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.