined you'd be taking it yourself... and yet, here you are! You glance around but there is nothing remarkable in the hallway, just a few battered old chairs.

After a few minutes, a door down the hall opens and you see someone leaving the opposite direction. Their body language indicates a little frustration, but before you can tell any more or have any chance to guess who they were, they disappear around a far corner. A minute or two later, the closer door opens slightly and you hear your name called. You're up!

You enter the room and glance around. The room is dominated by a large desk in the middle, which you note holds an assortment of items, and a solid if not particularly comfortable looking chair. At the front of the room is a simple desk atop which a timer clock is conspicuously placed; seated behind the desk is someone whose officious looking name badge proclaims to be the testing moderator. Your brief survey of the room concludes with the observation that there is only one thing left worth noticing. On one of the walls is a "scoreboard" of sorts, listing names and, you guess, exam completion times:

      Wildly Devious      14:13
      Ror al'Thand         9:17
      Gandalf              7:39
      Houdini             13:04
      Bestial Hamster     10:27

A moment later the testing moderator leads you to the large desk. You take stock of the items on it:

• A laptop computer



• A "That Was Easy!" button


• A "Hello, My Name Is" sticker


• A black fine-point Sharpie marker


• Three pens, colored black, blue, and red


• A box of 64 Crayola crayons


• Two #2 pencils


• A pencil sharpener


• A pencil eraser


• Four rubber bands


• Two paper clips



• A thumbtack


• A pitcher of water


• A glass


• A packet of Skittles


• A packet of what may be a generic brand of Swedish Fish

The testing moderator instructs you to write your name on the sticker and put it on. When you've done so, the moderator adds your name to the "scoreboard" and then goes over a few simple rules.

1. Once you begin the exam you may not leave your desk.


2. The laptop provides access to content on Google, Wikipedia, YouTube, IMDB, Reference.com, Merriam-Webster, OEIS, SparkNotes, imgur.com, and Puzzling Stack Exchange only; any other site will be blocked.


3. The entire setup here is standardized for all assessment exams, and any specific exam may use all, some, or none of both the items on the desk and the websites accessible from the laptop. You may freely use none, any, or all of any of the items or sites available to you to complete the exam.


4. The testing moderator may assist with technical issues, but cannot assist with the exam itself.


5. When you have completed the exam, press the "That Was Easy!" button. This will signal the testing moderator to stop the timer and collect your exam paper.

The testing moderator next indicates the exam will begin in a few minutes, so you may take that time to familiarize yourself with the items provided. You spend a minute looking them over, and another making sure there's nothing unusual about the laptop (there isn't), before your mind wanders. Your eye catches the Skittles packet and, inexplicably, this bizarre vision leaps unbidden to mind.

You are stirred from your brief reverie by the approach of the testing moderator, who then places a single sheet of paper face-down on the desk in front of you, takes a step back, and with a click of the timer ¹ says, "You may begin."

You flip over the paper and begin.

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^{©2017 PSE Assessments, Inc.                               Test# A014612   Supplement# 1xA001358} $$\begin{array}{lcl} \bf{A}\small\textbf{CROSS}&& \bf{D}\small\textbf{OWN}\\ \begin{array}{rl} 2.&\text{Dismembered}\\ 6.&\text{Holy elements}\\ 8.&\text{New, ubiquitous}\\ 12.&\text{Both lock and key}\\ 15.&\text{Language genie}\\ 17.&\text{42?}\\ 19.&\text{Not odd, not even}\\ 20.&\text{Keys; no locks}\\ 21.&\text{This might kill you}\\ \end{array}&\ \ \ \ \ & \begin{array}{rl} 1.&\text{Owned by millions}\\ 4.&\text{If I described my name ...}\\ 5.&\text{A burning sensation}\\ 7.&\text{Dessert without tea}\\ 10.&\text{Hearken! Listen!}\\ 13.&\text{Everyone has and knows one}\\ 14.&\text{Brothers' riddle}\\ 18.&\text{Short, brutal}\\ \ \\ \end{array} \end{array}$$

Missing Clues (3 out of 4):

• Search in/around egg for headless, confused, older ugly duckling concerning backwards PSE posting. (8 2 6)


• Hearing, smell or touch around ten, maybe twenty years. (8)


• Week (or Day, in the New Order). (4)

Final Clue

The three cryptic clues are needed to fill in the final clue correctly.
You also need to know:

• Order is important - first things first.


