What is Each Man's Artistic Field?

Boronoff, Pavlow, Revitsky, and Sukarek are four talented creative artists, one a dancer, one a painter, one a singer, and one a writer (though not necessarily respectively).

1. Boronoff and Revitsky were in the audience the night the singer made his debut on the concert stage.
2. Both Pavlow and the writer have sat for portraits by the painter.
3. The writer, whose biography of Sukarek was a best-seller, is planning to write a biography of Boronoff.
4. Boronoff has never heard of Revitsky.

What is each man's artistic field?

(Source: 101 Puzzles in Thought and Logic by Clarence Raymond Wylie, Jr.)

Let's take stock of the available facts one by one and rule out what is impossible in a table.

"Boronoff and Revitsky were in the audience the night the singer made his debut on the concert stage" means that neither Boronoff nor Revitsky are the singer:

	Dancer	Painter	Singer	Writer
Boronoff			✗
Pavlow
Revitsky			✗
Sukarek

"Both Pavlow and the writer have sat for portraits by the painter" means that Pavlow is not the painter. It also means that he is not the writer:

	Dancer	Painter	Singer	Writer
Boronoff			✗
Pavlow		✗		✗
Revitsky			✗
Sukarek

"The writer, whose biography of Sukarek was a best-seller, is planning to write a biography of Boronoff" means that neither Sukarek nor Boronoff are the writer (on the assumption that neither biography is an autobiography, which turns out to be justified later):

	Dancer	Painter	Singer	Writer
Boronoff			✗	✗
Pavlow		✗		✗
Revitsky			✗
Sukarek				✗

The meaning of the fourth fact will come up shortly, but notice that everyone but Revitsky has been ruled out as the writer, making him the writer and excluding all other possibilities:

	Dancer	Painter	Singer	Writer
Boronoff			✗	✗
Pavlow		✗		✗
Revitsky	✗	✗	✗	✓
Sukarek				✗

Now we can come to the significance of "Boronoff has never heard of Revitsky". Initially, this confused me. The trick is to realize that this interacts with the second fact: "both Pavlow and the writer have sat for portraits by the painter". In this context, "sitting for a portrait" means that the painter has "heard of" the subjects of his paintings, something that doesn't seem to be the case necessarily. But, taking this into account, this means that Boronoff cannot be the painter, because the writer (i.e. Revitsky) has been a subject of the painter. This simultaneously entails that Boronoff is the dancer and excludes that field for everyone else:

	Dancer	Painter	Singer	Writer
Boronoff	✓	✗	✗	✗
Pavlow	✗	✗		✗
Revitsky	✗	✗	✗	✓
Sukarek	✗			✗

Boronoff's field being nailed down then means that Pavlow is the singer and excludes this for the only remaining individual for whom this possibility has not already been ruled out (Sukarek):

	Dancer	Painter	Singer	Writer
Boronoff	✓	✗	✗	✗
Pavlow	✗	✗	✓	✗
Revitsky	✗	✗	✗	✓
Sukarek	✗		✗	✗

The only remaining possibility is for Sukarek to be the painter, completing the table:

	Dancer	Painter	Singer	Writer
Boronoff	✓	✗	✗	✗
Pavlow	✗	✗	✓	✗
Revitsky	✗	✗	✗	✓
Sukarek	✗	✓	✗	✗

Bayesian Bags

There are 3 bags; one containing a white counter and a black one, another two white and a black, and the third 3 white and a black. It is not known in what order the bags are placed. A white counter is drawn from one of them, and a black from another. What is the chance of drawing a white counter from the remaining bag?

(Source: The Mathematical Recreations of Lewis Carroll: Pillow Problems and a Tangled Tale)

The key to solving this one—my way, not Carroll's way, at least—is knowing that, although the order in which the bags is placed is not known outright, the draws give evidence about what order might have been used and thus about what bag remains. More concretely, if $P(R = n \mid D)$, where $R$ stands for the remaining bag, $n$ is which bag it is, and $D$ stands for the particular draws observed in this case, can be determined for each value of $n$ from one to three inclusive, that is, the conditional probability that bag number $n$ remains, given that a white counter was drawn from one and a black counter from the other, then the problem is an issue of multiplying each respective probability by the more obvious probability of drawing a white counter from that possibly remaining bag, and then adding all three terms.

