Sane knights believe true statements and say true statements. Sane liars believe true statements, but say false statements. Insane knights believe false statements and say false statements. Insane liars believe false statements, but say true statements. You meet two people, A and B. Each is either a sane knight, a sane liar, an insane knight, or an insane liar.
A:I believe that B is a knight.
B:A is insane.
A:We are both sane.
What are A and B?
Belief is a funny thing.
Consider two statements: "2 is prime" and "4 is prime" and let's see who can say what.
Sane knights can only say "2 is prime"
Insane knights believe that 4 is prime and 2 is not prime and so can only say "4 is prime"
sane liars know which numbers are prime, but lie and so can only say "4 is prime"
insane liars believe as insane knights, but lie and so can only say "2 is prime"
Now what happens with statements like "I believe 2 is prime"?
Sane knights *do* believe 2 is prime, so this statement is true and sane knights speak the truth
Insane knights do NOT believe 2 is prime (because it's true) so the statement is false. Since it's false, they believe it, and since they tell the truth, they say it.
So all knights say "I believe 2 is prime" regardless of sanity.
By a similar argument, all liars say "I believe 4 is prime". sane liars are lying about their belief, while insane liars don't believe they believe 4 is prime and try to lie and say they do.
The bottom line is that for any statement like "I believe X" X is true when a knight makes the statement and false when a liar makes the statement, regardless of sanity.
So when A says "I believe B is a knight" that means if A is a knight so is B and if A is not a knight, neither is B. So A and B are of the same type.
Now, we know that A isn't an insane liar (since A would believe that they're both sane and have to lie about it). But A also isn't a sane liar, otherwise B would be lying about A being insane and hence B would also be a sane liar, and then A's final statement would be true. So A isn't a liar which means A and B are both knights.
A and B differ in their assessment of A's sanity, and are both knights, so they must have different sanity. Then they are not both sane, and A's last statement is mistaken. So A is an insane knight, and B is a sane knight.
Let's double-check:
B is a knight, so we know that only knights can say "I believe B is a knight", and A is indeed a knight.
B is sane, and A is insane, so B's statement is indeed true.
A, insane, incorrectly believes A and B are both sane and so says so.
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Posted by Paul
on 2023-11-13 12:42:39 |
revised solution
