[Logic Puzzle] Pinocchio's Hats \ stacker news ~science

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1725 sats \ 22 comments \ @SimpleStacker 26 Nov science

Since @south_korea_ln is on hiatus from the puzzles, I will try to fill in the gap :) My puzzles will likely lean more towards logic, probability, and algorithms, and less towards geometry and algebra. I'll also probably not be able to post daily :(

I'll also be posting this with bounties. Bounty will go to whoever has the first correct answer that includes a complete explanation. So just guessing the correct answer without an explanation won't count.

But anyway, here it is:

Assume that both the following statements about Pinocchio are true:

Pinocchio always lies
Pinocchio says, "All my hats are green."

Which of the following, A, B, C, D, or E, can we conclude from the above statements?

A. Pinocchio has at least one hat B. Pinocchio has only one green hat. C. Pinocchio has no hats. D. Pinocchio has at least one green hat. E. Pinocchio has no green hats.

1,000 sats paid 2 times

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1000 sats \ 9 replies \ @Undisciplined 26 Nov

I believe A is the only statement that must be true. If Pinocchio had no hats, then his statement would not be false. I think Bertrand Russel referred to statements like that as “null”.

B is wrong, since he could have multiple green hats and one non green hat.

C is wrong, because he could have any number of hats as long as one is not green.

D is wrong, because only non green hats would be a lie.

E is wrong, because any number of green hats combined with one non green hat would be a lie.

29 sats \ 4 replies \ @SimpleStacker OP 26 Nov

This would be the most complete answer, and thus I rewarded it the bounty.

Statement C would make Pinocchio's assertion of "I have no green hats" a vacuously true statement. Thus, he would not be lying.

If the statement, "All my hats are green", is false, then what must be true is that Pinocchio has at least one non-green hat, which means he has at least one hat.

Thus we can conclude that statement A must be true.

23 sats \ 2 replies \ @SimpleStacker OP 26 Nov

Woops, I made a mistake in my writeup and can't edit.

It should say: Statement C would make Pinocchio's assertion of "All my hats are green" a vacuously true statement.

0 sats \ 1 reply \ @Undisciplined 26 Nov

Ha! I didn't read close enough to catch your mistake. I just inserted the correct statement there in my mind.

28 sats \ 0 replies \ @SimpleStacker OP 26 Nov

It's amazing how our brains can automatically fill in the blanks when there are mistakes, such that we don't even see the mistakes.

0 sats \ 0 replies \ @Undisciplined 26 Nov

Good point about C, that's a better reason.

22 sats \ 0 replies \ @south_korea_ln 26 Nov

I believe A is the only statement that must be true. If Pinocchio had no hats, then his statement would not be false. I think Bertrand Russel referred to statements like that as “null”.

Ok, so "All my hats are green" would not be a meaningful lie... it would be neither true nor false (a null statement, referring to a non-existent set).

Nice puzzle :)

21 sats \ 2 replies \ @Fiat_Revelation 26 Nov

If A is true then so is E.

He could have just 1 hat according to A, which says

at least 1

Then if it's only 1 then E also has to be true.

33 sats \ 1 reply \ @Undisciplined 26 Nov

E does not have to be true. If he has one red hat and one green hat, then his statement is a lie and E is false.

34 sats \ 0 replies \ @Fiat_Revelation 26 Nov

Haha touché!

1000 sats \ 1 reply \ @B_rian 26 Nov

A. In order for Pinocchio to fulfill his obligation to lie about the color of his hats, he must have at least one hat.

0 sats \ 0 replies \ @SimpleStacker OP 26 Nov

I rewarded this one the bounty too since technically it was earlier and the explanation is also correct. I was just waiting for a fuller more detailed explanation.

21 sats \ 2 replies \ @Fiat_Revelation 26 Nov

We can conclude A and E.

A. Pinocchio has at least one hat

True. If he has less than 1 hat, i.e. zero, then it could be construed as being true that "all (zero) are green," but he doesn't say things that are true. Like, if I said, all my Lambos are in the garage. It is not false if I have zero Lamborghinis. It is true that all zero of them are in the garage along with my zero Ferraris.

B. Pinocchio has only one green hat.

Is false because to have "only one green hat" is effectively all his hats, which we know is a lie.

C. Pinocchio has no hats.

Is false. See explanation of A.

D. Pinocchio has at least one green hat.

Is false. If he has 1 or more green hats then it's possible all are green, in which case he would have told the truth, which he doesn't do.

E. Pinocchio has no green hats.

Is necessarily true. By saying all his hats are green, Pinocchio has implied that he has hats, but is lying about the quantity and colour.

He has at least one hat, none of which is green.

0 sats \ 1 reply \ @Fiat_Revelation 26 Nov

Shit, I was too slow.

0 sats \ 0 replies \ @SimpleStacker OP 26 Nov

Better luck next time! You did have a great explanation though.

Edit: Oh wait, E is technically wrong though.

E is wrong because he could have 1 green hat and 1 red hat, and he'd still be lying about "All my hats are green."

21 sats \ 2 replies \ @south_korea_ln 26 Nov

Arf... I feel like I will end up spending more time solving puzzles than sourcing them ;) Great initiative. I like the bounty plan. Just a fair warning, lots of your sats will likely go to @Scroogey~~

Will give this one a try at night, if no one has figured out before then.

0 sats \ 1 reply \ @SimpleStacker OP 26 Nov

Yeah 1,000 sats is a lot to give out especially if I do this often. I may have to rethink my plan quite soon :)

0 sats \ 0 replies \ @south_korea_ln 26 Nov

In my experience, you'll approximately break even with the sats you get from the zaps. However, I refrained from doing an official bounty as it would not give me the discretion to adjust the reward amount based on the specifics of the moment. I would give more sats when I felt the person had to put more effort. Or when it was clear it wasn't a ChatGPT answer.

0 sats \ 3 replies \ @south_korea_ln 26 Nov

Well, damn, ended up focusing on this instead of my calculations.

Well, here we go.

If he always lies, either he has no hats, either he has hats but some are not green.

A. is wrong, because it's possible he has no hats. B. is wrong. This statement doesn't tell us how many hats he has nor their color. C. could be wrong, could be right. He could have hats where not all of them are green. D. can also not be concluded. He could have no hats at all or he might have hats of other colors than green. E. This is the only one that is right. If he has no hats, it is correct. If he has a green hat as well as other hats, it is also correct.

Answer: E

0 sats \ 2 replies \ @south_korea_ln 26 Nov

Arf, not sure after all. "no green hats" is not the same as "non-green hats". Not sure how to interpret this last sentence, but really need to get to some other stuff now.

EDIT: So my conclusion is that none of the statements can be definitely concluded.

0 sats \ 1 reply \ @SimpleStacker OP 26 Nov

See above for why A is the right answer. It's because C must actually be false. If C were true, then the statement "All my hats are green" would be a vacuously true statement, meaning Pinocchio didn't lie.

21 sats \ 0 replies \ @south_korea_ln 26 Nov

Got it :) Nice puzzle!