From Bayes' theorem, which, along with the concept of conditional probability, needs to be understood before going further, it can be known in this case that:

\[ P(R = n \mid D) = \frac{P(D \mid R = n) \times P(R = n)}{P(D)} \]

The first step towards going further is working out the numerators for each of the three probabilities. The denominator will be the sum of all those numerators. It will be assumed that any order of bags is equally possible. This means that the prior probability $P(R = n)$ is $\frac{1}{3}$ in each case. For the first bag to remain, one of two mutually exclusive events had to have happened, given the evidence: either a white tile was drawn from the second bag and a black tile from the third, or a white tile was drawn from the third and a black tile from the second. So in this case:

\begin{align*}
P(R = 1 \mid D) &= \frac{P(D \mid R = 1) \times P(R = 1)}{P(D)} \\
&= \frac{\left(\frac{2}{3} \times \frac{1}{4} + \frac{3}{4} \times \frac{1}{3}\right) \times \frac{1}{3}}{P(D)}
\end{align*}

As it has been said, $P(D)$ is not immediately relevant here. Another thing worth pointing out is that, because each prior probability is $\frac{1}{3}$, that value can be factored out of the summation that will take place in the denominator and then allowed to cancel with the same value in the numerator in each instance. So this problem is in fact the same as finding all the likelihood terms $P(D \mid R = n)$, then dividing each by the sum of all the likelihoods to find each posterior probability. Having said that, $P(D \mid R = 1)$ is equal to $\frac{5}{12}$. Repeating this process for bags #2 and #3 results in $P(D \mid R = 2)$ and $P(D \mid R = 3)$ both being $\frac{1}{2}$. This means that the sum of all the likelihoods is $\frac{17}{12}$. Then:

\begin{align*}
P(R = 1 \mid D) &= \frac{\frac{5}{12}}{\frac{17}{12}} \\
&= \frac{5}{17} \\
P(R = 2 \mid D) &= \\
P(R = 3 \mid D) &= \\
&= \frac{\frac{1}{2}}{\frac{17}{12}} \\
&= \frac{6}{17}
\end{align*}

Lastly, given the chain rule of probability, it can be known that:

\[ P(R = n, N = \text{white} \mid D) = P(N = \text{white} \mid R = n, D) \times P(R = n \mid D) \]

If, for example, $n$ is one, then the value of this equation is: $\frac{1}{2} \times \frac{5}{17}$, because the first bag contains one white and one black counter. Summing over all possible values of $n$ results in an expected value for the probability of drawing a white counter from the remaining bag (whether that one might be):

\[ \frac{1}{2} \times \frac{5}{17} + \frac{2}{3} \times \frac{6}{17} + \frac{3}{4} \times \frac{6}{17} \]

Or, rearranging:

\[ \frac{1}{17} \times \left(\frac{5}{2} + 4 + \frac{9}{2}\right) \]

So now it can be known that the final answer is $\frac{11}{19}$.

Chariot Race Winners (Outliers of Hyperborea Part IX)

The winners of chariot races were among the heroes of the Hyperboreans. Five such honored ones were discussing the number of chariot races they had won. Each has had more than three wins; no two have had the same number of wins; and each chariot racer's number of wins is divisible by three. The one with the most wins is the Grand Champion. Of the five chariot race winners, little is known except that exactly one of them is an Outlier.

From their statements below, what is the standard of veracity of each, how many chariot races did each win, and which one was the Grand Champion?

• Agathon
  1. Lysis is the Grand Champion.
  2. Protagoras has had 15 wins.
  3. Lysis is a Sororean.
  4. Phaedrus is not the Outlier.
• Lysis
  1. Sosias is the Outlier.
  2. Phaedrus is a Sororean.
  3. I am not the Grand Champion.
  4. Agathon is not the Grand Champion.
• Phaedrus
  1. I have had 18 wins.
  2. Agathon is the Grand Champion.
  3. Lysis is the third highest in the number of wins.
  4. Protagoras is the Outlier.
• Protagoras
  1. Agathon has had fewer wins than Lysis.
  2. Sosias has had 15 wins.
  3. Lysis is the Outlier.
  4. Agathon has had more wins than I have had.
• Sosias
  1. I have had more than six wins.
  2. Agathon has had more wins than Protagoras.
  3. Phaedrus is the Outlier.
  4. Phaedrus's third statement is false.

