# PanGalactic Timber

Here at PanGalactic Timber Company, we've rescued an Alien Artifact from the ruins of New York City, caused by those so-called "Avengers." This allows us to traverse the universe in search of high-quality woods for your pleasure!

While the Artifact in question may still unfortunately have some alien remains sullying it, we can assure you that each harvested tree is fastidiously cleaned before being converted into the finest hardwood floors you can imagine.

Construction on a new U.N. building to memorialize the attack (of course using our marvelous planking) started two weeks ago. Unfortunately, our first three days of production were lost to a bore weevil infestation. More fortunately, production measures have been put into place, and we no longer have to worry about bore weevil troubles. This was possibly a good thing, as we faced an unusual Artifact-related problem the first day, which was a logistical nightmare.

Starting on the fourth day, we were able to produce 14 floors, with 12 the next, proceeding in the following fashion:

?, ?, ?, 14, 12, 11, 10, 9, 9, 8, 8, 8, 7, 7

We have a deadline for this skyscraper on Day 25, after which we have to let the electricians in. How big will this posh new U.N. building actually be?

The Artifact enabling space travel from "The Avengers" is an 'Infinity Stone' called the 'Tesseract'. My apologies, I thought that was more widely known.

Looking through our records, we have discovered that on our third day of construction, should the timber have been usable, we would have been able to produce eighteen floors.

• Do I get a free Pan Galactic Gargle Blaster if I answer? – Mithrandir Aug 29 '16 at 18:47
• Do the "?" entries indicate that the fourth-day production is the same as the fourth-day production would have been without the weevils, rather than everything having been started 3 days later -- so that what we have is entries 4-14 of a sequence that we're requested to extrapolate as far as entry 25? – Gareth McCaughan Aug 29 '16 at 19:58
• @GarethMcCaughan That is correct, and I will say that f(x) is a non-negative integer for all x>=4. – Sconibulus Aug 29 '16 at 20:08
• The question refers to "hardword" floors. Should that be "hardwood" or is this a clue? – Gareth McCaughan Aug 29 '16 at 21:15
• It's nice to see a number sequence with a bit of flavor. Actually, more than a bit. Well done. +1 – axavio Aug 30 '16 at 1:38

[I had a different conjectural formula here before, but it is inconsistent with what we have since been told about f(3). Note that some discussion in comments below pertains to that formula.]

It has become clear that the problem we have to solve is a combination of the "overt" problem (here are the numbers, work out how to extend the sequence) and a "covert" problem made up of cryptic hints in the question. But let's see what progress if any we can make on the overt problem given the newly revealed value of f(3). Presumably the covert hints are meant to guide us to some formula (maybe involving counting trees on the edge-graph of a tesseract, or something) and it seems like it will have to be a fairly simple one.

We have f(3..14) = 18,14,12,11,10,9,9,8,8,8,7,7. In particular, note f(3)=18, f(3^2)=9=18/2. So perhaps we have something of the general form f(x) ~= a/log(x) (where the ~= indicates that we might be taking floors, or there might be some kind of offset, or "log" might mean something more like bitcount, or who knows what). That would explain the problems on day 1, and perhaps "logistical" is a hint at logarithms too.

Well, this quickly yields another simple formula that agrees with the given information:

f(n) = floor(20/log(n)). [We get the same output if we replace 20 with any number from 19.9 to 20.5, but I don't see any particular reason to find any such number preferable to 20.]

These are

natural logs, presumably made from natural hardwoods rather than mysterious alien artefacts.

I will be moderately surprised if this turns out not to be the intended formula -- but I am still some way from figuring out all the cryptic clues.

If this is right, the total up to and including day 25 will be

172 floors made from the finest natural logs.

logs and floors

but it seems like maybe there is more. Let's take a look at some features of the story that might possibly be part of the covert puzzle.

PanGalactic Timber Company

Curious name. Might just be a Douglas Adams reference for fun, but maybe there's something more.

an Alien Artifact [...] the ruins of New York City, caused by those so-called "Avengers."

Sconibulus has indicated that this is a reference to a Marvel movie. The artifact is also called an "infinity stone" and "the tesseract" (the latter normally means a 4-dimensional analogue of a cube). There isn't anything in my answer that particularly reflects any of that. (In particular, the single magic number in my formula isn't the number of edges in a tesseract or anything like that.)

traverse the universe

Slightly odd turn of phrase. Is the answer supposed to involve some algorithmic thing involving e.g. traversal of binary trees?

the Artifact in question may still unfortunately have some alien remains sullying it

That's a bit weird. Also, this bit of the question was edited for no very obvious reason. So there may be a clue here, but I'm entirely at a loss to figure out what sort of clue.

each harvested tree is fastidiously cleaned before being converted into the finest hardwood floors you can imagine.

Sconibulus has already indicated that "floors" was meant to be a clue that the floor function features in the answer (to which I have to say: really? I mean, it's not like floors enter the question in some unnatural way that actually indicates something cryptic going on). But "fastidiously cleaned" and "the finest ... you can imagine" are interestingly hyperbolic, and "converted" wouldn't be the usual word here. Does the intended answer involve converting something to something else? Does it involve imaginary numbers?

