# King cake with shy guests

Very distant inspiration from what a family member asked me some years ago on epiphany, this might be a classical logic exercise, but due to the flavor it's quite hard to find the same problem under a different formulation, let alone confirm an original. Additional note: This is a fairly easy one, and intended as such, more of the kind you can retell to your friends after you find the solution

Alice, Bob and Charlie meet on epiphany to eat the traditional king cake. The rule says that the youngest goes under the table and name the recipient of each piece as they are cut. However, our three guests are very shy, and revealing their age is unthinkable as none wants the other to know if they are the elder of the group. They've never shared it before between themselves, it's not going to change now. The cake is slowly getting colder, and we're not anywhere close to finding out who's the youngest. So, they agree to call Zorro, a common friend, only problem being that Zorro is logician, and he will not reveal who is the youngest without a little twist of his own. Instead, he asks to speak privately with each one of them and this is what he said:

• "I can tell you whether or not you are the oldest", to Alice;
• "I can tell you whether or not Charlie is older than Alice", to Bob;
• "I can tell you whether or not you are in the middle", to Charlie.

Zorro tells each one of them their piece of information (for example now Alice knows if she is the elder or not) and hangs up. Very satisfied with his trick, he is confident that after a little discussion they will get to eat the king cake hot enough and without anyone feeling uncomfortable. Keeping in mind that Zorro knows the ages of Alice, Bob and Charlie, is Zorro a good logician?

Every agent in this problem is rational, they all want to please their friends so they won't voluntarily reveal the oldest one just for the fun.

If there is a way for them to succeed then they will succeed (i.e. in some configurations if they can mess up but also succeed, then they will do the latter). The power of friendship will guide them.

Order doesn't matter, each one chooses to reveal or not their piece and they all share it at the same time and the discussion stops. After the exchange, in order to send someone under the table, every one should have the certitude of who's the youngest but no one should have the certitude that someone knows they are the oldest.

• If there's still loopholes, feel free to point them out, preferably under the question, or mention me if you do it under an answer Commented May 2 at 12:31
• @PDT I feel like your second point is answered by my last note, "every one should have the certitude" means that. And the first point I don't agree, this is how the question is intended to be asked. Commented May 2 at 12:41
• I guessed that by ‘a good logician’ or not you just meant if his trick works or not. And really what the question is asking is ‘under what circumstances does his trick work.’
– PDT
Commented May 2 at 12:46
• @PDT yeah there is an extra step in proving if he is a good logician. No need to reveal it beforehand Commented May 2 at 12:47
• My point is the definition of ‘good logician’ is under specified.
– PDT
Commented May 2 at 12:50

If the arrangement is

ABC or BAC

And

Only Alice doesn’t state her information.

Then:

Since if ABC, B states C is not older than A and the configuration could be ABC, BAC, ACB for their minds. C states that C is not in the middle and the only possible configurations are ABC or BAC. This way they know C is the youngest however A and B’s position is not known.

Or

If BAC they will know C is younger than A and is not in the middle. This reveals his position. Although A would know B’s age B would not be embarrassed because he wouldn’t know this fact and for C it can either be BAC or ABC in his eyes.

Then

Zorro is a good logician.

Zorro is a not good logician if the youngest be:

BOB or ALICE because ACB means that they would work out A is the oldest if they were to work out B even if A doesn’t state her information- one has to know C is younger than A and in the middle to know B. For CAB there are insufficient clues for the position B. If BCA it has to be known that A is younger than C and C is in the middle but then they can deduce B is the oldest. Finally if CBA it means again A has to be known to be younger than C but still A’s position is not specified.

• I edited in some clarification, Zorro tells them what he can answer for each one privately and answer that question for each one (what i meant by piece of information), also privately. Keep in mind that some might not want to share their piece Commented Apr 30 at 11:10
• I don't follow how BAC is a valid arrangement. Everyone learns that Charlie is the youngest, and Alice learns that Bob is older than her, therefore the oldest. But if I understand correctly, we require that no one other than the oldest person can figure out who is the oldest. Commented Apr 30 at 18:21
• @PDT I see, it's not that no one can know who's oldest, it's that no one can know that anyone else knows who's oldest. But if the solution is BAC, can't Alice simply blurt out "Bob is oldest!" once the cake is cut and ruin Zorro's careful logic? "None wants the other to know if they are the elder" - Alice doesn't care about concealing the fact that someone else is oldest, only herself. Commented Apr 30 at 19:56
• Another issue with BAC is that from Alice's perspective, she has no problem immediately stating her bit of information that she is not the oldest (I see no particular reason she should wait for Bob or Charlie to speak first). But if she does that, once Charlie is revealed as the youngest, everyone knows Bob is oldest. There is no reason Zorro should rely on Alice remaining silent in BAC, he can only count on that with ABC. If the solution is BAC, the players can obey all the rules and accidentally wind up making it impossible to cut the cake without revealing the eldest to everyone. Commented May 1 at 12:31
• @NuclearHoagie I think the OP was intending that there has to be a unanimous agreement made before they send anybody under the table. They therefore all have to know who is the youngest before they send him under the table. If you ask me I feel the question is in desperate need of an edit.
– PDT
Commented May 1 at 19:31

If they can trust Zorro has done his work well,

at the end, everyone knows who is the youngest.

No matter the information given (to Bob), Bob can conclude he is not the youngest. This is because who is the youngest will become common knowledge, thus known by him, and then he will know who is the oldest. -- Bob can share this conclusion to everyone, no matter what the others are told, since it will become common knowledge anyway. (And thus, even if he is the oldest, sharing it will not tell anyone he is the oldest, if the logician did his job well).

Charlie will not be told he is the middle one, since then at the end he will know the eldest. -- However, he cannot tell he is not the middle one as long as there is a chance he is the oldest. (Since the youngest then will find out he is) -- Thus, Charlie cannot say anything safely.

If Alice is the oldest, she can safely share she is not the youngest. (since that will become common knowledge anyway) -- If Alice is not the oldest, she may still be the middle one, but only if the logician made an error (since she will learn who is the youngest (not herself) and then herself not being the oldest will tell her who the oldest is). So, in that case she can conclude (and tell) she is the youngest.

Thus, assuming Zorro did his work right, there are two possibilities:

1. Alice is the oldest, and Charlie the youngest. Then, after Bob and Alice both have told the others they are not the youngest, they can start the process of dividing the cake happy.
2. Alice is the youngest and says so, Charlie is the oldest.

• The second possibility doesn't work, there is no way for Alice to figure out she's youngest with CBA. All anyone knows is the Charlie is older than Alice, and that Charlie isn't in the middle (Alice not being oldest is redundant with knowing that Charlie is older than her). CBA and CAB both satisfy 1) Charlie is older than Alice and 2) Charlie isn't in the middle. No one can figure out if Bob or Alice is youngest, and they'll out Charlie as oldest along the way anyhow if it's revealed that he's both 1) not the middle and 2) not the youngest. Commented May 2 at 13:59
• @NuclearHoagie : I have to disagree. I don't think I can explain better than above, but CAB would result in A knowing C is oldest, and thus is impossible, assuming Zorro did his work properly. Commented May 2 at 20:18