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

This is the last and, by far, the hardest of all the series. It took me 15 pages in my composition book to solve it, and a lot of mistakes were crossed out. But it can be solved! Let's go to it.

There may have been a better way to go about this, but I went very methodically and started by testing the assertion that each of the champions is a Sororean, starting with Agathon. Agathon can be ruled out quickly for the following reason: if Agathon is a Sororean, then Lysis is also a Sororean. But Agathon also claims to that Lysis is the Grand Champion, while Lysis claims not to be. While it is still possible that Lysis is a Sororean, Agathon is not.

Now on to Lysis. What if Lysis is a Sororean? Well if Lysis is a Sororean then so is Phaedrus. If Phaedrus is a Sororean, then Agathon is the Grand Champion. But Lysis claims that Agathon is not the Grand Champion, so Lysis is not a Sororean.

What if Phaedrus is a Sororean? Remember that this possibility is not ruled out by Lysis not being a Sororean. (Here's why.) Here things get a little more complicated. If all of what Phaedrus says is true, then Protagoras is the Outlier, meaning no one else can be, and Agathon's statements are false, true or false, true and false. Because the Outlier has already been chosen, this state of affairs is only possible if Agathon is a Midrorean whose statements are false, true, false and true. This means that Protagoras has had 15 wins. Then, evaluating the statements of Protagoras, they can be believed to be false, because Phaedrus claimed that Agathon is the Grand Champion; false, because each number of wins is unique and 15 has already been assigned to Protagoras; false, because Protagoras is believed to be the Outlier; and true, of necessity, because Protagoras would be a Nororean otherwise. This actually looks good because, under the previous assumptions, Agathon is the Grand Champion. But not so fast! What are the truth values of Lysis's statements? They are false, true, true and false, which makes Lysis another Outlier, when there is only one. Because the chain of reasoning from Phaedrus being a Sororean only had one possibility every step of the way, that chains rolls back all the way and Phaedrus is not a Sororean.

Now how about Protagoras being Sororean? Protagoras says that Lysis is the Outlier. Lysis's statements are then, under this assumption, false, false, either true or false, and true (because Protagoras claimed that Agathon had fewer wins than someone else). Not enormously helpful in itself. But what about Sosias's claims? They are either true or false, true, false, and either true or false. This is only possible if Sosias is a Midrorean whose statements are false, true, false and true. This then means that Sosias has had exactly six wins, which contradicts Protagoras's claim that Sosias has had 15 wins. Protagoras is not a Sororean either.

The last possible individual to consider in this case is Sosias. If Sosias is a Sororean, then Phaedrus is the Outlier. When I solved this initially, there were three ways for Phaedrus to be the Outlier, and I evaluated all of them painstakingly. But I looked for an easier way and found one when as I was writing this up: if Phaedrus is the Outlier, then Agathon's third and fourth statements are false. This is only possible if Agathon is a Nororean, and Lysis is then not the Grand Champion. This then means that Lysis's third statement is true. But Lysis's first statement is false. This is only possible if Lysis is another Outlier, but there can only be one. All of this means Sosias is not a Sororean.

At this point a sort of "theorem" can be derived: no one is a Sororean.

Now it's time to make less charitable assumptions. What if Agathon is an outright Nororean? If so, then Lysis is not the Grand Champion and so it can be inferred that Lysis's second statement is false and his third statement true. At first, this seems compatible with him either being the Outlier or a Midrorean whose statements are true, false, true and false. However, recall that, because Agathon is believed to be a Nororean, then Phaedrus is in fact the Outlier. This limits the possibilities for Lysis to only the latter case. But Lysis's first claim—putatively true—is that Sosias is the Outlier, which contradicts the earlier inference that Phaedrus is the Outlier.

So now, the somewhat more charitable assumption that Agathon might be some kind of Midrorean can be considered. Agathon can't be the true-false-true-false kind because his third statement is that Lysis is a Sororean, and no one here is a Sororean. The only other kind is false-true-false-true. All told this would mean that:

1. Lysis is not the Grand Champion.
2. Protagoras has had 15 wins.
3. Lysis is not a Sororean.
4. Phaedrus is not the Outlier.

