Succinct Answer: All members of the Wisdom Council will correctly identify their dot color at Dinner. Shocked at his abrupt fall from power, PHB retreats to the countryside as a commoner, where he spends the rest of his days musing to himself:

"For a moment, wasn't I the king. If I'd only known how the king would fall..."

Explanation: There are two sources of "hidden" information that allow the members of the Wisdom Council (as perfect logicians) to solve the puzzle:

 I. PHB's statement that at least one of them has a Red dot.
II. The statement (or lack thereof) that each member of the Wisdom Council may make at each meal.

In terms of hidden information source (I): While PHB's statement appears at first glance to be innocuous -- since all members already knew that there was at least one Red dot (since they each could see the two Red dots on the other members' foreheads) -- it turns out that it provides the key piece of information that allows them to identify their own dot color. Understanding why this apparently information-less statement actually does provide information is no simple task. To begin with, we observe that the possible rainbow of colors that PHB can use is a deliberate complication that is actually meaningless. Instead, we can simplify the riddle by only introducing two possible colors (say, Red and Black), whereby here "Black" can be thought of as a replacement for "Not Red."

Next, we enumerate all possible scenarios (the "universe" of possible events, in terms of thinking about this mathematically):


Scenario 0: 0 Red dots, 3 Black dots.
Scenario 1: 1 Red dots, 2 Black dots.
Scenario 2: 2 Red dots, 1 Black dots.
Scenario 3: 3 Red dots, 0 Black dots.


(Scenario 0 is not possible based on PHB's hint that there is at least one Red dot, but let's ignore that for the moment.)
For each scenario, we can separate out the perspectives of the people with Black dots versus the perspectives of the people with Red dots. Since PHB's hint references people with Red dots, we'll focus on everyone's perspective on the total number of Red dots.


For Scenario 0: Everyone has Black dots, so everyone sees 0 Red dots.
For Scenario 1: The person with a Red dot sees no other Red dots; while the two people with Black dots each see 1 Red dot.
For Scenario 2: The two people with Red dots each see one other person with a Red dot; while the person with a Black dot sees 2 Red dots.
For Scenario 3: Everyone has Red dots, so everyone sees 2 Red dots.


The key point here is that there is a separation between the perspectives of people with Red dots versus people with Black dots. (This separation is perhaps hard to see, since half of the scenarios -- namely, Scenarios 0 and 3 -- only have one dot color involved; but the separation becomes more apparent as you increase the number of members on the Wisdom Council.) Namely, the people with Red dots on their own foreheads observe one less Red dot than the people with Black dots observe. It turns out that PHB's hint actually emphasizes this separation, with the property that it allows people with Red dots on their own forehead to more quickly identify their own dot color than people who have a Black dot on their own forehead. To understand why this is the case, we consider the following simplification:


Imagine that PHB had only painted one Red dot (i.e. this is Scenario 1). Then with PHB's hint that there is at least one Red dot, the person who has the Red dot can see two Black dots, and thus immediately know they must have the Red dot, and assert this knowledge at the first opportunity (Breakfast). Meanwhile, the two people with Black dots each observe one Red dot and one Black dot, and are unable to identify their own dot color at Breakfast time.

We can see above how PHB's hint can be leveraged differently by people with Red dots versus people with Black dots; and it is precisely this initial "separation" that kicks off a cascade of information, which brings us to the second key source of "hidden" information...

In terms of hidden information source (II): The iterative nature of the riddle -- in terms of having distinct "rounds" in which the members simultaneoulsy pronounce (or not) their dot color -- provides the second key piece of information. Namely, we observed above the separation of people with Red dots versus people with Black dots: those with Red dots will observe one less Red dot than those with Black dots. Of course, members don't know which group they are in -- indeed, this is the whole point of what they're trying to figure out.
As another simplification to the puzzle, suppose that at each meal, if no member of the Wisdom Council put forth a guess as to their own dot color, PHB provided an additional hint. Specifically, PHB's initial hint was "There is at least one Red dot." Suppose that after Breakfast, when nobody guessed yet, PHB provided the hint: "There are at least two Red dots." Then at Lunch, when still nobody provides a guess, PHB again offered more information: "There are at least three Red dots." In this case, it is easy to see (since there are only 3 members) that every member must have a Red dot, and would correctly state this by the Dinner meal.

