Even Bilbo Baggins would have a hard time tackling this week's riddles.
The third installment in The Hobbit trilogy debuts this week. We're celebrating with brainteasers inspired by the riddles exchanged between Gollum and Bilbo in the fifth chapter of J.R.R. Tolkien's The Hobbit. You'll find them below, under the sub-heading "Sunday Puzzle(s) #11: Riddles From a Recluse." What follows here is an explanation of where these riddles originated, and a brief introduction by Tolkien scholar Adam Roberts, author of The Riddles of The Hobbit.
Some weeks ago, io9 reader and puzzle enthusiast Wallace Pustinjak sent me a copy of Riddles From a Recluse, collection of fifty riddles of Pustinjak's own creation. The book's contents immediately called to mind the wit, wordplay, and poeticism of The Hobbit's "riddles in the dark," which appear in the the eponymous fifth chapter of Tolkien's tale. When I asked Pustinjak if this similarity was intentional, he informed me that he had, in fact, found inspiration in The Hobbit's riddles. I knew at once that I had to include a selection of his puzzles in our series. The release of Peter Jackson's Battle of the Five Armies seemed to me an ideal opportunity to do just that.
I've selected seven of Pustinjak's riddles. They appear below, and I hope you enjoy them. However, I did not want these puzzles to appear without some additional context on Tolkien's use of riddles in his work. For that, I turned to Adam Roberts. An English professor and author of several science fiction novels, Roberts also wrote a book examining Tolkien's use of riddles in The Hobbit, titled, rather appropriately, The Riddles of The Hobbit. Roberts was gracious enough to supply us with a brief introduction, which precedes Pustinjack's riddles, below.
"Riddles and The Hobbit," By Adam Roberts
Though it may look like a relatively small-scale, straightforward YA adventure narrative, The Hobbit has always struck me as a rather more complex and riddling text than that. It is, for instance, rather more riddling, indeed, than the more linear Lord of the Rings. Riddles interpenetrate the earlier novel in a way not really true of the later. It’s not just the celebrated ‘riddle game’ between Bilbo and Gollum (although it’s no coincidence, I think, that that scene was the highlight of the rather overlong first movie in Peter Jackson’s trilogy, The Hobbit: An Unexpected Journey). Riddles are everywhere in the book, from Gandalf treating Bilbo’s ‘good morning’ as a riddle (‘what do you mean?’) to the Trolls asking ‘what’s a burrahobbit’. The larger form of the novel expresses a riddling, or ironic, juxtaposition of ‘pagan’ Northern European mythology and ‘respectable’ eighteenth-century bourgeois Christianity. This, strangely, is much less a feature of The Lord of the Rings, where the Christian symbolism is much more straightforwardly worked into the fabric of the nobel. Frodo travels through the sort of imagined secondary world familiar from a thousand Fantasy novels written in the genre that Tolkien effectively invented. Bilbo, though, walks through a rather stranger place, figured somewhere between Tolkien’s restless worldbuilding (the invented ‘history’ of Middle Earth) and a more archaic world out of Grimm’s fairy tales or folk stories.
Kids love riddles; but then, most adults do too. Tolkien wrote The Hobbit in part because he was himself deeply immersed in Anglo-Saxon culture—and the Anglo Saxons loved riddles. Many of our earliest examples of the form date from then, and collections like The Exeter Book continue to entertain and delight today. I think that the Old English love of riddles captures something of the way they approached life: life is a puzzle, but one to be encountered with joy and wit rather than despair. A lot of art is mimetic, but the relationship between riddles and reality is ironic, playful, tricky. Something similar can be said of the two modes of art, ‘realism’ (mimetic) and Fantasy (ironic).
The Anglo-Saxon world from which Tolkien took so much inspiration saw the universe as a riddle, and prized an ironic stance with respect to it. Not that courage and loyalty and strength were unimportant (of course, they were vitally important), but that a warrior hold his strength lightly, that he face death with a smile, that he fight more fiercely in the teeth of certain defeat. I am not talking about flippancy, or a more clumsy disrespectful. I am talking about accepting that there is a mismatch between our human abilities to understand and the brute fact of the cosmos. All we have to decide, to quote somebody, is what to do with the time that has been given us. Riddles teach us what Keats, in a different context, called negative capability. That's more important now, in some ways, than it has ever been.
