And I wonder whether the gods can predict Random's answers. I suppose they have to be able to; if I ask True "Will Random admit to being Random if I ask him if he is?" he either needs to be able to answer Da or Ja, or answer with the equivalent of "I don't know".

Hint: Gur xrl vf gb nfx gur svefg dhrfgvba va fhpu n jnl gung lbh xabj bar bs gur tbqf gung vf abg Enaqbz. Gura nfx gur arkg gjb dhrfgvbaf bs gung tbq.

Abj, V unir n cnve bs dhrfgvbaf gung pna va 2/3 bs cbffvoyr fvghngvbaf trg zr n aba-Enaqbz tbq gb nfx gur guveq dhrfgvba gb, ohg gung'f abg rabhtu gb fbyir gur ceboyrz, V guvax.

V qba'g guvax V pna vzcebir ba zl svefg dhrfgvba, ohg gurer znl or n orggre jnl bs pubbfvat jub gb nfx gur frpbaq bar guna whfg nfxvat n qvssrerag enaqbz (ohg abg arprfnevyl Enaqbz) tbq.

I can easily do it with 4 questions, BTW. The question "vf gur nafjre lbh'er tbvat gb tvir zr gb guvf dhrfgvba gehr?" vf nyjnlf nafjrerq ol nyy tbqf jvgu gur jbeq sbe lrf, which enables me to take the strategy that Matthew's talking about.

My brain hurts.

I believe the random god will always give you the same answer as the other two, not just half the time - I think the rule that "whether Random speaks truly or falsely is a completely random matter" means just what it says, that Random will answer as if she were either True or False, not that Random will give answers that have nothing to do with the question.

(I am so hoping the sequels get translated. Logic puzzles in a steampunk scifi plot with a Miyazaki-esque art style made for a very pleasant rainy afternoon.)

Pbafvqre gur dhrfgvba: "Vs V nfxrq lbh K, jbhyq lbh ercyl wn?" Guvf dhrfgvba, jura nfxrq bs rvgure n gehgu-gryyre be n yvne, jvyy cebqhpr gur nafjre "wn" vs gur nafjre gb K vf "lrf" naq "qn" vs gur nafjre gb K vf "ab". Ol senzvat nyy lbhe dhrfgvba va guvf sbez, lbh ner erqhprq gb gur rnfvre ceboyrz bs gjb gehgu-gryyref naq bar enaqbzvmre, nyy bs jubz nafjre va Ratyvfu.

Incidentally, Haley, in Order of the Stick, came up with a great solution to many Knight/Knave puzzles (not counting ones like the one Jim cites above, or any other puzzle that doesn't have the knights/knaves speaking English).

Has a solution been found to the latter case?

Just for lulz, here is a Full Monty solution. Here there be spoilers. First, the easier problem where the gods answer yes/no:

Gurer ner frireny cbffvovyvgvrf sbe gur svefg dhrfgvba, juvpu lbh fubhyq nfx bs N. Urer ner n srj jvgu avpr jbeqvatf gung lbhe nirentr tbq jbhyqa'g dhvooyr jvgu:
* Ubj jbhyq lbh erfcbaq vs V jrer gb nfx lbh "Vf O Enaqbz?"
* Vf O "zber yvxryl" gb gryy gur gehgu guna P?
Sbe rvgure bs gurfr dhrfgvbaf, nffhzr N nafjref lrf. Vs N vf abg Enaqbz, gura lbh pna fubj gung O vf Enaqbz. Va bgure jbeqf, vs N nafjref lrf gura rvgure N be O vf Enaqbz. Lbh pna fvzvyneyl frr gung vs N nafjref ab gura rvgure N be P vf Enaqbz. Ohg gur vzcbegnag cneg vf gung rira gubhtu lbh qba'g xabj jub vf Enaqbz, lbh xabj fbzrbar jub qrsvavgryl vfa'g. Gur bgure gjb dhrfgvbaf tb gb gung crefba, naq lbh pna frr gung "Vf gjb cyhf gjb rdhny gb sbhe?" naq "Vf N Enaqbz?" jvyy tvir lbh rabhtu vasbezngvba gb gbgnyyl fbeg bhg gur vqragvgl bs nyy guerr.

and to adapt it to the ja/da problem at hand:

Sbe rnpu bs gur dhrfgvbaf D va gur cerprqvat fbyhgvba, nfx "Vs V jrer gb nfx lbh D, jbhyq lbhe erfcbafr or qn?" Vs gur tbq nafjref qn, gura gur tbq jbhyq unir nafjrerq lrf gb D vs fur fcbxr Ratyvfu, naq fvzvyneyl wn zrnaf ab. Nzhfvatyl, nsgre fbyivat gur ceboyrz va guerr dhrfgvbaf lbh fgvyy qba'g xabj ubj gur erfcbafrf genafyngr vagb Ratyvfu.

