Knights, knaves, and birthdays

If Zahara says It's Xeno's birthday, and we know only 1 person is lying, then it's Yves birthday?

Hi Winston, If Zahara says that it's Xeno's birthday, then it could be actually Xeno's birthday (and 2 people are lying). We don't know whether 1 or 2 people are lying, so Zahara saying it's Xeno's birthday isn't sufficient information to know whose birthday it is. (Therefore, Zahara didn't say it was Xeno's birthday.)

But in the problem you said you will tell the number of truth statements before revealing Zahara's statement?

No, in the problem you are only told that the number of true statements is not sufficient to determine who's birthday it is. Now that you know that fact, it is sufficient to know what Zahara said. (If at the beginning of the puzzle, you were told what Zahara said, that would not be good enough. Perhaps Zahara said it was Yves' birthday and all three statements were true.)

My interpretation was the same as Winston's.

From an information standpoint, you first say that we are told how many true statements there are, and that is not enough information to determine whose birthday it is. As you correctly note, this implies that we are neither told that the number of true statements is 3 or 0. Thus we must be told that there is exactly one true statement or exactly two true statements.

If we are told there are exactly two true statements, the birthday could be Yves' or it could be Zahara's. It cannot be Xeno's birthday, since that would falsify both Yves and Zahara. It could be Yves' birthday if Zahara said "Xeno" or "Zahara". Or it could be Zahara's birthday if Zahara said "Zahara". If the second piece of information provides enough information, then Zahara could not have said "Zahara". If Zahara had said "Zahara" we would not have enough information even at the end to determine whose birthday it was. But if Zahara said "Xeno" and we knew two statements were true, then we have enough information to determine that it was Yves' birthday.

But we can be told that there was exactly one true statement. Clearly this is not possible if it's Yves' birthday. However, it is possible if it is either Xeno's birthday or Zahara's birthday.
If it's Xeno's birthday and Zahara says it's Xeno's birthday, we have exactly one true statement. (Zahara's) If it's Zahara's birthday and Zahara either says it's Yves' birthday or Xeno's birthday, we have exactly one true statement. (Xeno's) Thus if we are first told that there is exactly one true statement, and we then are told that Zahara's information is sufficient to determine the birthday, Zahara must have said that it's Yves' birthday. Because Zahara saying 'Xeno' would not have provided information to split these two possibilities. Thus if Zahara's information is sufficient to narrow the possibilities down to one, it is Zahara's birthday.

So, based on the sequence of events, with no information either about how many true statements there were, or what Zahara said, we don't have enough information to determine whose birthday it was. We might have had two true statements + Zahara saying "Xeno", in which case it's Yves' birthday. Or we might have had one true statement, and Zahara saying "Yves", in which case it's Zahara's birthday.

I now see what you intended to say, but your language was not really clear.

"If I told you how many true statements were made, then you still wouldn't know whose birthday it is." This is clear.

"Now that you know that , if I told you what Zahara said then you'd know whose birthday it is."

Winston and I took "that" to mean "how many true statements were made". But apparently you meant "that" to only mean "the number of true statements isn't sufficient."

I like your solution, it's very concise!

Xeno says, "It's Yves' birthday!"
Yves says, "It's not Xeno's birthday!"
Zahara whispers, "It's _ 's birthday", but so quietly that you can't make out which of the three names she said.

If I told you how many true statements were made, then you still wouldn't know whose birthday it is. Now that you know that, if I told you what Zahara said then you'd know whose birthday it is.

X and Y aren't saying contradictory things here, but you might have 3 different situations depending on the truth values of their statements.

Birthday boy Xeno --> two lies FF --> possibilities of 0 or 1 true statement(s).
Birthday girl Yves --> two truths TT --> possibilities of 2 or 3 true statement(s).
Birthday girl Zahara --> one of each FT --> possibilities of 1 or 2 true statement(s).

Thus, it can't be FFF nor TTT or we'd know whose birthday it was straight away, which would be Xeno's or Yves' respectively. Then, the number of true statement made must be either 1 or 2. Even then, you cannot distinguish further whether the birthday person is X or Z for 1 true statement made or Z or Y for 2 true statements made.

But we're told that Z's statement is crucial information for decoding whose birthday it was, so let's look into that deeper.

We already agreed that no variation possibilities in the truth values of their 3 statements is invalidated by the "If I told you how many true statements were made, then you still wouldn't know whose birthday it is" statement made by the poster, so we only have to consider those of 1 or 2 truths cases.

Case 1 :
If it was X's birthday, Z must have told the supposed only truth and uttered "X".

Case 2 :
If it was Y's birthday, Z must have told the supposed only lie and uttered "X" or "Z".

Case 3 :
If it was her own birthday, Z could have said anything, a supposed truth of "Z" or supposed lies of "X" or "Y" so that the total truths can still remain 1 or 2.

So, how could this help determine whose birthday it was? It lies in the possible replies by Z. There would be multiple possibilities for replies of "X" (3 in total) or "Z" (2 in total), but an answer of "Yves" by Zahara is uniquely hers (just for her own birthday).

Answer : It's Zahara's birthday by the false statement "Yves".