An enterpeton has arrived at the bathhouse. Enterpetons are strange reptiles with exposed organs. Four of these organs can answer questions, and follow certain rules. These organs are: the heart, which says yes if the answer is yes, and answer randomly if not; the gallbladder, which says no if the answer is yes, and answer randomly if not; the lung, which answers no if the answer is no, and answer randomly if not; and the spleen, which says yes if the answer is no, and answer randomly if not. The organs can only answer yes-no questions. Unfortunately, you don't know enough about anatomy to tell which organ is which. What is the quickest (fewest questions on average) strategy to determine the identity of the organs through questioning?
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1$\begingroup$ Is there an intended "nice" solution to this? You could make any number of these puzzles with different answerers and different characteristics, but does something make this one special? $\endgroup$– Deusovi ♦Commented Aug 4, 2021 at 18:47
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2 Answers
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First, it's helpful to create a truth table for the four organs.
| Organ | Yes? | No? |
|---|---|---|
| Heart | Yes | Rand |
| Gallbladder | No | Rand |
| Lung | Rand | No |
| Spleen | Rand | Yes |
Once we have that, we can ask our first question:
Are you the gallbladder?
The organs will answer like this:
Heart: Either yes or no
Gallbladder: No
Lung: No
Spleen: Yes
Here's what that tells us:
Could-be-heart: heart, gallbladder, lung, spleen
Could-be-gallbladder: heart, gallbladder, lung,spleen
Could-be-spleen: heart,gallbladder,lung, spleen
Could-be-lung: heart, gallbladder, lung,spleen
We can ask this second question:
Are you the heart?
Here's how the organs will answer:
Heart: Yes
Gallbladder: Either yes or no
Lung: No
Spleen: Yes
What does that tell us?
Could-be-heart: heart, gallbladder,
lung, spleen
Could-be-gallbladder: heart, gallbladder, lung,spleen
Could-be-spleen: heart,gallbladder,lung, spleen
Could-be-lung:heart, gallbladder, lung,spleen
For the third and final question:
Is the year 2021? Only this question, we'll ask multiple times.
Here's how the organs will answer:
Heart: Yes
Gallbladder: No
Lung: Either yes or no
Spleen: Either yes or no
What does that tell us?
Well, after the first time, here's what we'd know:
Could-be-heart: heart,gallbladder,lung, spleen
Could-be-gallbladder:heart, gallbladder, lung,spleen
Could-be-spleen: heart,gallbladder,lung, spleen
Could-be-lung:heart, gallbladder, lung,spleen
And how does that get us to our answer?
We've effectively split each organ into one of two organs, and by asking a question that's always true, if we ask it enough times, the organ that answers randomly will eventually answer "wrong" thus eliminating it from the running for the heart/gallbladder (which always answers the same) and proving that it's actually the other organ out of the possibilities. It's faster to ask these three establishing questions first because then the randomization afterwards needs only a 50/50 chance to succeed - we'll probably only need to ask about 4 questions total.
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$\begingroup$ The problem with this solution is that you cannot be absolutely sure that a random organ will not continue to answer the exact same thing. While the chances are absolutely miniscule; what if you continued asking the question until it was no longer 2021? For that matter, you can't be sure that the organs won't answer whichever answer makes the puzzle most difficult for you and therefore will never change their answer! $\endgroup$ Commented Aug 4, 2021 at 22:17
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$\begingroup$ I could definitely replace "Is the year 2021" with "Are you an organ?" or some other always-true answer if we're concerned about that. And, since the original question says that they answer randomly, they cannot answer whichever one makes the puzzle most difficult. $\endgroup$ Commented Aug 6, 2021 at 15:05
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$\begingroup$ @TakingNotes since ther organs are allowed to answer randomly, it's possible for them to take the worst-case path, and impossible to rule such cases out. However, the question asked concerns the average case. $\endgroup$ Commented Aug 14, 2021 at 4:22
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To narrow down four possibilities into two, ask this question first:
"Will Spleen lie and Lung tell the truth?"
If yes is correct: Heart says yes, Spleen says no, Lung says yes, and Gallbladder says no.
If no is correct: Spleen tells the truth (which is impossible for a "no" question) and/or Lung lies. So Heart can say anything, Spleen must say yes, Gallbladder can say anything, and Lung says yes (but must say no according to its own rule, which creates a contradiction).
Because of the contradiction, only Yes can be correct, so Heart and Lung will be differentiated from Spleen and Gallbladder.
Then:
To differentiate Heart from Lung, ask "Will Lung or Heart say no?":
If yes: Heart says yes, Lung says no.
If no: Both Heart and Lung say yes, leading to a contradiction.
Because of the contradiction, only Yes can be correct.
Or:
To differentiate Gallbladder from Spleen, ask "Will Gallbladder or Spleen say yes?":
Yes: Gallbladder says no, spleen says yes.
No: Gallbladder says no, spleen says yes, leading to a contradiction.
Because of the contradiction, only Yes can be correct.
The best solution:
Ask two organs the first and second questions. That makes 4.
