Was Humpty Dumpty right?

Raymond Smullyan's What is the Name of This Book? contains a puzzle which I'll paraphrase here:

• Of the identical twins Tweedledum and Tweedledee, one of them lies on Mondays, Tuesdays and Wednesdays and tells the truth on the other days of the week; the other lies on Thursdays, Fridays, and Saturdays and tells the truth on the other days of the week.
• One of the twins owns a rattle; the other does not. Alice wants to restore the rattle to its rightful owner.
• Alice comes across one of the twins and asks him: "Who really owns the rattle?"
• He replies: "The true owner of the rattle is telling the truth today." [Edit: this previously incorrectly said "is lying today."]
• Humpty Dumpty claims that the probability this twin owns the rattle is thirteen fourteenths. How did he come up with that number?

My question: is Humpty Dumpty right?

• Are you assuming the "how did he come up with this number" part to be trivial/obvious, and then go on to ask if that logic is sound? Commented Aug 28, 2023 at 11:43

NOTE: Answer pre-dates correction to the question. Humpty Dumpty is now correct.

Humpty Dumpty is wrong.

Both twins tell the truth on Sunday, so neither of them could give that reply. Hence it is definitely not Sunday.

As it is not Sunday, one of the twins is telling the truth and the other is lying.

If the owner is telling the truth, the owner won't give that response on that day because it would be a lie.

If the owner is lying, then they wouldn't give that answer because it would true.

If the non-owner is lying, then the owner would be telling the truth, so the statement would be a lie and the non-owner could say it.

If the non-owner is telling the truth, then the owner would be lying, so the statement would be true and the non-owner could say it.

From this we can conclude it is definitely not Sunday and the other twin owns the rattle.

There is a variation of this puzzle where Humpty Dumpty is correct and that is where the reply is "The true owner of the rattle is telling the truth today".

Monday to Saturday, the true owner can always say this and the non-owner cannot as the twins are doing opposite things. On Sunday they are both telling the truth, so either of them could make that reply and that means there is a 50:50 chance we are talking to the owner.

This means that in this scenario, where the reply was the opposite, there is a 13/14 chance we are talking to the owner. The remaining 1/14 is the scenario where we are talking to the non-owner on a Sunday.

• Oops, I accidentally mistyped the original question - it should have said "is telling the truth today"! Thanks for pointing this out. Commented Aug 28, 2023 at 13:02
• Does Humpty Dumpty not know the day of the week? Commented Aug 28, 2023 at 21:06
• Downvoted due to this answer being wrong now that the question has been edited.
– Karu
Commented Aug 29, 2023 at 3:41
• @Karu this seems like a weird choice given that even before the question was edited this answer went above and beyond and answered BOTH cases (the previous and the new question). Commented Aug 30, 2023 at 1:00
• Of course you do you, I've upvoted for the reason I stated so it balances out. Commented Aug 30, 2023 at 1:01

Humpty Dumpty is wrong. (well, depending on what assumptions you make interpreting the puzzle)

Humpty's answer assumes that this is equally likely to have occurred on any given day regardless of the other information in the puzzle, but that isn't justified. Since either twin could make that statement on Sunday, but only the owner could on any other day, the fact that the statement was made suggests that it's more likely to be Sunday.

So, we have eight equally likely possibilities: that the owner said this on any given day, or that the non-owner said it on Sunday. So the probability that the statement was made by the owner is seven eighths.

To see why this makes sense, consider the situation before the twin answers Alice's question. At this point, Alice has chosen a day at random, and picked a twin at random, so we have fourteen equally likely possibilities. When the twin answers, that lets us eliminate six of those possibilities, but it doesn't change the relative likelihood of the remaining options - Alice is still just as likely to have chosen the owner on Sunday as on Monday.

I would say that Humpty Dumpty is

Incorrect

This is because

At the time these people are having this conversation, it is either Sunday, or it's not Sunday. The day of the week is an external fact which is presumably known to the people in the conversation, even if it's not known to us, reading this conversation.
As a result, the probability that this twin is the owner is either 1/2 (if it's Sunday) or 1 (if it's not Sunday).

I would say that Humpty Dumpty is

Correct

We know that both twins speak truth on a Sunday, it would leave us with 4 possible scenarios on one of the other 6 days:

• He is the owner and he speaks the truth
• He is the owner and he is lying
• He is not the owner and he speaks the truth
• He is not the owner and he is lying

Let's split the 4 possibilities further:

1. If he is the owner and he is speaking the truth, then his statement is true and he is the owner
2. If he is the owner and he is lying, then by saying "The owner tells the truth" he lies and he is the owner
3. If he is not the owner and he speaks the truth, then by saying "The owner tells the truth" he lies and therefore this is not a possible option
4. If he is not the owner and he lies, then "The owner tells the truth" statement would be a lie, which makes him the owner of the rattle, but he is not. So this is again a not possible option

Now we have the following:

Either it's Sunday and he owns the rattle, it's Sunday and he doesn't own the rattle or it's a different day of the week and he is the rightful owner(we ruled out that he can not be the not-owner and give this reply). The chance of him being the owner on a random day is thus: $$(1+1+1+1+1+1+0.5)/7 = 6.5/7 = 13/14$$