(Perhaps fastidiously cleaning alien remains = subtracting off the remainder after taking the floor, in which case we've found a way to use more of the cryptic hints but it doesn't actually change what we're doing.)

a new U.N. building to memorialize the attack

Mention of the UN might be unnecessary detail -- is there a hint here?

(of course using our marvelous planking)

"Planking"? Unusual word. Reference to Planck's constant somehow??

our first three days of production were lost to a bore weevil infestation.

I haven't heard the term "bore weevil" before (though there are "borer weevils"); is there some coded hint here?

More fortunately, production measures have been put into place,

Curiously vague; some suggestion of "production systems" in theoretical computer science?

we faced an unusual Artifact-related problem the first day, which was a logistical nightmare.

This may be intended to hint at (1) a singularity at n=1 and (2) logs.

My guess is that there are more cryptic hints I haven't identified or decoded correctly yet.

• This is one of probably many formulae that fit, and correctly pulled the "floor" clue from the story, however there are several more clues hidden in the story that are not used in this formula. – Sconibulus Aug 29 '16 at 20:07
• So, er, are you saying that my answer which correctly fits the given sequence is a wrong answer because it doesn't include details that match up with implicit hidden clues in the question? – Gareth McCaughan Aug 29 '16 at 20:23
• Isn't that what puzzles are all about? In most, each line has multiple possible answers, with the group as a whole admitting only one? – Sconibulus Aug 29 '16 at 20:51
• Usually either (1) the requirements for something to be a solution are actually stated in the question, or else (2) it's made explicit that the question is one of steganography. Otherwise the danger is that the question ends up effectively being "guess what I'm thinking of" which is no fun and liable to get it closed as "unclear what you're asking" or "too broad"... – Gareth McCaughan Aug 29 '16 at 21:14
• The difficulty I have with this particular puzzle is that it seems to consist of an explicitly-asked question with multiple possible solutions, plus some hidden cryptic hints that the questioner understands as constraints on what counts as an acceptable answer. Eventually I suppose we'll find out what all the hidden constraints are, and then maybe it will be clearer how good a puzzle it is :-). – Gareth McCaughan Sep 1 '16 at 15:55

The other answer is close enough to being what I was going for to warrant the checkmark, but I thought I should explain my thoughts fully, so that I can get feedback of where to improve for future puzzles.

My thought was that the Timber Company, and the various mentions of converting wood to planks would clearly signify

Logging, and therefor the Log() function.

The fact that the alien artifact was covered in alien remains, to the point that everything coming through it was tainted enough to require special cleaning, was an attempt to point solvers at

Gross

And the name of the artifact was also relevent.

It was a Gross Tesseract. -> In the vein of a 4-square being used in a puzzle as 16, I was trying to imply 144^4.

I chose U.N. because

Parts of the full function graph are UNdefined, and Complex...

I'm not sure that I should have done that, perhaps a generic building would have been better.

Unusual artifact related problem that was logistically challenging was referencing
!> The fact that the Tesseract was also an 'Infinity Stone', and that limx->1+(f(x)) was infinity.

There were also references to cleaning off the remain(der)s, and floors, both pointing at the floor function which @Gareth got.

All in all, the goal was to get solvers to

Sum(4:25,floor(log(144^4))) which they could check against the numbers to fill in where X goes, which, in my formulation, was floor((logx)144^4) although I intended to accept any that fit the numbers, and got at least a couple of the clues.

20/ln(x)

Which is very close to an alternate formulation of my intended solution,

ln(144^4)/ln(x) ~= 19.88/ln(x)

And so he is awarded the check.

• Please don't hesitate to leave feedback on how I can make better puzzles in the future. – Sconibulus Sep 1 '16 at 18:05
• For my taste, the central idea needs too many consecutive big leaps without any confirmation. I mean, you need to go from "sullied with alien remains" to "gross" meaning 144, which is quite a leap; then you need to go from "gross" + "tesseract" to $144^4$, which isn't so big a leap but note that there's nothing in the question to indicate combining the elements in this way; and then you need to guess that what you're computing is $\log_{\mathrm{day}}(144^4)$, which again is clued only by the fact that if you think of doing that then you get the right answers. ...continues – Gareth McCaughan Sep 1 '16 at 18:12
• I don't want to say "no one would ever think of that" -- the denizens of p.s.c are awfully clever and maybe someone would have -- but generally puzzles where you have to make several unconfirmed leaps in the dark are really hard because the implied search space is so large. – Gareth McCaughan Sep 1 '16 at 18:13
• On the other hand, the answer is obtainable by a nice simple formula (and I am willing to believe that given the values from 3 to 14 there really aren't any essentially different ones that match) and you were willing to give credit for a version of the formula that didn't involve guessing every single thing you were thinking; clearly the puzzle wasn't impossibly hard on those terms since it was in fact solved :-). – Gareth McCaughan Sep 1 '16 at 18:15
• I think the UN thing is much too obscure to expect solvers to find it, but it doesn't do any harm. – Gareth McCaughan Sep 1 '16 at 18:16