Under these circumstances, Lysis's second statement is false and his third true. This assignment is compatible with Lysis either being a true-false-true-false Midrorean or the Outlier. If Lysis is a Midrorean of the type described, then:

1. Sosias is the Outlier.
2. Phaedrus is not a Sororean.
3. I am not the Grand Champion.
4. Agathon is the Grand Champion.

Sosias is believed to be the Outlier, but we're not going to turn his attention to him right now. Instead, we're going to look at Phaedrus. According to what we have so far, Phaedrus's second statement, that Agathon is the Grand Champion, is true. Because no one is a Sororean, and because the position of Outlier has already been assigned to Sosias, Phaedrus has to be a false-true-false-true Midrorean. However, this would mean that Protagoras is the Outlier when that role has already been assigned to Sosias.

The assumption that Agathon is a false-true-false-true Midrorean can still possibly be preserved—if and only if Lysis is the Outlier. (And, if Lysis turns out not to be the Outlier, then Agathon must be the Outlier.) Lysis's first three statements are, under prior assumptions, false, false and true, respectively. The truth or falsity of the fourth statement can be determined through much the reasoning that was done with Phaedrus in the paragraph before this one: briefly put, if Protagoras is not the Outlier, then Agathon is not the Grand Champion. This makes Agathon's fourth statement true if the assumptions check out. Not very useful yet, but it does complete the assignment of truth values to Agathon.

Now, let's turn our attention to Protagoras. If Lysis is the Outlier, then Protagoras's third statement is true, making him a true-false-true-false Midrorean. All told, Protagoras, when read critically, is saying:

1. Agathon has had fewer wins than Lysis.
2. Sosias has not had 15 wins.
3. Lysis is the Outlier.
4. Agathon has not had more wins than I have had.

Now we turn to Sosias. In the current framework, Sosias's second and third statements are both false. This requires Sosias to be a Nororean. If Sosias is a Nororean, then Phaedrus's third statement is really true, which means that Phaedrus is a true-false-true-false Midrorean.

So it looks like we have a complete assignment. Does it work? Can we finally answer the questions about these mysterious characters? Let's try. According to what we claim to know, Sosias is a Nororean. This means that Sosias has not had more than six wins. Additionally, because of how the problem is set up, every champion's number of wins is some number $3n$ where $n >= 2$ and $n$ is unique. The only number of wins Sosias can possibly have if we're right is six. Phaedrus's putatively true statements as a Midrorean are that he has had 18 wins and that Lysis is third highest in wins. One of Agathon's putatively true statements as a Midrorean is that Protagoras has had 15 wins. Altogether, Sosias has had six wins; Protagoras, 15; and Phaedrus, 18. From the truth of Lysis's fourth statement (and the falsity of Phaedrus's second), Agathon is not the Grand Champion. This means that Agathon's number of wins is less than 18. Indeed, it's less than 15, because 15 has already been assigned to Protagoras. On the other hand, it's more than six because of Sosias. It can only be nine or 12 by these facts, but is further constrained by the fact that Lysis was said to have been in third place. Lysis can take third place if he has 12 wins and Agathon has nine. All of this is fully consistent with all prior assumptions.

Final answer: Agathon is a Midrorean with nine wins, Lysis is the Outlier with 12 wins, Phaedrus is a Midrorean with 18 wins (Grand Champion), Protagoras is a Midrorean with 15 wins and Sosias is a Nororean with six wins.

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.

Four for the Races (Outliers of Hyperborea Part VII)

Four Hyperboreans are practicing for the upcoming chariot races. They are known to be a Sororean, a Nororean, a Midrorean, and an Outlier. They make the following statements:

1. 1. This race track is slow.
   2. D is doing so well in practice that he will win.
   3. C is the Outlier.
2. 1. A's first statement is true.
   2. I am the Midrorean.
   3. D is the Nororean.
3. 1. This race track is fast.
   2. I agree with B's second statement.
   3. I am not the Nororean.
4. 1. I am doing so poorly in practice that I will lose.
   2. B's second statement is false.
   3. A is the Sororean.

Which one is the Sororean, which one is the Nororean, which one is the Midrorean, and which one is the Outlier?

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

Consider the notion that B is the Midrorean. If so, then B2 is true and his other statements are false. This means that A1 is false and that D is not the Nororean. It follows from the first of these implications that A is either the Nororean or the Outlier because of the falsity of at least one of his statements and that the race track is in fact fast.

