These solution notes are **very** detailed! Reading them before tackling the challenge yourself will dispossess you of the masochistic joy of working through the best worst use of your spare time mankind has ever devised! Read on at your peril!
After falling down the bridge, there is an `empty lantern` to the `east`, which should be `take`n. To the `west`, there is a passage, where one can take the `ladder` down or venture into the `darkness`. Attempting to venture into the `darkness` at this point will result in being eaten by Grues.
Returning to the fork at the passage and venturing to the `darkness`, we now `continue``west`ward to the ruins. The `north` door is locked, but dotted elsewhere around the ruins are a `red coin` (2 dots), `corroded coin` (triangle = 3 sides), `shiny coin` (pentagon = 5 sides), `concave coin` (7 dots) and `blue coin` (9 dots) which should be `take`n and `look`ed at, and the equation:
In other words, we seek a solution to the equation *a* + *bc*<sup>2</sup> + *d*<sup>3</sup> - e = 399, where {*a*, *b*, *c*, *d*, *e*} = {2, 3, 5, 7, 9}.
The Synacor challenge being, of course, a programming-orientated exercise, the usual response to this problem is to code up a quick program to loop through all 5! = 120 permutations of the coins to find which one satisfies the equation. Being a mathematician, however – could you tell from the italics? – I will solve this the [*proper* way](https://xkcd.com/435/): with a scientific calculator and some thinking (\*insert insufferably smug expression here\*).
Firstly, note that -7 ≤ *a* − *e* ≤ 7, so 392 ≤ *bc*<sup>2</sup> + *d*<sup>3</sup> ≤ 406. Furthermore, since *bc*<sup>2</sup> is always positive, *d*<sup>3</sup> ≤ 406. We can thus rule out *d* = 9, so *d* ≤ 7, *d*<sup>3</sup> ≤ 343 and *bc*<sup>2</sup> ≤ 63, also ruling out *c* = 7 and *c* = 9. Furthermore, since 392 ≤ *bc*<sup>2</sup> + *d*<sup>3</sup>, *d*<sup>3</sup> ≥ 329. Since we already knew *d* ≤ 7, *d* = 7.
Consequently, 49 ≤ *bc*<sup>2</sup> ≤ 63. It is trivial to check that no solutions satisfy this for *c* = 2 and *c* = 3, so *c* = 5 and *b* = 2. The values of *a* and *e* are now easily found.
The solution is:
9 + 2 * 5^2 + 7^3 - 3 = 399
**BOOM!**
(Well, actually all I did the first time was get up to *d* ≤ 7, then use guess and check.)
The coins should therefore be `use`d in the order: `blue`, `red`, `shiny`, `concave`, `corroded`.
Proceed to the `north` door and `use` the `teleporter` to obtain the code:
== Ruins ==
Things of interest here:
- teleporter
> use teleporter
You activate the teleporter! As you spiral through time and space, you think you see a pattern in the stars...
............
After a few moments, you find yourself back on solid ground and a little disoriented.
At this point, you will almost certainly need to delve into the code of the challenge, if you haven't already. The code in `challenge.bin` past the self-test is encrypted, so disassembling and analysing the code is most easily done based off a memory dump from a running copy:
1809 data "You find yourself standing at the base of an enormous mountain. ...
(Accompanied by some other familiar-sounding but somewhat garbled strings.) Searching the code for references to `1809` yields nothing, but searching for `1808` yields the following as the only result:
090d data 17fe 1808 6914 6917
0911 halt # (i.e. data 0000)
The first address referred to in the `090d` line has value `0009`, which looks suspiciously like a length. Indeed, this is the exact length of the following string `Foothills`. Continuing in this manner,
17fe data 0009 "Foothills"
1808 data 00b7 "You find yourself standing at the base of an enormous mountain. ...
6914 data 0002 18c0 18c8
6917 data 0002 0917 0912
Following the rabbit hole,
18c0 data 0007 "doorway"
18c8 data 0005 "south"
0917 data 1929 1933 691e 6921 0000
0912 data 18ce 18d8 691a 691c 0000
And unsurprisingly, the `0912` line leads to the data for the screen reached when venturing `south` from the beginning.
Scrolling through the list of rooms beginning `090d`, we notice a peculiarly long line, `0949`:
Aah, so it looks like each room is stored as a block of 5 words, the first four pointers to lengths of words: a string (the title), a string (the text), a list of pointers to strings (the exit names) and a list of pointers to more rooms (the exits), followed by a memory location to `call` (or `0000`).
