Logic puzzles for you!

[blimix]

I wrote a puzzle today, and in correcting it, I discovered that it lends itself to a series of similar puzzles with intriguing variations. Have a crack at them yourself, and expect to see some spoilers if you view comments. I know that some of you won't be able to resist this.

Puzzle 1:

The sphinx crossed a secret chamber deep in the castle's basement, trying to keep a level head and ignore the sounds of battle coming from outside. Her friends would survive only if she could quickly find and disable their enemy's source of power. There was, she had been informed, only one path to that source. As she approached the three doors, each one in turn formed a mouth and spoke, then went silent and became again a plain, wooden door with a number on it.

The first door said, "Door two speaks truly and leads to doom."
The second door said, "Door three speaks truly and leads to power."
The third door said, "Door one lies and leads to power."

An inscription above them read, "At least one door speaks the truth."


Puzzle 2:

"Good thing they sent a sphinx to handle this," she thought, passing through the doomless door.

There followed a similar array, but this time, the inscription read, "The person who knows which door leads to power still doesn't know whether it lies."

The first door said, "Door three lies and leads to power."
The second door said, "Door one lies and leads to doom."
The third door said, "Door two speaks truly and leads to doom."


Puzzle 3:

"Cute," thought the sphinx, leaving the door open behind her. "There was a fast way and a slow way to think about that, and for my friends' sake, I'm glad I found the fast way." But the next set of doors made her pause.

The inscription read, "The person who knows how many doors lie will know which door leads to power."

The first door said, "Door two lies and leads to power."
The second door said, "Door three speaks truly and leads to power."
The third door said, "Door one speaks truly and leads to doom."


Puzzle 4:

"I'll have to use that one later," the sphinx noted as she passed through the door. "Though I can't help wondering whether by 'power' they mean 'more doors'."

The next wall displayed the inscription, "The person who can discern between truth and lies will know the door to power."

"Well, of course they will," she scoffed. "Someone who could detect lies wouldn't have to use logic here at all! What kind of clue is that?" But her friends needed her, so she approached the doors.

The first door said, "Door three lies and leads to doom."
The second door said, "Door one lies and leads to power."
The third door said, "Door two speaks truly and leads to power."

She pondered this. "Huh. Maybe I do need to consider that inscription."


Afterward:

Carefully aiming her flashlight, the sphinx flipped the circuit breakers labeled "Robot pirates" and "Weasel mind control ray". The wiring wasn't up to code anyway, so really, she was doing them a favor.

Which doors did she pass through?


Author's note (because I was asked): "And" is used formally here. So if door 4 lies, it could still tell a half-truth, like "Bunnies are fuzzy and emeralds are soft." The word "and" makes the entire sentence false if any part of it is false.

Also, yes, you can assume that one and only one door in each puzzle leads toward power, and the other two indeed lead to doom. I intend "power" and "doom" to be negations of each other, so don't bother with doors that lead to both or neither. (Save that idea for a lateral thinking puzzle set.)

Comments

[dolohov.livejournal.com]

Hmm. I think I have all of them, but that last one has me partly stumped -- I'm not 100% sure how to apply the inscription.

[blimix.livejournal.com]

Hmm. The fourth one's inscription could be reworded as, "The person who can discern between truth and lies will know the door to power." That should be less ambiguous. If that doesn't address the issue that you have with it, please let me know your thoughts on it. A puzzle's writer is its worst tester, so I may need help finding the problem.

And thanks for tackling these!

[dolohov.livejournal.com]

Huh. That's not at all what I was thinking. I had found three mutually-consistent scenarios for that one, and only one of the doors is lying in all three of them, so I had been thinking that the inscription pointed to that one. Now I'm not sure.

[blimix.livejournal.com]

Well, I just triple- and quadruple-checked the fourth puzzle, and your result ("only one of the doors is lying in all three of them") is off. Off far enough that perhaps you should double-check that you have transcribed the doors' statements correctly.

Meanwhile, I should edit the wording of the inscription.

[dolohov.livejournal.com]

Feel free to screen that comment, BTW

[blimix.livejournal.com]

Correct! Good job.

[wotw]

For Puzzle 1:

First, there's no particular reason to trust (or distrust) the inscription, so we might as well ignore it.

Possibility A: Power lies behind Door One. The doors all utter random strings of words that are chosen with no regard for logical consistency or where the power actually lies.

Possibility B: Power lies behind Door Two. The doors all utter random strings of words that are chosen with no regard for logical consistency or where the power actually lies.

Possibility C: Power lies behind Door Three. The doors all utter random strings of words that are chosen with no regard for logical consistency or where the power actually lies.

Since all three of these are possibilties, Power could lie behind any of the three doors.

[blimix.livejournal.com]

Fair enough. Logic puzzles like this involve a tacit assumption that statements are true unless there is a clear implication that they might not be. These puzzles rely on that. The inscriptions essentially present the puzzles, so solving the puzzles without trusting the inscriptions is like solving a sudoku while changing the starting numbers: An exercise in ignoring the intended puzzle.

The sphinx has been given (in the story) no reason to trust the inscriptions or whomever wrote it, so acting in accordance with the first puzzle was a leap of faith. The fact that the room contained a puzzle, instead of a pit trap full of black mambas, is evidence that whoever created it did not intend to simply doom anyone trying to get through, which itself is weak evidence that the inscription is reliable. The fact that trusting it did not lead to any apparent doom is evidence that the further inscriptions are reliable.

But without that exercise in trust, psychology and vaguely Bayesian reasoning, you're right. It could all be tigers and no ladies.

[wotw]

Despite my nitpicking, these really are quite wonderful puzzles. Thank you for posting them.

However, while I buy your response to my nitpicks with puzzle 1, I don't think the same response will work for Puzzle 2. Because this time, in the spirit of the puzzle, I *will* trust the inscription. Now, because I don't start out trusting any of the doors, and indeed don't have any reason to believe that their statements have anything at all to do with what's behind them, I cannot possibly discern which door leads to power. (Surely the doors could have made these exact statements no matter where the power lay. This makes the inscription vacuously true: Every person who knows which door leads to power (i.e. nobody) also fails to know whether it lies. (And also *does* know whether it lies, incidentally.)

So I can start by assuming (correctly!) that the inscription is true and *still* discern no information at all about where the power lies.

[blimix.livejournal.com]

Excellent thought. I think it would be more convincing had I worded it, "Every person who knows which door leads to power still doesn't know whether it lies." But since it says, "The person," that implies that it's referring to a person, real or hypothetical, who does know which door leads to power.

Since it's not explicitly stated, that implication can perhaps be ignored, leaving room for your interpretation.

Heck, it's even open to a more strict interpretation that's still open, if "The person who knows which door leads to power" is the illiterate barbarian Fred, and he only knows which door is correct because he guessed correctly and didn't get eaten by a grue. He has no clue whether that door lies. In that case, there's no useful information available for the puzzle.

[wotw]

RIght. But let's not lose sight of the fact, as I said before, that these are nitpicks and your puzzles are wonderful

[blimix.livejournal.com]

Thanks! I appreciate it.