This is one reason why The Hobbit is so fascinated by riddles. There are others, I think. Working on ‘riddles’ in Tolkien’s fable has helped me see something clearer about the way the book is structured. It is, of course, a story of good versus evil, like a Grimm's tale or a parable; but it understands something of the appeal of wickedness as well as its delinquency. The two most memorable characters in the book (after Bilbo maybe; or perhaps to a greater extent even than him) are Gollum and Smaug. The tale really comes alive when those two are on the page. What differentiates them from the other threats the travellers face, like the goblins, wolves or spiders, is that they are riddling, they have an extra layer of complexity to them. They both invite us not only to dismiss them as bad, but to understand them as conflicted, or glamorous, or other. In other words, the moral universe for Tolkien includes both the patently good and the patently bad, but also a riddling middle ground.
Sunday Puzzle(s) #11: Riddles From A Recluse
Unlike other rulers I am strengthened by neglect.
Over my jurisdiction, great taxes I elect.
When routine dues are paid, I am weakened severely.
Yet if too long delayed, it may cost you quite dearly.
My vault is of the flesh, my tellers silver and bone.
The upkeep of the kingdom is financed with a loan.
A pallid visage in plain sight revealed by its foe.
It does not fast, it does not feast, and yet does shrink and grow.
Much grander and yet more minute than those it stands beside.
Unendingly stalking around, a circuitous stride.
I'm left behind yet never taken, set down in a row.
Seldom seen in isolation, captured by the snow.
Set apart by haste, though immobile all the same.
When left un-defaced, a betrayal of the game.
Born of the cold and born of the heat.
Pacing the world on legs oh so fleet.
Swiftest up high, lethargic down low.
The actions are seen, the form does not show.
With twelve eggs on order, the cook sat and thought.
"One at a time if I like it or not."
With three in the freezer and three in the pot,
Three in each hand neither too cold nor hot.
The first two were airy, or so he remembers,
The last two both burnt up and ended in embers.
A bound serpent dances on the cave floor.
Writhing guardsman of the windy door.
Sharp yet supple and mute yet speaking.
Devoid of hunger, nourishment seeking.
Ceaseless blur or rigid banner varying by host.
On the smallest scale less substantial than a ghost.
Commonplace yet pivotal to rise above it all.
As a pair they triumph yet once separated fall.
We'll be back next week with the solutions – and a new puzzle! Got a great brainteaser, original or otherwise, that you'd like to see featured? E-mail me with your recommendations. (Be sure to include "Sunday Puzzle" in the subject line.)
SOLUTION To Sunday Puzzle #10: Six Fearsome Heroes
Last week, I asked you to tackle this "extremely difficult" Star Trek puzzle, designed by University of Kentucky professor Raphael Finkel.
I'm going to come right out and say it: This puzzle is, in fact, "extremely difficult." It is not so much one puzzle as it is several logic puzzles. Some of those puzzles are nested, such that certain conclusions cannot be made until one has accurately arrived at some other conclusion or conclusions. This kind of puzzle can get very complicated very quickly, and solving it typically involves the use of a spreadsheet or some kind of table to keep track of all the relationships in play.
It's also the kind of puzzle that begs for a programmatic approach. And, in fact, that's exactly how Finkel solves it himself. " This admission may come as a surprise," he writes me by email, "but I have no idea how to attack this puzzle with pen and paper!"
Raphael Finkel's Solution
Before you yell at me: Yes, it is possible to deduce without the aid of a computer. That solution appears below. Here is how Finkel says he approaches it:
The way I solve it is to express it in my own notation, which I call Puzzle Lingo (Software ― Practice and Experience, Volume 34, number 15, pages 1481-1504, December 2004). I then use a script I wrote that converts this notation into an answer-set program (a kind of logic notation, derived from Prolog). I then use standard solver software (such as clasp) to search for a solution, which I then post-process, into something that looks like this:
The solver follows a sophisticated version of this method: Select some statement (such as "Data fears Geordi") and assign it a truth value (true or false). Then see what other statements are logical consequences of that assignment. If there is a contradiction, backtrack. If all statements now have an assigned truth value, stop. Otherwise, pick another statement and continue recursively. What distinguishes one solver from another is how they pick the sentences to assign, what truth value they assign, and what they learn from contradictions. On my computer, the clasp solver takes 0.05 seconds (including input and output!), and including pre-processing and post-processing, the entire process uses less than 0.1 CPU seconds.
