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        riddles >> medium >> A puzzle about apples from the Guardian
        
(Message started by: WombatDeath on Nov 24th, 2017, 4:16pm)
Title: A puzzle about apples from the Guardian
Post by WombatDeath on Nov 24th, 2017, 4:16pm
Hello!  I haven't posted here for many years, and I was a comparative dunce then so I dread to think how turgid and flabby my neurons are these days.  Anyway, you may be aware that the Guardian newspaper posts a riddle every fortnight here:

https://www.theguardian.com/science/series/alex-bellos-monday-puzzle

(The hyperlink button doesn't appear to work, sorry)

This week's puzzle was fun, though that's not why I'm posting.  Here it is:

***

You and your two friends Pip and Blossom are captured by an evil gang of logicians. In order to gain your freedom, the gang’s chief, Kurt, sets you this fearsome challenge.

The three of you are put in adjacent cells. In each cell is a quantity of apples. Each of you can count the number of apples in your own cell, but not in anyone else’s. You are told that each cell has at least one apple, and at most nine apples, and no two cells have the same number of apples.

The rules of the challenge are as follows: The three of you will ask Kurt a single question each, which he will answer truthfully ‘Yes’ or ‘No’. Every one hears the questions and the answers. He will free you only if one of you tells him the total number of apples in all the cells.

   Pip: Is the total an even number?

   Kurt: No.

   Blossom: Is the total a prime number?

   Kurt: No

You have five apples in your cell. What question will you ask? We can assume that you and your friends are all perfect logicians.

***

I'd expect that puzzle to go in the Easy forum. The reason I'm posting it in this one is because it occurred to me that Pip and Blossom may have been in on the kidnapping. They are stated to be perfect logicians, but what if there had actually been, say, 16 apples?  If they truly had no advance knowledge of the correct answer, are those really the optimal questions for them to have asked?

So my question is: in the scenario described, what should each person ask?  I realise that there may be no perfect solution that guarantees a correct answer at the end of the process, but I'm guessing that there is a set of three questions that provides the greatest likelihood of getting the right answer.
Title: Re: A puzzle about apples from the Guardian
Post by dudiobugtron on Nov 25th, 2017, 1:36am

on 11/24/17 at 16:16:08, WombatDeath wrote:
If they truly had no advance knowledge of the correct answer, are those really the optimal questions for them to have asked?

So my question is: in the scenario described, what should each person ask?  I realise that there may be no perfect solution that guarantees a correct answer at the end of the process, but I'm guessing that there is a set of three questions that provides the greatest likelihood of getting the right answer.

Pretty interesting thoughts!!

I think the perfect question depends on how many apples are in their cells, so the answer will probably be pretty complex.  But as for the question, I prefer to interpret it this way: Since they are perfect logicians, the questions they asked would have been optimal given their knowledge of the number of apples in their room.  Therefore, the fact that they asked that question gives you a clue as to how many apples are in their room as well.

PS: Of course, it's possible that the correct answer is [hide]always: 0 apples.
They realise the best solution is to eat all the apples.[/hide]
Title: Re: A puzzle about apples from the Guardian
Post by rmsgrey on Nov 25th, 2017, 6:19am
Yeah, unless the even and prime questions also lead to always being able to resolve the total (which seems unlikely) there's a blindingly obvious set of questions that would guarantee all 3 knowing exactly how many apples are in each room:

Pip: [hide]Do I have exactly n apples in my room?[/hide]
Blossom: [hide]Do I have exactly m apples in my room?[/hide]
Me: [hide]Do I have exactly 5 apples in my room?[/hide]

As for the questions as asked, if the total is even, we can assume Blossom would ask a different question, so the first case to investigate is when the total is prime:

7, 11, 13, 17, 19, 23

Depending on people's actual numbers of apples, everyone can eliminate one or both of 7 and 23, and possibly 11 or 19 as well, so your question has to at least distinguish between 13 and 17.

I can't see any options that work regardless of Blossom's actual number of apples, so the next question is whether there's a specific number of apples Blossom could have where the problem is always going to be solvable for the right question from yourself? If he only has one apple, he's still looking at 7, 11, 13, 17 and I can't see any question that would let him reliably pick the correct one, nor one that would allow someone else to identify one and still let him pick the correct one from the remainder if they can't.
Title: Re: A puzzle about apples from the Guardian
Post by dudiobugtron on Nov 26th, 2017, 5:29pm

on 11/25/17 at 06:19:32, rmsgrey wrote:
Yeah, unless the even and prime questions also lead to always being able to resolve the total (which seems unlikely) there's a blindingly obvious set of questions that would guarantee all 3 knowing exactly how many apples are in each room:

Pip: [hide]Do I have exactly n apples in my room?[/hide]
Blossom: [hide]Do I have exactly m apples in my room?[/hide]
Me: [hide]Do I have exactly 5 apples in my room?[/hide]


Good spotting! (Despite your claim that this was blindingly obvious, I still think it is quite impressive to have thought of it.)  I wonder if there is a way to reword the question to avoid this trivial solution.
Title: Re: A puzzle about apples from the Guardian
Post by towr on Nov 26th, 2017, 10:24pm
You could forbid them to say any number other than the total number of apples.

But then you can just describe each number in a different way.
"Is the number of apples
in my room equal to
the square root of the
number of words in this
question that I now ask?"
Title: Re: A puzzle about apples from the Guardian
Post by rmsgrey on Nov 27th, 2017, 1:42pm

on 11/26/17 at 17:29:51, dudiobugtron wrote:
Good spotting! (Despite your claim that this was blindingly obvious, I still think it is quite impressive to have thought of it.)  I wonder if there is a way to reword the question to avoid this trivial solution.

I then went and checked the comment section on the Guardian article, and quite a few people posted the exact same idea, so I stand by my "obvious".

Several commenters also pointed out that, in the original problem, your apple count is redundant.


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