• Activity is important - let it order the answers.


• Democracy is more important than activity - but only when indicated. $*$


• Location is important - remember to stay where you are.


• Some of you will need to remember - let the dead be dead to you.

For each number below, apply the three cryptic clues and the information provided.
They will show you the path to success. The taste of victory will soon be yours!

$~~~~ 186 ~~~ \rlap{*} ~~ 99 ~~~ 62:$
$~~~~ 231 ~~~ 65 ~~~ 25 ~~~ \rlap{*} ~~ 455 ~~~ 105.$
$~~~~ \rlap{*} ~~ 70 ~~~ 22!$

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You take a deep breath, and begin the exam.
When you're done, don't forget to press the "That Was Easy!" button¹ and turn your exam paper in to the testing moderator.

Good luck!

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^{1 Feel free to actually time how long you take to complete the exam, and include it in your answer :)}

 
—HINTS—
_{helpfulness level 0:}

_{Pretty much everything on the exam form is important. Outside the exam form, anything in italic is flavor text and can be ignored. (And the "scoreboard" is just for fun and has no relevance.)}

_{helpfulness level 1:}

_{Across/Down clues are used only to fill in the grid; beyond that, these clues and what they reference are not needed to correctly fill in the final word.}

_{helpfulness level 2:}

_{The final clue numbers are used in two different ways.}

_{helpfulness level 3:}

_{?⇒{cc1}⇒{cc2}⇒{cc3}⇒word
To do this for each final clue number; how many values are needed for each, and where do they come from?
The test tells you more or less how to get what you need, if you can figure it out ...}

_{helpfulness level 4:}

_{There's two key pieces of information hiding in plain sight. You'll have to use one of the permitted websites to understand what they mean.}

Answer

Most of the answers for the crossword have already been given in Deusovi's answer, so I'm going to focus on the...

Final Clue

To save some arithmetic trouble, hint 4 is probably referring to

the test number being an OEIS sequence: A014612 is the sequence of numbers that are products of three primes. I feel ashamed that we actually needed a hint to see this!

The sequence in the supplement, A001358, is the sequence of numbers that are products of two primes. Furthermore, the fact that we have "1xA001358" in the supplement suggests we should complete the factorization of these numbers by prefixing a 1 before them, getting three factors.

The natural next step is then to

Factor all the numbers given in the final clue:

  186: 2 3 31
   99: 3 3 11
   62: 1 2 31 
  231: 3 7 11
   65: 1 5 13

   25: 1 5 5
  455: 5 7 13
  105: 3 5 7
   70: 2 5 7
   22: 1 2 11

These probably are

The indices to the "question or answer" - "sentence" - "word" cipher we need to decode the final clue. It remains to find which questions we should apply this to: I tried using the first questions on PSE but they didn't have sentences 31 words long.

As it turns out, an idea I had discarded earlier because it would mean insane dedication on part of the setter,

The numbers in the final clue are also the number of the question needed! As theorized earlier, the first factor indicates the number of the answer we want (1 is for the question, 2 is for the first answer, and so on), the second factor indicates the sentence number, and the third factor indicates the word in that sentence. Some of the questions have very messy punctuation, but the final decoded message is:

For 3 down: quickly follow the day dreams. Color it!

According to OP, "follow the day dreams" is a reference to our vision of the Skittles commercial: "taste the rainbow", which gives us the final word for 3 down. So I guess we found some use for our crayons to color our rainbow? :)

Filling in the final answer in the crossword:

Now we can just press the "That Was Easy!" button and walk away proudly! After all, it only took us... 50 days, right? Piece of cake!