A Meta-Puzzle: who was the Mathematician, who the Philosopher, who was Normal, and in what order did they...
logic/smullyan/election New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
A Meta-Puzzle: who was the Mathematician, who the Philosopher, who was
Normal, and in what order did they finish in the election?
[ Editor: the full text of Gareth Rees's puzzle is not in the collected
postings; the following is pieced together from the parts quoted in the
replies. The logician Alice tells a friend about an election among
three candidates, A, B and C, each of whom is a Mathematician (who
always tells the truth), a Philosopher (who always lies) or Normal (who
may do either). ]
"A said, 'I am a Mathematician. Elect me!'
"B said, 'A is a Philosopher. Elect me instead!'
"C said, 'B is lying. Elect me instead!'
"Then the voting took place, and the results were announced. A, B
and C came first, second and third, though not necessarily in that
order. The candidates then made further short speeches:
"A said, 'I came higher than C.'
"B said, 'C came first.'
"C said, 'The person who came first is not a Philosopher.'
"Now, I still didn't have enough information to tell who was the
Mathematician, who the Philosopher, and who was Normal. I therefore
asked C if he was a Philosopher. He answered 'yes' or 'no', and then I
knew which candidate was which type of faculty."
[ Editor: the puzzle ends with two more paragraphs, not quoted in the
postings, in which Alice says she does not know whether asking the same
question of the candidate who came second would have told her anything. ]
The Puzzle: who was the Mathematician, who the Philosopher, who was
Normal, who came first in the election, who came second, and who came
third?
[ Editor: problem statement reconstructed from the postings. ]
Show solution
I guess a meta-puzzle requires meta-analysis ... Since all three types can answer "no" to the question "Are you a Philosopher?", a "no" answer yields no information about the answerer's type. Since Alice gained information from C's answer to this question, C must have answered "yes", and therefore C is Normal. Since Alice does not know whether the answer to the same question (addressed to the candidate who came in second) would yield information, she must not know what the second-place finisher's answer would be (although she knows his/her type). Only a Normal's answer to this question is unknown beforehand, so the second-place finisher must be Normal (and therefore is C). B claims that C came in first, but C came in second. Therefore B is the Philosopher (and A is the Mathematician). Mathematician A's claim that (s)he came in ahead of second-place C must be true, since (s)he is a Mathematician. Conclusions: Mathematician A won the election Normal C came in second Philosopher B was the big loser -- J. Miller <jmil@hogpc.ho.att.com> The only useful answer to "are you a Philosopher" is "yes" (since anyone can say "no"), which indicates that the speaker is Normal. So C is normal. Now there's a problem: If Normals are consistent (that is, a given Normal will either always lie or always tell the truth), then C is a lying Normal (since he claimed to be a Philosopher). So in the first exchange B must be telling the truth, so he's the Mathematician, and in the second exchange B came first (he said so) and a Philosopher came first (from C). That is a contradiction, so: Normals are not consistent, so we can ignore C's statements above, which leaves A and B disagreeing with each other. So how did the logician know which one of them was the Mathematician and which the Philosopher? -- Benjamin Rhinelander Thompson <thompson@spot.Colorado.EDU> Alice knows the result of the election; the thing she did not know was the faculty of each candidate, until she asked C if he was a philosopher. Thompson is right: the answer must have identified C as Normal. From that and the knowledge of the result of the election, Alice could easily identify the faculty of each candidate. Alice's friend (as well as the puzzle reader) does not know the result of the election. Any result would have allowed Alice the identification of the faculties (if C came first, B tells the truth; otherwise, B lies), however, there are several outcomes which are compatible with the available information. Therefore, the last two paragraphs of the puzzle are needed. If (and only if) the puzzle teller, by his/her hypothetical question, can identify the follow-up runner as Normal (note that C has been already identified as Normal), identity and election result can be sorted out: B is a lying philosopher, coming third; A the honest mathematician won. BTW, knowing that the original poster works in a Computer Lab, it is easy to deduce that no one else than the mathematician can have won! -- Andreas Luckhoff <luckhoff@mayo.edu>