What if A is Nororean? Then A2 and A3 are also false. This means that D is not doing well in practice, not well enough to win anyhow and also that C is not the Outlier. Because, in addition to this, B is believed to be the Midrorean and A the Nororean, only D can be the Outlier. Also, C must be the Sororean under these assumptions.

Do the assumptions work? It can be seen easily that all of C's statements can be true. Additionally, D can be the Outlier if his statements true, false and false.

Final answer: A is the Nororean, B is the Midrorean, C is the Sororean and D is the Outlier.

Olive Picking (Outliers of Hyperborea Part VI)

Olives are an important staple in Hyperborea, and olive picking is an occupation engaged in by many inhabitants. Four olive pickers are having a discussion. One is known to be a Sororean, one is known to be a Nororean, one is known to be a Midrorean, and one is known to be an Outlier. Their statements follow:

1. 1. I picked more olives today than anyone else.
   2. I am the Midrorean.
   3. C is the Outlier.
2. 1. I am the Midrorean.
   2. I would have picked more olives today than A, except that I did not start until after lunch.
   3. C is the Outlier.
3. 1. D dropped more olives than he picked.
   2. I picked more olives today than A did.
   3. B picked olives all day today.
4. 1. C picked more olives today than A did.
   2. I did not drop any olives today.
   3. B would have picked more olives than A today, except that he did not start until after lunch.

Which one is the Sororean, which one is the Nororean, which one is the Midrorean, and which one is the Outlier?

The key to getting started on this one is to note that neither A nor B can be the Sororean because they each claimed to be something else. This leaves C and D. If C is the Sororean then A's first statement, at least, is false. The possibility that A might be a Midrorean with order FTF is tempting and we will run with it. This leaves either B or D to be the Outlier.

If B is the Outlier then his first statement is false and so is his third statement. This constrains his second statement to being true. If B is the Outlier then all that is left for D to be is the Nororean. However, the falsity of D3 is not compatible with the putative truth of B2, because the statement negated in one instance and accepted as true in the other is essentially the same. This means B is not the Outlier.

If D is the Outlier than his first and second statements are true and false, respectively. The truth value of the third can be determined by recognizing that it is now B who is constrained to be the Nororean. Because this entails that B2 is false, then so is D3. Additionally, B1 and B3 also check out as false statements, confirming this impression.

Final answer: A is the Midrorean, B is the Nororean, C is the Sororean and D is the Outlier.

One Is an Outlier (Outliers of Hyperborea Part V)

Outliers, although few in number, occasionally make their presence known. Hyperboreans are either Sororeans, Nororeans, Midroreans, or those few Outliers. As to the four inhabitants who make statements below, little is known as to their standards of veracity, except that exactly one of them is an Outlier.

1. 1. B is the Outlier.
   2. D is a Sororean.
   3. C is a Midrorean.
2. 1. I am not the Outlier.
   2. A is a Nororean.
   3. C is not a Midrorean.
3. 1. I am not a Midrorean.
   2. B is not the Outlier.
   3. D is a Sororean.
4. 1. I am a Sororean.
   2. B is the Outlier.
   3. A is a Midrorean.

To which group does each of the four Hyperboreans belong?

First, entertain the proposition that A is a Sororean. If A is a Sororean, then D is also a Sororean, but D tells us that A is a Midrorean, which is a contradiction. So A is not a Sororean.

What if B is a Sororean? Then A is a Nororean and the possibilities for Outlier are now restricted to either C or D. If C is the Outlier than his statements are, respectively, true, true and, by implication, false. (Three true statements in a row would make for a Sororean.) Lastly, all of D's statements can be assigned a value of false without contradicting anything else, making him a Nororean.

Final answer: A is a Nororean, B is a Sororean, C is a Outlier and D is a Nororean.

Olympic Games (Outliers of Hyperborea Part IV)

It is not commonly known that the original Olympic Games occurred not in Olympia, Greece, but in Hyperborea. Three Hyperborean Olympic athletes are discussing the results of the recent competition. Inhabitants of Hyperborea belong to three groups: Sororeans, Nororeans and Midroreans. There are also those few Outliers.

Their groups are unknown except that exactly one of the athletes is an Outlier.

1. 1. I was the winner of the one-half league run.
   2. You can count on what C says to be truthful.
   3. I am not a Midrorean.
2. 1. A did not win the one-half league run.
   2. I entered three events.
   3. C is the Outlier.
3. 1. I am not the Outlier.
   2. B's second statement is truthful.
   3. A did not win the one-half league run.