The process Finkel describes – of testing a statement's validity by seeing whether other statements are logical consequences, and backtracking when contradictions arise – is a logic puzzle solving strategy known as "Ariadne's Thread." It requires a lot of organization and, in an instance like this, a lot of time. It's the perfect approach for a computer, which can rapidly run through different scenarios and keep track of their outcomes.
Reader John Bohannon took precisely this approach, creating a program of his own that solves the puzzle in well under a minute. Those of you fluent in Python should check out Bohannon's solution, which he has has posted to GitHub.
For a person, however, brute-forcing a problem with Ariadne's thread can be maddening. Maybe even impossible. Which raises the question: How best to approach this problem with pen and paper?
Raphael Ordoñez's Solution
I think ejconer was the first commenter to provide a correct response. However, the solving process ejconer supplied was difficult for me to follow and compare with my own (admittedly sloppy) problem-solving process.
Some days later, however, I came across a beautifully documented solution by math professor Raphael Ordoñez that, I think, embodies the clearest, most organized approach to this puzzle. Here, reproduced with Ordoñez's permission from his blog, is that solution. (Ed. Note: For clarity, I have replaced Ordoñez's text-based tables with something a little cleaner.)
Raphael Ordoñez's Solution to Six Fearsome Heroes
Upon closer examination, we find that there are five statements dealing with only heroes or fear (Statements 3, 5, 6, 7, 10), five statements dealing with only Tri-D Chess or Fizzbin (Statements 1, 2, 8, 9, 11), and exactly one statement dealing with both (Statement 4). Let's renumber, and also break up Statement 10:
1A. Troi is feared by the person Geordi fears.
1B. Riker is feared by the person Picard fears
1C. Picard's hero fears Geordi.
1D. Riker is the hero of Worf's hero.
1E. Data is the hero of Riker's hero.
1F. Data's hero is not Geordi.
2. Worf's hero ranks 3 times lower at Tri-D Chess than the crew member who is best at Fizzbin.
3A. Picard ranks two positions behind Troi at Fizzbin.
3B. The person who is worst at Fizzbin is better than Troi at Tri-D Chess.
3C. The person ranked number 3 at Tri-D Chess is ranked 4 positions higher than Data at Fizzbin.
3D. Riker is ranked 2 lower at Tri-D Chess than the crew member ranked 2 at Fizzbin.
3E. Geordi ranks 2 at Tri-D Chess.
The puzzle of fears and heroes can be solved on its own, using only Statements 1A – 1F. The fact that each crew member is feared by some other member means that the correspondence is surjective; therefore, it must bebijective as well, and each must be feared by exactly one other member. So we can't have two members fearing the same member, etc. The same goes for heroes.
First, for fears, Statement 1A yields
F: Geordi → (???) → Troi
and Statement 1B yields
F: Picard → (???) → Riker
For heroes, Statements 1D and 1E yield
H: Worf → (???) → Riker → (???) → Data
Now, Data must adulate someone; he can't adulate Riker or the two unknowns in the hero chain, because their admirers are accounted for, and he can't adulate Worf, as this would leave out our third unknown. So he must adulate the third unknown, who in turn adulates Worf:
H: Worf → (???) → Riker → (???) → Data → (???) → Worf
Next, Statement 1F states that Data doesn't adulate Geordi, so Data must adulate Picard or Troi. If Picard, then Statement 1C implies that Worf fears Geordi. This would give us
F: Worf → Geordi → (???) → Troi
F: Picard → (???) → Riker
Five crew members are written explicitly here. So at least one of the unknowns already appears in one of these chains, and the only way the two chains fit together is if Picard is feared by Geordi, so
F: Worf → Geordi → Picard → Troi → Riker
But this creates an impossible situation. Worf must adulate Geordi or Troi. If Geordi, then this violates his fear of Geordi, since no one can both adulate and fear the same person. If Troi, then Troi would adulate Riker, violating her fear of Riker.