The internet time-waster I've been using lately is Kingdom of Loathing. Luckily there's a built-in limit to the amount of time you can waste on it per day.

Vg frrzf jbegu cbvagvat bhg rkcyvpvgyl gung guvf jvyy or gehr bs nal fbyhgvba gb gur chmmyr: gurer ner fvk cbffvoyr pbasvthengvbaf bs tbq, gjb cbffvoyr genafyngvbaf bs gur ynathntr, naq ab jnl gb qvfgvathvfu orgjrra gjryir qvssrerag cbffvovyvgvrf jvgu bayl guerr lrf-be-ab dhrfgvbaf (hayrff lbh purng).

(rot13'd because, while it doesn't contain a solution, it contains the realization that put me on the path to a solution)

Proof: Vs gurl qvqa'g, gura gurer jbhyq or bayl 4 cbffvovyvgvrf (fvapr punatvat gur genafyngvba jbhyq or cbffvoyr), naq gurer ner 6 nafjref gb pubbfr nzbat.

That's what comes from doing too much work on resolution proofs: assume the opposite (impossibility), try to prove it, and see where the unprovable lemma (or the hole in the proof) is.

I have no idea how that got dropped. The semicolon separating the two parts of the sentence came through fine and it was meant to be a straight cut-n-paste.

I've now corrected the problem.

(I wrote a story about that once)

(Gleaned from usage: handing a glass of booze to an alcoholic. Official meaning may differ.)

Note to Self: Never follow those links. (Probably shouldn't read the posts.)

Chris, as far as I can see, that's not what that rule says. Answering "truly" is not the same as "answering as if you were the one named 'True'", and answering "wrongly" is not the same as "answering as if you were the one named 'Wrong'". As far as I can see, if you ask Random "Are you True", then she will either truly answer "No" (well, in the gods' language), or falsely answer "Yes", while, if she would always answer as if she were either True or False, she would always answer "Yes".

Or am I wrong here?

My feeling is that it ought to be possible to solve the general problem, and derive a formula for angle "x" based on any values of the four angles down at the base. I've looked at that a little, but not gotten much done so far.

Yes, and the general solution is "use trigonometry."

The Law of Sines makes solving this one a snap. The trick comes in solving it using a straightedge and the knowledge that the interior angles of a triangle add up to 180.

Qenj n yvar guebhtu cbvag Q cnenyyry gb NO.

Does it have a definite answer, or a family of answers? I think I have two fully consistent solutions which aren't the same.

I don't know the official answer to the problem.

I've found the same answer two different ways, if that helps. That doesn't rule out some other, but it's good enough for me.

I would think it would be very hard not to glimpse X when you're drawing it.

Wouldn't that be measuring the angle instead of solving for it? You can print the diagram he provides to avoid getting spoiled instead. And I must say, I'm kinda startled by the idea that the accuracy of the diagram matters. My geometry class back in the mists of time didn't allow for any assumptions that an accurate diagram would make or break. You don't look at something and decide that it is parallel or congruent, you prove so, or are provided that info. Otherwise, it ain't so.

The geometry problem is the one I latched on to, and I'm utterly stumped at the moment. I'm pretty discouraged by logic puzzles most of the time because I don't seem to do well enough at them to get any better, so I have to resort to using hints and cheats. When I'm not using the hints and cheats (and sometimes even when I am), then I end up giving up for lack of progress. I don't seem to ever get better. *sigh*

Also, Fantastic Contraption rocks my socks. These are my two favorites from my solutions:

Chaingang: Beautiful in its hideousness
Friends: It just gives me that warm fuzzy feeling

And I like Alley Oop for being so contrary. Okay, my favorite three.

(I like Hook too. I'll stop now, I promise.)

