D&D.Sci (Easy Mode): On The Construction Of Impossible Structures

[abstractapplic]

This is a D&D.Sci scenario: a puzzle where players are given a dataset to analyze and an objective to pursue using information from that dataset.

Duke Arado’s obsession with physics-defying architecture has caused him to run into a small problem. His problem is not – he affirms – that his interest has in any way waned: the menagerie of fantastical buildings which dot his territories attest to this, and he treasures each new time-bending tower or non-Euclidean mansion as much as the first. Nor – he assuages – is it that he’s having trouble finding talent: while it’s true that no individual has ever managed to design more than one impossible structure, it’s also true that he scarcely goes a week without some architect arriving at his door, haunted by alien visions, begging for the resources to bring them into reality. And finally – he attests – his problem is definitely not that “his mad fixation on lunatic constructions is driving him to the brink of financial ruin”, as the townsfolk keep saying: he’ll have you know he’s recently brought an accountant in to look over his expenditures, and he’s confirmed he has the funds to keep pursuing this hobby long into his old age.

Rather, his problem is the local zoning board. Concerned citizens have come together to force him to limit new creations near populated areas, claiming they “disrupt the neighbourhood character” and “conjure eldritch music to lure our children away while we sleep”. While in previous years he was free to – and did – support any qualified architect who showed up with sufficiently strange blueprints, the Duke is now forced to be selective: at present, he has fourteen applicants waiting on his word, and only four viable building sites. He finds this particularly galling, since about half the time when an architect finishes their work, the resulting building ends up not distorting the fabric of spacetime, and instead just kind of looking weird. It’s entirely possible that if he picks at random, he’ll end up with no new impossible structures at all this month, which – he asserts – would utterly break his heart.

This is where you come in. Using his records from previous years, he wants you to evaluate his current crop of architects’ plans and pick out the four most likely to successfully defy the laws of Nature. (If there are any ties, he’d like you to resolve them in favour of whichever option is cheapest; however, his primary concern remains the instantiation of as much impossibility as possible.)

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I’ll post an explanation of how I generated the dataset, and the resulting optimal strategy, sometime on Monday 20th May. I’m giving you three days, but the task shouldn’t take more than an hour; use Excel, R, Python, the malevolent mutterings of men dressed as birds, or whatever other tools you think are appropriate^[1]. Let me know in the comments if you have any questions about the scenario.

If you want to investigate collaboratively and/​or call your choices in advance, feel free to do so in the comments; however, please use spoiler blocks or rot13 when sharing inferences/​strategies/​decisions, so people intending to fly solo can look for clarifications without being spoiled.

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Note: This challenge was originally commissioned by a mysterious sponsor who—after concluding their own use of it—graciously permitted me to make it public domain on the condition that I never explain who paid for it or why. In the unlikely event that you think you’ve seen this one somewhere before, I humbly request that you not publicly discuss this fact.

1. ^

   Though, to be honest, using anything more advanced than spreadsheets for this one would probably be overkill.

[aphyer]

Every structure produced by an architect who apprenticed under B. Johnson or P. Stamatin is impossible. No structure produced by an architect who apprenticed under M. Escher, R. Penrose or T. Geisel is impossible. Slightly under half of self-taught architects produce impossible structures. Materials, blueprints, etc. have no visible effect on this.

There are 5 structures proposed by apprentices to B. Johnson or P. Stamatin (D, E, G, H and K), so we don’t need to risk any of the self-taught people.

Cost is based on materials: Nightmares are by far the most expensive, Silver a distant second, the others seem comparable and cheap. G is the only one of our candidates who plans to use Nightmares, so we leave them out and fund D, E, H and K.

[Unnamed]

I got the same result: DEHK.

I’m not sure that there are no patterns in what works for self-taught architects, and if we were aiming to balance cost & likelihood of impossibility then I might look into that more (since I expect A,L,N to be the the cheapest options with a chance to work), but since we’re prioritizing impossibility I’ll stick with the architects with the competent mentors.

[dr_s]

I find a pattern in that buildings using Dreams together with either Wood or Silver have an 80% chance of being Impossible when made by a Self-Taught architect, but honestly this seems irrelevant since the other two types of background are a 100% guarantee so they’re better value for money anyway.

