From f6ac40234c0e9b366e1c8a012b040f6880890508 Mon Sep 17 00:00:00 2001 From: RunasSudo Date: Sun, 29 May 2016 21:32:32 +0930 Subject: [PATCH] Update notes to code 6 --- notes.md | 39 ++++++++++++++++++++++++++++++++++++++- 1 file changed, 38 insertions(+), 1 deletion(-) diff --git a/notes.md b/notes.md index 58bad1e..28d62ba 100644 --- a/notes.md +++ b/notes.md @@ -42,7 +42,7 @@ At the foothills, the first area of the RPG: Required opcodes: All except `halt` ## Code 5 (Twisty passages) -After falling down the bridge, there is an empty lantern to the `east`. 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. +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. Taking the `ladder` down, then traversing `west`, `south`, `north`, a code is obtained: @@ -53,3 +53,40 @@ Taking the `ladder` down, then traversing `west`, `south`, `north`, a code is ob You take note of this and keep walking. There is also a can, which can be `take`n and `use`d to fill the lantern, which can then be `use`d to become lit, and which will keep away Grues. + +## Code 6 (Dark passage, ruins and teleporter) +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: + + _ + _ * _^2 + _^3 - _ = 399 + +### Mathematical! +In other words, we seek a solution to the equation *a* + *bc*2 + *d*3 - e = 399, where {*a*, *b*, *c*, *d*, *e*} = {2, 3, 5, 7, 9}. + +Synacor 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*2 + *d*3 ≤ 406. Furthermore, since *bc*2 is always positive, *d*3 ≤ 406. We can thus rule out *d* = 9, so *d* ≤ 7, *d*3 ≤ 343 and *bc*2 ≤ 63, also ruling out *c* = 7 and *c* = 9. Furthermore, since 392 ≤ *bc*2 + *d*3, *d*3 ≥ 329. Since we already knew *d* ≤ 7, *d* = 7. + +Consequently, 49 ≤ *bc*2 ≤ 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.