We probably could have reached these same conclusions by analysing the suspicious-looking block of code following the room definitions, but assembly makes my head spin so ¯\\\_(ツ)\_/¯
4695 data 0088 "The tablet seems appropriate for use as a writing surface but is unfortunately blank. Perhaps you should USE it as a writing surface..."
090d ... # the foothills from earlier
1270 ... # a subroutine that presumably prints code 4
### Code 7 (Synacor Headquarters and Teleporter 2: Electric Boogaloo)
Examining the data for the teleporter:
0a94 data 4a55 4a60 099f 1545
4a55 data 000a "teleporter"
4a60 data 0048 "This small device has a button on it and reads \"teleporter\" on the side."
099f ... # the north room in the ruins
1545 ... # aha!
Now, let's see what this `1545` does. C-style, because assembly makes my brain spin.
Phew, so in other words, we seek an `R8` such that `178b(4, 1, R8)` equals 6. Let's see if we can't rewrite that function. Note that adding 0x7fff = 32767 to a number modulo 32768 is equivalent to subtracting 1. Thinking through the code, then,
A(R1, R2) = R2 + 1 , if R1 = 0
A(R1 - 1, R8) , if R1 ≠ 0 and R2 = 0
A(R1 - 1, A(R1, R2 - 1)) , if R1 ≠ 0 and R2 ≠ 0
Recognise anything? Well, neither did I the first time, and I'd [already seen the video](https://www.youtube.com/watch?v=i7sm9dzFtEI). It's the [Ackermann function](https://en.wikipedia.org/wiki/Ackermann_function)! With the slight twist that instead of the second line being `A(R1 - 1, 1)`, it's `A(R1 - 1, R8)`.
No mathematical wizardry here, just implementing this and run a brute-force on all possible values of `R8`. And as much as it pains me to admit this, this is a tool for the raw processing efficiency of C, which I am not very proficient in, so I based my solution on [this wonderful code](https://github.com/glguy/synacor-vm/blob/master/teleport.c) by [glguy](https://github.com/glguy). My only contribution is the parallelisation of the computation. (500% speed-up! Whoo!)
Running the algorithm, the correct value is revealed to be `0x6486`. Now we simply set `R8` to `0x6486` and patch the code to skip the check, before `use`ing the `teleporter`:
A strange, electronic voice is projected into your mind:
"Unusual setting detected! Starting confirmation process! Estimated time to completion: 1 billion years."
You wake up on a sandy beach with a slight headache. The last thing you remember is activating that teleporter... but now you can't find it anywhere in your pack. Someone seems to have drawn a message in the sand here:
............
It begins to rain. The message washes away. You take a deep breath and feel firmly grounded in reality as the effects of the teleportation wear off.
Arriving at the beach, traverse `north`ward until you reach a fork. To the `east` lies a `journal` containing clues as to the upcoming puzzle.
With the ability to map the puzzle (possibly with the help of our knowledge of the map data format), and (unlike the adventurers, it seems) a grasp of basic arithmetic, this puzzle shouldn't be too difficult to solve. I couldn't do better than [paiv](https://github.com/paiv)'s [solution](https://paiv.github.io/blog/2016/04/24/synacor-challenge.html) ([code](https://github.com/paiv/synacor-challenge/tree/master/code/src/vault), [map](https://github.com/paiv/synacor-challenge/blob/master/notes/vault-locks.svg)), and it doesn't look like there's much parallelisation to do, so I'll leave you with those links.
The solution is:
* start: **22**
*`north`, `east`: 22 + 4 = 26
*`east`, `north`: 26 − 11 = 15
*`west`, `south`: 15 × 4 = 60
*`east`, `east`: 42 − 18 = 42
*`west`, `north`: 31 − 11 = 31
*`north`, `east`: 30 − 1 = **30**
Navigating through the maze:
As you approach the vault door, the number on the vault door flashes white! The hourglass is still running! It flashes white! You hear a click from the vault door. The orb evaporates out of hour hands.
> vault
== Vault ==
Things of interest here:
- mirror
> use mirror
You gaze into the mirror, and you see yourself gazing back. But wait! It looks like someone wrote on your face while you were unconscious on the beach! Through the mirror, you see "............" scrawled in charcoal on your forehead.
Congratulations; you have reached the end of the challenge!
Given that you've made it this far (You didn't cheat, did you? I did warn you at the beginning!) this last challenge should be no problem.