[aphyer]

This is true, but ’80%′ here means only ¹⁶⁄₂₀. A result this extreme is theoretically p=0.005 to show up out of 20 coin flips...if you treat it as one-tailed, and ignore the fact that you’ve cherry-picked two specific material-pair options out of 21. Overall, I’d be very surprised if this wasn’t simply randomness.

[dr_s]

I admit it’s cheating a bit the spirit of the challenge, but in practice, I guess it’s the round amount that makes me suspicious that it might be intentional. But it’s true there doesn’t seem to be a broader materials related pattern, so it may just be as you say.

[Lorxus]

Finally, I get to give one a try! I’ll edit this post with my analysis and strategy. But first, a clarifying question—are the new plans supposed to be lacking costs?

First off, it looks to me like you only get impossible structures if you were apprenticed to “Bloody Stupid” Johnson or Peter Stamatin, or if you’re self-taught. No love for Dr. Seuss, Escher, or Penrose. Also, while being apprenticed to either of those two lunatics guarantees you an impossible structure, being self-taught looks to do it only half the time. We can thus immediately reject plans B, C, F, J, and M.

Next, I started thinking about cost. Looks like nightmares are horrifyingly expensive—small wonder—and silver and glass are only somewhat better. Cheaper options for materials look to include wood, dreams, and steel. That rules out plan G as a good idea if I want to keep costs low, and makes suggestions about the other plans that I’ll address later.

I’m not actually sure what the relationship is between [pair of materials] and [cost], but my snap first guess—given how nightmares dominate the expensive end of the past plans, how silver and glass seem to show up somewhat more often at the top end and wood/​dreams/​steel show up at the bottom end fairly reliably—is that it’s some additive relation on secret prices by material, maybe modified by the type of structure?

A little more perusing at the Self-Taught crowd suggests that… they’re kind of a crapshoot? I’m sure I’m going to feel like an idiot when there turns out to be some obvious relationship that predicts when their structures will turn out impossible, but it doesn’t look to me like building type, blueprint quality, material, or final price are determinative.

Maybe it has something to do with that seventh data column in the past plans, which fell both before apprentice-status and after price, which I couldn’t pry open more than a few pixels’ crack and from which then issued forth endless surreal blasphemies, far too much space, and the piping of flutes; ia! ia! the swollen and multifarious geometries of Tindalos eagerly welcome a wayward and lonely fox home once more. yeah sorry no idea how this got here but I can’t remove it

Regardless, I’d rather take the safe option here and limit my options to D, E, H, and K, the four plans which are: 1) drawn up by architects who apprenticed with either of the two usefully crazy masters (and not simply self-taught) and 2) not making use of Nightmares, because those are expensive.

For a bonus round, I’ll estimate costs by comparing to whatever’s closest from past projects. Using this heuristic, I think K is going to cost 60-80k, D and H (which are the same plan???) will both cost ~65k, and E is going to be stupid cheap (<5k). EDIT: also that means that the various self-taught people’s plans are likely to be pretty cheap, given their materials, so… if this were a push-your-luck dealie based on trying to get as much value per dollar as possible, maybe it’s even worth chancing it on the chancers (A, I, L, and N)?

[abstractapplic]

Response to clarifying question:

Yes. The Duke has learned the hard way that his architects’ guesses as to how much their projects will end up costing are consistently worse than useless; if you want to optimize on cost as well as impossibility, that’s another thing you’ll have to deduce from the record of finished projects.

[James Bishop]

Apprentices of B. Johnson and P. Stamatin always design impossible structures. There are five such architects among the fourteen options. Nightmares are extraordinarily expensive, and, conveniently, required for only one of the five designs. Excluding that design leaves D, E, H, and K.

[simon]

Looks like architects apprenticed under B. Johnson or P. Stamatin always make impossible structures.

Architects apprenticed under M. Escher, R. Penrose or T. Geisel never do.

Self-taught architects sometimes do and sometimes don’t. It doesn’t initially look promising to figure out who will or won’t in this group—many cases of similar proposals sometimes succeeding and sometimes failing.

Fortunately, we do have 5 architects (D,E,G,H,K) apprenticed under B. Johnson or P. Stamatin, so we can pick the 4 of them likely to have the lowest cost proposals.