While the riddle doesn't have PHB explicitly adding these hints at each meal, it turns out that effectively, that is what is happening. To see why this is true, we need to come back to the key observation above, which is that the perspective of people with Red dots is different than the perspective of people with Black dots - namely, people with Red dots on their own forehead see one less Red dot on the foreheads of the others. Returning to the simplification above, and thinking about the enumerated scenarios: Notice that for Scenario 1 (just one Red dot): The person who has the one Red dot will be able to declare their dot color at Breakfast, while the people who have Black dots won't know until Lunch. Similarly, suppose we are in Scenario 2 (two Red dots): Then nobody will know their dot color at Breakfast, but then with PHB's (hypothetical) updated hint "There are at least two Red dots", then now at Lunch both the people with Red dots will be able to guess their dot color (since they each observe one Red and one Black dot; and hence know they must have the 2nd Red dot); while the one person with a Black dot still won't know at Lunch if they are Red or Black.
The pattern here is that Red-dotted members will learn their forehead color one meal before the Black dotted members. This in turn effectively makes the absence of anyone guessing at a meal provide the same information as if PHB had updated his hint by going one higher in the minimum number of Red dots that he painted.

While this intuition is more formally captured in the discussion below, we first solve the riddle directly, i.e. providing a direct argument that does not (explicitly) rely on understanding the full generalized solution.
Imagine you are the first member on the Wisdom Council. Use 'B' to denote Black dot, 'R' to denote Red dot, and then looking around at the other two members, denote your view of the universe as:


{?, R, R}

where '?' represents the fact that you are not certain of whether your own forehead is 'B' or 'R' (but you are certain that the other two members have a 'R' dot). I.e. you know there are only two possible scenarios:


Scenario 2: {B, R, R}
Scenario 3: {R, R, R}

Next you conduct the following thought experiment:

"Let's suppose PHB put us in Scenario 2, so that my own dot color is Black: {B, R, R}. In this case, what would the other two members of the Wisdom Council observe?"

For example, thinking about the perspective of the 2nd member, if we were in Scenario 2, then they would see: {B, ?, R}. But then this 2nd member would be able to reason as follows:

"If I am Black, then we would be: {B, B, R}. And in this case, Member 3 would proclaim knowledge at Breakfast that they know they must be Red."

And so Member 2 would have the above thought, and wait for Breakfast. And when the third Member does NOT proclaim to know their own dot color, Member 2 knows that as a group they are not {B, B, R}. In particular, Member 2 would be able to proclaim at Lunch that they must be "R", not "B". Thus, at Lunch, Member 2 would proclaim their dot color as Red.

As Member 1 of the Wisdom Council, you have reasoned this all out in your head. Namely, you know that you are in either scenario {B, R, R} or {R, R, R}, and you recognize that if you are in scenario {B, R, R}, that Member 2 (and indeed, also Member 3) will be able to declare at Lunch that they know their own dot color. Since Lunch comes and goes without anyone saying anything, you recognize by Dinner that you must not be in Scenario 2: {B, R, R}, and so you can safely proclaim (as will Members 2 and 3) that you know your own dot color must be R.

*****
While the above argument just involves Red and Black dots, it turns out that using "Black" as a stand-in for "Not Red", the same argument will hold true even for the actual scenario where the six basic colors of the rainbow could be used: {Red, Orange, Yellow, Green, Blue, Purple}. Indeed, this riddle can be further generalized in three ways:
1. Any number of colors (e.g. not limiting to the six rainbow colors) can be available for PHB to use.
2. Any number of Wisdom Council members are possible; the only difference being that the number of opportunities members have to assert their own color (this was 3 in the original riddle, for the 3 meals in a day) must (at least) match the number of members.
3. PHB need not paint all foreheads the same color; indeed, he can paint them any distribution of colors he wants (the hardest case is the case of using all the same color; all other cases will lead (some of) the Wisdom Council members to learn their own color sooner). The only condition is that PHB publicly announces one of the colors he used.
The first generalization should be apparent via the logic above about replacing all non-Red colors by "Black" (a.k.a. "Non-Red"); where Red is special as the color announced by PHB. For the other two generalizations, another way to reason about the answer is via an inductive argument, based on the following statement:

Lemma: If there are exactly R people with a Red dot on their forehead, then all the Red-dotted people will correctly identify their dot color in the Rth round.