What are the groups of the three athletes?

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

This one actually unraveled for me pretty quickly.

If A is a Sororean then you can take him at his word that what C says is truthful, i.e., that C is also a Sororean. However this cannot be the case because A1 and C3 contradict.

A might be a Midrorean and in this case the order must be FTF and C is still a Sororean, still leaving B to be the Outlier. B's statements are, respectively, true, either true or false, and false. Whether B's claim that he entered three events is true or not is immaterial for the purposes of solving the puzzle; either a true or false assignment can leave him an Outlier.

Final answer: A is a Midrorean, B is the Outlier and C is Sororean.

One of Each (Outliers of Hyperborea Part III)

Among four Hyperboreans one is a Sororean, one is a Nororean, one is a Midrorean and one is an Outlier. They make the following statements:

1. 1. I am the Outlier.
   2. D is not more truthful than I am.
   3. B is the Sororean.
2. 1. A is the Outlier.
   2. C is not the Sororean.
   3. I am not the Midrorean.
3. 1. A is not the Outlier.
   2. B is not the Sororean.
   3. I am more truthful than D is.
4. 1. B is not the Nororean.
   2. A's second statement is false.
   3. C's third statement is false.

Which one is the Sororean, which one is the Nororean, which one is the Midrorean, and which one is the Outlier?

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

A obviously cannot be the Sororean. If that were the case, then he wouldn't have said he is the Outlier.

If A is the Midrorean, then the order is FTF. If, per A2, D is not more truthful than A is, then D is either the Nororean or the Outlier. Additionally, because A3 is false, then B is also either the Nororean or the Outlier, leaving only C to be the Sororean. All of C's three statements check out under this assumption. So far, so good, now on to the other two.

What if B is the Nororean? B1 and B2 check out as false but B3 does not. This means D is the Nororean then, right? Well that can't be the case either, because D1 is true and Nororeans always lie.

So the assumption that A is the Midrorean has to be thrown out now. How about A being the Nororean? A's statements can all feasibly be false. Also, the falsity of A3 means that B is not the Sororean, allowing only C or D to be Sororean.

What if C is the Sororean? All of C's statements check out. This means that either B or D is the Midrorean. B, however, cannot be the Midrorean because the only possible order is FTF and B2 is false under these assumptions. If D is the Midrorean, then the order of statements has to be TFT. However, this doesn't pan out either, because C3 is false.

So the assumption that C is the Sororean is necessarily false. This doesn't immediately defeat the assumption that A is Nororean. This assumption still has one more chance at life if D is the Sororean. Indeed, all of D's statements are feasible. All this leaves is who is the Midrorean and who is the Outlier.

Consider the possibility that B is the Midrorean. The order is constrained to FTF and this works out. Lastly C must be the Outlier. Under all the previous assumptions, C's statements are true, true and false, respectively, behavior befitting an Outlier.

Final answer: A is the Nororean, B is the Midrorean, C is the Outlier and D is the Sororean.

An Outlier (Outliers of Hyperborea Part II)

Four Hyperboreans are engaged in conversation. One is a Sororean, one is a Nororean and one is a Midrorean. The fourth speaker is an Outlier.

From their statements below, which one is the Sororean, which one is the Nororean, which one is the Midrorean, and which one is the Outlier?

1. 1. I am the Outlier.
   2. D is the Nororean.
2. A's first statement is true.
3. 1. I am not the Outlier.
   2. B is not the Midrorean.
4. C's first statement is true.

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

For me, the first assumption that seemed especially productive was assuming that A is the Midrorean. If A is the Midrorean then A1 is false and A2 is true. This means that D is the Nororean and, in turn, that C's first statement is false, making C the Outlier. The only remaining possibility for B is being the Sororean, but this cannot be the case because B said that A's first statement is true and it was assumed that A was the Midrorean.

So, now, consider the implications of A being the Nororean. A1 and A2 both check out as false. B's statement is false and so B is constrained to either being the Outlier or the Midrorean. Consider what happens if B is the Outlier. Both of C's statements check out, which makes C the Sororean. Lastly, D can feasibly be the Midorean.

Final answer: A is the Nororean, B is the Outlier, C is the Sororean and D is the Midrorean.