Cost appears to depend primarily (only?) on the materials used.

dreams < wood < steel < glass < silver < nightmares

Throwing out architect G’s glass and nightmares proposal as too expensive, that leaves us with D,E,H,K as the architect selections.

(edit: and yes, basically what everyone else said before me)

[Bohaska]

I tried to visualize the entire dataset and look for patterns...

I decided to try mapping the entire group, by connecting all structures where they only differ by one column. After I did so and played around with the graph, I realised that lots of structures by the same person are connected and either all fail or all succeed. To investigate, I decided to only connect structures that are by the same person, and the pattern appears very nicely:

See the image at https://​​39669.cdn.cke-cs.com/​​rQvD3VnunXZu34m86e5f/​​images/​​2d52f50b48a8badc77102018aefe03ad8f36e2a05be8f15c.png\ (same image host at LessWrong, and I can’t find out how to spoiler images)

[Greg D]

I’m not a data scientist, but I love these. I’ve got a four-hour flight ahead of me and a copy of Microsoft Excel; maybe now is the right time to give one a try!

!It seems like the combination of materials determines the cost of the structure.

!Architects who apprenticed with Johnson or Stamatin always produce impossible buildings; architects who apprenticed with Geisel, Penrose, or Escher NEVER do. Self-taught architects sometimes produce impossible buildings, and sometimes they do not.

!This lets us select 5 designs from our proposals which will certainly produce impossible buildings. To do better, we need to understand how to tell when a proposal by a self-taught architect is likely to produce an impossible building.

!~44% of designs by self-taught architects are impossible. This more-or-less matches the ²⁄₅ of masters whose apprentices reliably produce impossible buildings. So I hypothesize that self-taught students pick a favorite master at random and crib their style, acting (illegibly) like a typical apprentice thereafter. So now I need to see if there are particular materials, structure types, or blueprint types which are favored by students of any of the known master architects. By choosing designs by self-taught architects which have those properties, maybe I can tease out whose style they’re probably using.

!A structure can contain either dreams or nightmares, but not both.

!I’m too smooth-brained to tease out complex correlations on this flight while just using Excel: if there’s something weird going on (like, buildings made with either Dreams -or- Glass are likely to be impossible, but if you use both at once they cancel one another out somehow), I don’t know how to find it. So I’ll just assume everything is independent of everything else and do a Bayes to it.

!We can down-select our variables to match those which appear in the Self-Taught proposals; it does us no good to learn whether the “good” architects make use of Nightmares or not, if none the proposals before us make use of Nightmares.

!Good properties: Towers; buildings of Dreams and /​ or Glass; Hastily-Sketched blueprints. Bad properties: Mansions, Mechanisms; buildings of wood and /​ or Steel; Obsessively Detailed blueprints.

!So I choose proposals D, E, G, H, and K (probability 1); and also proposal A (probability ~62%) if we’ve got room.

!Ok, I just got off the plane and checked the puzzle description. Turns out we only get to choose 4 buildings, and there was no reason to try and tease out what Self-Taught architects are doing. In that case, I need to rank proposals D, E, G, H, and K by likely price.

!Structure price looks vaguely exponential, so I’ll take do a linear fit to minimize RMS(log10(error)). If I minimize RMSE directly then it always screws up the low-price structures to get marginally better fits on high-priced ones.

!It really looks like for each structure, you pick two materials; each material contributes a random amount to the price, with every material having its own distribution of price contributions. I can’t figure out what dice or whatever are being rolled for each material, but the fit gives me the average contribution for each one.

!So I choose proposals K, E, D, and H, with expected prices 30k, 73k, 78k, and 78k. Proposal G should be impossible too, but it’ll probably cost about 572k.

[Yonge]

Those apprenticed under Escher, Geisel, and Penrose never produce impossible structures. Those apprenticed under Johnson and Statmin always do. Self Taught architects sometimes do.

I couldn’t find a reliable way of determining which Self-Taught architects would produce impossible structures, so I will have to go with four of D, E, G, H, K.

All the really cheap structures are made out of wood and dreams. Unfortunately none of the 5 architects proposals have these materials. Excluding these the next cheapest ones are all made out of 2 of wood/​dreams/​steel, the next set have Glass and one of the other 3, this suggests K will be cheapest. Silver is the next cheapest, so we want D, E and H.

My submission is therefore : D, E, H, K