To verify the truth of the above statement, we use induction (if you are not familiar with inductive arguments -- you should learn! -- the idea is to prove the statement for some initial "base" value, and then argue that as soon as the statement is true for one value, it is also true for the "next" value, and so on forever). So, starting with the base case of R = 1 (since R = 0 is impossible, given PHB's statement that at least one person has a Red dot):


Base Case (R = 1): If exactly one person has a Red dot, then that person will hear PHB's statement, look around to see nobody else has a Red dot, and conclude the Red dot must be on their own forehead. Thus, they will correctly identify their dot color in the 1st round.

Induction Step (R > 1): Fix any R > 1, and assume by induction that the Lemma is true for all values of r < R. Namely, for any r < R, the Lemma states that if there were r Red dots in total (for r < R), then all those r people with a Red dot on their forehead would be able to correctly identify their dot color in round r (and consequently, any non-Red dotted-foreheads will be able to identify their dot color the very next round). Namely, we can assume the lemma is true for r = (R-1), so that if there are (R-1) Red dots, then those members with a Red dot on their own forehead will be able to announce knowledge of their own dot color in round (R-1). Now suppose that there are R people with a Red dot. Then round (R-1) comes and goes, and nobody is able to announce their own dot color. This indicates to everyone that there must be more than (R-1) Red dots in total. But then all the people with Red dots on their own forehead will look around, and observe that there are only (R-1) dots that they can see on everyone else's forehead, and conclude that their own forehead must have a Red dot in order to bring the total up to (at least) R. Thus, in the Rth round, all people with a Red dot on their forehead will correctly identify their dot color as Red.

*****
One interesting thing about this riddle (all variants) is that the information that is revealed by PHB at the start of the contest (that there is at least one Red dot) appears at first glance to be completely innocuous, with no actual information conveyed (that the Wisdom Council members didn't already know anyways). For example, the statement PHB made is essentially: "At least one of you has a Red dot." But since they all have Red dots, they can each see that the other two have Red dots, and thus each member already knew that at least one of them had a Red dot. So at its surface, it doesn't seem like PHB said anything of value, since they already knew the truth of his statement. However, the information that PHB provided is nonetheless crucial: without it, the members of the Wisdom Council would never have been able to guess their own dot color, leaving Einstonia to be ruled by PHB (of course, as perfect logicians, the Wisdom Council never would have agreed to accept this challenge in the first place had they not been guaranteed that PHB would provide them with this crucial hint).

Succinct Answer: Yes, take the host up on his offer, and declare that your friend did not even clear the first obstacle.

Explanation: Each time you (and your friend) are presented an opportunity by the host, extra information is revealed (to the person who was not given the opportunity) based on the fact that you each decline the offer:

1. First, you are given the opportunity to claim you made it further than your friend. If you had cleared all 3 obstacles, you would know your friend hadn't (since otherwise you both would've earned the Grand Prize for each clearing the course). Therefore, you would've taken the host up on his offer. Since you don't, your friend learns that you must not have cleared all 3 obstacles.
2. Next, your friend is given the opportunity to claim they made it further than you. If they had cleared all 3 obstacles, they would know that you hadn't (again, since otherwise you both would've earned the Grand Prize for each clearing the course). Therefore, your friend will take the host's offer if they cleared all 3 obstacles. Since they don't, you learn that your friend did not clear all 3 obstacles.
3. Next, you are presented the offer to still win your money back if you know you at least tied. Since your friend rejected their previous offer, you know they cleared at most 2 of the 3 obstacles. Therefore, if you also cleared (at least) two obstacles, you would take the host up on his offer. Since you don't, your friend learns that you must not have cleared two obstacles.
4. Next, your friend is given the opportunity to claim they made it at least as far as you. By the previous point, your friend knows you cleared at most 1 of the three obstacles. Therefore, your friend would take up the host's offer if they had cleared even 1 of the three obstacles. Because they fail to do so, you know that your friend must not have cleared any obstacles.

As a result, you can safely reason your friend cleared none of the obstacles, and get your money back when the host makes his final offer.

This riddle, as with the two other versions of this riddle presented on this page, can be generalized to a higher number of (in the present version of the riddle) obstacles. The only thing that needs to happen is that you and your friend need to continue to be offered alternating opportunities to claim that they made it further (or at least as far) as the other. There is an iterative argument, that each time you and your friend reject their opportunity, that you each can be certain that the other cleared one fewer obstacles.

See the Answer section of the "Colored Forehead Dots" version for more explanation.

*****
One interesting thing about this riddle (all variants) is that it needs some extra "catalyst" information in order to kick off the iterative argument/process. For example, in this version of the riddle, the catalyst is the knowledge that:

"At least one of you failed to clear at least one obstacle."