I consider that when you ask Random a question, he turns her back, flips a god-coin, and depending on whether the coin comes up "da" or "ja" tells the truth or doesn't.

Now that I've remembered about parallel lines...

V znqr n zvfgnxr ba gur zngu, naq vg pnzr bhg evtug naljnl orpnhfr OQR naq NRQ ner rdhny natyrf.

Now I need to PROVE it.

... rot13 may not be strong enough of a cypher.

I would suggest that the answer is Mu. In theory, there are questions where the distinction is important. For instance, any god who is capable of answering "Are you going to answer (your language's word for) 'no' to this question?" is Random and not bound to lie or tell the truth. But here you should assume that "yes-no question" means a question that any god *can* answer with (their word for) yes or no. A similar non-issue would be whether False lies out of delusion or contrariness (and, hence, whether his beliefs are correct even if his statements are not).

And wow. I like playing Fantastic Contraption this much. But I like watching other people's fun designs THIS much!

No Mu for you! "Are you going to answer 'no' to this question?" is an easy question for False: he can say "yes" (lying) or say "no" (still lying). However, True would have a seizure, or hit you with lightning, or something.

Now, I think this discussion has attributed *three* possible definitions for Random. This is compounding the confusion. Let us imagine three different aspects of the Random god. (I am ignoring the language issue, obviously.)

Random-Answer: Flips a coin when asked a question. Heads: say "yes"; tails: say "no".
Random-Valence: Flips a coin when asked a question. Heads: give the answer which is true; tails: give the answer which is false. (This is what Jim@45 describes.)
Random-Copycat: Flips a coin when asked a question. Heads: give the answer which True would give if asked that question. Tails: give the answer which False would give if asked that question.

Note that Random-Answer is the only one who is really ignoring the question.

Also note that you can invent three *more* aspects -- same as the above, except they flip their coin once per *questioner*, or once per conversation, or once per lifetime -- not once per question. (Was this what Chris@14 meant by "not just half the time"? I'm not sure.) However, once-per-question is the traditional model for these puzzles.

Now, none of these is a "Mu" distinction. I'm pretty sure you can tell them all apart by asking the right questions. (Sorry, I'm not actually great at these puzzles.)

The question of delusion versus contrariness isn't all that "mu" either. It's perfectly possible to turn to False (once located) and ask him, "Do you *believe* that two is an even number?" If False is sane and lying out of contrariness, he will say "no". If he lies because all his beliefs are false, he will say "yes" (in fact he doesn't, but he is deluded about that fact!) There's a whole set of puzzles based on this distinction. (You also have the god who is deluded *and* contrary, so he accidentally answers all questions with the truth...)

(Although the puzzle setup doesn't have to support this stuff. The writer could declare that gods are miserably unintrospective, and really don't think about their own beliefs, in which case these questions fall into the "I don't know" bucket.)

I got to that eventually but haven't had the time to poke at it with my decaying algebra and get anything resembling a correct answer. (I'm relatively certain X =/= 0, which is what I got on my first pass through.) I'm not at all certain that it is a workable solution, but it is one I haven't given up on yet.

(Oh, hey, I just remembered that vs lbh fhogenpg n ynetr natyr sebz n fznyy bar, lbh qb trg n erny aba-artngvir ahzore - vg jencf onpx cnfg guerr uhaqerq naq fvkgl. Gung znl znxr n qvssrerapr! (Math is not one of my native skills, as you can see.)

WASD is a fairly standard alternative-cursor-keys layout, used amongst others by multi-person games and an awful lot of first-person shooter fans.

It was a lot harder than I expected to get an equation with x on only one side, and the formula came out a lot hairier than I thought it was going to.

Here it is:
tan x = sin a * sin c * sin (b+c+d) / ((sin (a+b+c) * sin (c+d)) - (cos a * sin c * sin (b+c+d)))

I put a few extra parentheses than strictly necessary in the denominator there, I hope they make it easier to read.

I checked the formula for the case we're given here. It was very exciting when I finally hit the "arctan" button on my calculator and the number 20 popped out.

The only thing you need to know to solve it is that the three angles of a triangle add to 180 degrees. Then start working out the unlabeled angles on some of the many nested triangles and you'll have it in no time. I think I'd worked it out in my head in under a minute the first time I looked at it.

Plus that doesn't solve the general case, which is what I was interested in.