Six Hyperboreans (The Outliers of Hyperborea Part VIII)

Six Hyperboreans hold the jobs of chariot maker, fishnet weaver, musician, olive grower, tax collector, and wine maker. One is known to be a Sororean; two are known to be Nororeans; two are known to be Midroreans; one is known to be an Outlier. They make statements as follows:

• Agenor
  1. As olive grower I have the most important job.
  2. Cadmus is the chariot maker.
  3. Philemon's third statement is true.
• Alphenor
  1. Philemon is not the fishnet weaver.
  2. I find my work as tax collector to be very satisfying.
  3. Now that Hesperus is the chariot maker, we have been winning more races.
• Cadmus
  1. Agenor claims to be the olive grower, but that is my job.
  2. Alphenor is the tax collector.
  3. Hesperus is the chariot maker.
  4. Everything that Philemon says is false.
• Callisto
  1. The last tour I took with my lyre was so successful I intend to schedule another one.
  2. Everything that Agenor says is true.
  3. Cadmus is not the chariot maker.
• Hesperus
  1. We have been winning very few chariot races lately.
  2. Agenor is the wine maker.
  3. Cadmus is the tax collector.
  4. I am the fishnet weaver.
• Philemon
  1. I am the tax collector.
  2. Callisto is the musician.
  3. Hesperus's statements are all true.
  4. Cadmus is the olive grower.

Which Hyperborean is the Sororean, which are the Nororeans, which are the Midroreans, which is the Outlier, and what is the job of each?

(Source: Challenging False Logic Puzzles by Norman D. Willis)

The knowledge that only one of the six is a Sororean is very helpful. Callisto's second statement that everything Agenor says is true precludes Callisto from being the Sororean, because then there would be two Sororeans. The same applies to Philemon given his third statement. Additionally, Agenor saying Philemon's third statement is true precludes him from being the Sororean as well by extension. Already, three can immediately be ruled out as being the Sororean.

So what if Alphenor is the Sororean? Then Philemon is not the fishnet weaver—not enormously helpful yet. What is more helpful are the two assumptions that Alphenor is the tax collector and Hesperus is the chariot maker.

Under these circumstances, Agenor might be the Outlier whose statements are true, false and false, respectively. However, that constrains Cadmus to being a Midrorean whose statements are false, true, false and true, respectively, for the reason that Alphenor is believed to be the tax collector and the Sororean and Outlier positions have already been used up. But Cadmus cannot be such a Midrorean because this would mean that Hesperus isn't the chariot maker anymore. This also means that Agenor is not the Outlier.

But Agenor could still be one of the two Nororeans. In such case, Cadmus's second and third statements can both be true—if he is the Outlier. One or both of his first and fourth statements are then false, allowing three possibilities to test. Assume that the first is true but the fourth is false. This makes Cadmus the olive grower and here the pieces start to fall into place. I made a few other successful assumptions all at once after this and I don't know exactly how they came to me, but what really matters is that they can be justified logically.

The notion that Cadmus is an Outlier whose fourth statement is false allows some of Philemon's statements to be true, but only as a Midrorean—remember that Sororean and Outlier have already been taken up. Because Alphenor is believed to be the tax collector, Philemon's first statement is false, so Philemon's statements are false, true, false and true respectively, if Midrorean. This is consistent with the assumption that Cadmus is the olive grower, and it also means that Callisto is the musician, as well as that at least some of Hesperus's statements are false—which follows from the assumption that Alphenor is the Sororean—and that Philemon is not the tax collector, which also follows from the belief that Alphenor is the tax collector.

Callisto then might well be a Midrorean whose statements are true, false and true respectively, meaning he is the musician. It also means that Cadmus is not the chariot maker—Cadmus is believed to be the olive grower—and that at least some things Agenor says are false—Agenor is claimed to be a Nororean.

Only Hesperus is unassigned but, fortunately, only one possibility is left for him: to be a Nororean. Does this final assumption work out? Well, Alphenor is believed to be the Sororean and has said that they've been winning a lot of chariot races lately. Hesperus says otherwise, as would be expected. The third statement is also false under the prior assumptions because Cadmus is believed to be the olive grower. The falsity of the other two statements is less obvious, but it is nevertheless possible now to attempt a complete assignment of who's who.

Up to now, Alphenor, Cadmus, Callisto and Hesperus are believed to be tax collector, olive grower, musician and chariot maker, respectively. Remember that Alphenor also tells us, presumably truthfully, that Philemon is not the fishnet weaver. Hesperus claims to be the fishnet weaver, but Hesperus is believed to be a Nororean and under this assumption is lying. Then Agenor is the fishnet weaver. All that remains for Philemon to be is the wine maker. These assignments are fully consistent with all previous assumptions. For example, Agenor being the fishnet weaver is consistent with him being a Nororean and then saying he is the olive grower. The others can be checked in a similar fashion.

Final answer: Agenor is a Nororean and the fishnet weaver, Alphenor is the Sororean and the tax collector, Cadmus is the Outlier and the olive grower, Callisto is a Midrorean and the musician, Hesperus is a Nororean and the chariot maker and Philemon is a Midrorean and the wine maker.