(This extra information is indirectly available via the failure to win the Grand Prize, which you both learn when the race concludes). At first glance, this key piece of information appears to be redundant, with no actual information conveyed: Both you and your friend already knew that at least one obstacle was not cleared (namely, since you each failed to clear even the first obstacle). However, the introduction of this piece of information (again, indirectly available through the failure to win the Grand Prize) is nonetheless crucial to the riddle: without it, neither you nor your friend would have been able to ever learn anything about how many obstacles the other had cleared.

For example, since you both failed all three obstacles, you both already knew the truth of the statement: "At least one of you failed to clear at least one obstacle." So at its surface, it doesn't seem like this piece of information adds any knowledge or changes anything. But this piece of information does tell you (and your friend) something that you did NOT already know: Namely, that the other person now necessarily also knows that there was at least one failed obstacle. For example, if your friend had cleared every obstacle, then before learning whether or not you collectively won the Grand Prize, your friend wouldn't know if the statement "There was at least one failed obstacle" was true or not. But from the moment it becomes public knowledge that you did not win the Grand Prize, now you know with certainty something about your friend's knowledge: namely, that your friend also definitely knows that at least one obstacle was not cleared. This turns out to be the crucial catalyst that allows you to earn your money back. Namely, the thing you learned from the fact that you failed to earn the Grand Prize is not that at least one obstacle was not cleared (which you already knew anyway); rather, the thing you learned is that your friend also is necessarily aware of the fact that at least one obstacle failed to be cleared.

For example, consider the statement: "I know if my friend knows that there was at least one failed obstacle." Then the this statement is false before you each learn that you failed to win the Grand Prize, but the statement becomes true as soon as you both learn you didn't win the Grand Prize. So the information that you (and your friend) learn by not winning the Grand Prize has to do with what your friend (necessarily) knows; and this in turn adds to your own knowledge. And then with each declined opportunity to gamble $10 to earn your money back, by both you and your friend, the information cascades, eventually leading to you both learning that the other has failed to clear all obstacles.

Another way to think about this is as follows:

• After you decline to gamble $10 at your first chance, your friend knows you didn't clear all 3 obstacles.
• After your friend declines to gamble $10 at his first chance, you know that your friend didn't clear all 3 obstacles.
• After you decline to gamble $10 at your second chance, your friend knows you didn't clear (at least) 2 of the 3 obstacles.
• After your friend declines to gamble $10 at his second chance, you know that your friend didn't clear any of the 3 obstacles.

So the key initial information that is made public (that not all obstacles were cleared), as well as the information that is provided by each declined opportunity to win money back, each provide a little bit more information about what the other person knows about the opposite person's performance. Namely, without the initial information about failing to win the Grand Prize, each person has no idea what the other person might know about their own performance. This then would mean that nothing is learned by the fact that either of you turn down the opportunity to gamble $10 to get back your initial $5 payment. For example, even if you had cleared all 3 obstacles, you wouldn't necessarily know that your friend didn't also clear them all, and so you would've declined the host's initial offer of the $10 gamble. But after both you and your friend learn the Grand Prize was not won (and you each know the other person also has learned the Grand Prize was not won), your friend can now gain information by the fact that you do decline the host's first $10 gamble (and likewise you gain knowledge when your friend declines).

Succinct Riddle:

1. You are on an island of 100 natives, whose eyes are always either brown or blue.
2. The natives are perfect logicians.
3. The natives become immediately hostile if they discover anyone has different eye color than themselves. As a result of this, long ago Priestess Half-Blind (PHB) implemented strict rules to never discuss eye color, nor learn the color of your own eyes.
4. If natives do learn the color of their own eyes, they must immediately drink the poisoned water of the Murky Pool, which (if they survive) will blind them and turn their eyes white.
5. Every morning, the natives gather for their morning ritual, where they take turns looking at all the others in the eyes and extending their customary greeting "you are beautiful."
6. As it turns out, all 100 natives happen to have brown eyes (not blue).
7. Your eyes are blue.
8. After the morning ritual, on the day you depart the island, you say goodbye to the natives:

   "Thank you, my new friends, for your hospitality. I will remain forever grateful to you for keeping me alive this past year, and for the wisdom of PHB, for I know that not all of you would have welcomed me here, were it not for her."

What will happen, and when will it happen, to the island natives after you depart.