It follows that Data adulates Troi. We have
H: Worf → (???) → Riker → (???) → Data → Troi → Worf
Now, either Worf adulates Picard and Riker, Geordi, or vice versa. But Worf can't adulate Picard, or else Picard would adulate Riker, with the implication that Riker would fear Geordi (Statement 1C), resulting in
F: Picard → (???) → Riker → Geordi → (???) → Troi
which violates Riker's adulation of Geordi. So Worf adulates Geordi, and Riker, Picard. We've completed our hero chain:
H: Worf → Geordi → Riker → Picard → Data → Troi → Worf
Next, Statement 1C implies that Data fears Geordi, so we have
F: Data → Geordi → (???) → Troi
F: Picard → (???) → Riker
The only way for these chains to fit together is for Picard to be the first unknown. So we have
F: Data → Geordi → Picard → Troi → Riker → Worf → Data
This completes our chain of fear. On to Tri-D and Fizzbin.
According to Statement 3A, Picard ranks 2 lower than Troi at Fizzbin, so he can't rank 5 or 6, and Troi can't rank 1 or 2. Similarly, Troi can't rank 6 at Tri-D (Statement 3B), and Data can't rank 3 at Tri-D or 3, 4, 5, or 6 at Fizzbin (Statement 3C); since Picard can't rank 5 or 6 at Fizzbin, Statement 3C also implies that Picard can't rank 3 at Tri-D. Also, Riker can't rank 5 or 6 at Tri-D (Statement 3D). Finally, we know for a fact that Geordi ranks 2 at Tri-D (Statement 3E). So far we have:
We know that Geordi is Worf's hero, so Geordi is ranked 3 times lower at Tri-D Chess than the crew member who is best at Fizzbin. Since we already know Geordi is ranked 2, the person who is best at Fizzbin must also be best at Tri-D. This eliminates Data and Picard from being best at Tri-D, and Geordi, Riker, and Troi from being best at Fizzbin. So Worf must be best at both. Worf does not rank 2 at Fizzbin, so Statement 3D implies that Riker does not rank 4 at Tri-D.
Now, Riker ranks either 1 or 3 at Tri-D. If he ranks 1, then Troi must rank 3. It would follow that Troi ranks 4 places above Data at Fizzbin (Statement 3C). With the spaces available, the only way this could happen is for Data to rank 1 at Fizzbin and Troi to rank 5. But Statement 3D would imply that Troi ranks 2 at Fizzbin, a contradiction.
So Riker ranks 3 at Tri-D. He therefore ranks 4 places above Data at Fizzbin, placing Data at 1 and Riker at 5. We now know that Data did better than Troi at Tri-D (Statement 3B), so he can't rank 1 at Tri-D, and Troi can't rank 5 at Tri-D. We have:
Statement 3D implies that the person who ranks 2 at Fizzbin also ranks 5 at Tri-D; this means that Data does not rank 5 at Tri-D, hence must rank 4.
So Troi is ranked 1 at Tri-D, and Picard is ranked 5. It follows that Picard is ranked 2 at Fizzbin (Statement 3D), and Troi is ranked 4 (Statement 3A). Geordi must therefore rank 3 at Fizzbin.
And we're done. To sum up:
- Data fears Geordi, adulates Troi, ranks 4 at Tri-D, and ranks 1 at Fizzbin.
- Geordi fears Picard, adulates Riker, ranks 2 at Tri-D, and ranks 3 at Fizzbin.
- Picard fears Troi, adulates Data, ranks 5 at Tri-D, and ranks 2 at Fizzbin.
- Riker fears Worf, adulates Picard, ranks 3 at Tri-D, and ranks 5 at Fizzbin.
- Troi fears Riker, adulates Worf, ranks 1 at Tri-D, and ranks 4 at Fizzbin.
- Worf fears Data, adulates Geordi, ranks 6 at Tri-D, and ranks 6 at Fizzbin.
*LONG, DEEP INHALE* – Did you get all that? Ordoñez tells me he solved this puzzle and wrote up the solution while his students were taking a final exam. As someone who poured untold hours into tackling this problem, I think I hate him for this.
This was hard. The way the rankings were ordered (with rank 1 being the worst and rank 6 being the best) had me constantly checking to make sure I had my positions correct, and I know from the comments that I was not alone. I asked Finkel if this was meant to be intentionally misleading, to which he replied:
Good question. I think the expression "twice as good" should mean "has a rank twice as high", so rank ought to increase. It's kind of strange that we don't generally do that.
Anyway – would you believe that Finkel has harder puzzles than this? According to him, this is a 4-star puzzle. There is has a harder version, he says, "where every crew member eats lunch with exactly one other crew member," that comes in at 6-stars.
Perhaps we'll see another of his logic puzzles in the future.