logic puzzle

[Zoot Allures]

You’ve been caught snooping around a spooky graveyard with your best friend. The caretaker, a bored old man fond of riddles (and not so fond of trespassers), imprisons each of you in a different room inside the storage shed, and, taking your phones, says, “Only your mind can set you free.” To you, he gestures toward a barred window. Through it, you can see 12 statues. Out of your friend’s window, which overlooks the opposite side of the graveyard, she can see eight. Neither of you know the other’s count.

The caretaker tells you each, individually, that together you can see either 18 or 20 statues. Unfortunately, there’s no way to tell your friend how many you can spot. The only way for you both to escape is for one of you to give the total number of visible statues. Get it wrong, and neither of you ever leave. The caretaker asks you each one at a time, once a day, and you can choose to answer or to pass. Both of you know that you’re always asked first.* If you both pass on a given day, the question—are there 18 or 20?—is posed to each of you again the next day, and the next, and so on, until you get it right or wrong. The caretaker cackles, “If you need me, I’ll be out preparing your graves.”

How do you escape?

 

[Mr Bret]

I don't know how you escape, but the key words/phrase seems to be "To you, he gestures toward a barred window. Through it, you can see 12 statues. Out of your friend’s window, which overlooks the opposite side of the graveyard, she can see eight. Neither of you know the other’s count.

So, my guess is that there are 20 statues.

You and I are on the same page so far.
There is an implication that you can not communicate with your friend, so it seems there's no way of knowing how many are visible on the other side.
So really all you know is there are at least 12.
There is a huge risk in guessing the wrong answer, so how to figure out the total.
I'm curious if there's any significance to the * after "always asked first.".

 

[LurkingInMyTent]

The question is poorly worded. "Both of you know that you're always asked first" is unclear, do you mean that the same one of the two prisoners is always given the first chance to answer and that both prisoners are aware of this fact?

What I would do is repeatedly knock 12 times (the number of statues that I can see) as hard as I could on the wall of the shed nearest where I guess the other prisoner is and hope that they figure out what I mean, and when the caretaker comes in to complain, kill him.

The less violent answer, assuming that neither of you starve to death, involves counting the number of times you've been asked the question and hoping that your fellow prisoner has also figured it out.

 

[MissCroft]

I don't know the answer but I got accidentally locked in a cemetery once. I only remember one angel statue. She watched over me while I slept.

 

[buttercup]

The caretaker tells you that the thing can be solved by using your mind. In other words, he tells you that the total can be deduced.

You see 12 statues, and you know there are either 18 or 20 in total. So you know your friend sees either 6 or 8.

Now, you note that the only way you can give an answer that you know to be correct is if you see 19 or 20 (in which case you would answer 20, of course.)

So, you can deduce that the problem is only capable of being solved by deduction , if there are 20 visible statues.

There are no circumstances under which you are able to deduce the correct total, if the correct total is 18.

You can only arrive at the answer by deduction is if the answer is 20.

QED.

 

[Zoot Allures]

the answer is 20 and it takes 5 days to figure it out

Why?

 

[oldjones]

Since the question was first asked on the 11th, and today's the 12th, there's only three more days to go. I can do that standing on my head.

But I did like buttercup's solution — even if I don't buy the logic — and unless the caretaker's a cheat the two of them are already enjoying their freedom.

On my own I'm as far as: After Day 1, both have passed and they each know the other's possibles: 10 and 12 for you, 6 and 8 for her. Both also know the combo can't be the two minimums. That would make the total 16 and the Caretaker a liar, which never happens in these puzzles.

The trick is to deduce by the other person's passes whether they see their max or min number in their pair. IF the other captive gets it right, you know what the answer is, but so what? Games over. And if you guess wrong, ditto. So no uncertain guesses, the only 'tool' available is to pass.

Next day they pass again. So at the end of Day 2, you know she might see 6, but you also know that she understands if she does, that means you cannot see 10. So on Day 2 she'll either correctly guess 18, or pass. And she passes.

Now you know she's not seeing 6, and so must be seeing 8, so you guess 20 on Day 3.

Wait… Is that it? Sump'ins wrong earlier on in my so-=called thinking, but TERB already timed me out once. I'll be back

 

[Zoot Allures]

Both parties are thinking about what the other guy is thinking so Old Jones is on the correct path but it takes 5 days not three

 

[Zoot Allures]

First, you and your friend have to realize that each “pass” counts as a signal. Given that you haven’t had time to consult with each other before being separated, this will require some degree of intelligence—and confidence in each other. And second, it will require five days of signalling to deduce the number of statues with certainty. In game theoretic terms, this establishes the needed “common knowledge” to escape. Many logic riddles rely on this concept, like “Blue Eyes,” described on the website xkcd.com—how can you figure out your eye color if other people won’t tell you and you can’t see your reflection? The social scientist Simon DeDeo describes another logic puzzle whose solution relies on building common knowledge in his Nautilus article “The Bitcoin Paradox.”

Day 1

If you saw 19 or 20 statues, then you could conclude there are 20 statues. But you only see 12, so you pass. This signals to your friend that you see at most 18 statues.

If your friend saw 0 or 1 statues, knowing that you see at most 18, she could conclude there would have to be 18 statues. But your friend sees 8, so passes. This signals to you that she sees at least 2 statues.

Day 2

If you saw 17 or 18 statues, then you could conclude there are 20 because your friend must see at least 2 statues. But you only see 12, so you pass. This signals to your friend that you see at most 16 statues.

Now if your friend saw 2 or 3 statues, knowing that you see at most 16, she could conclude there would have to be 18 statues. But your friend sees 8 statues, so she has to pass. This signals to you that she sees at least 4 statues.

Day 3

If you saw 15 or 16 statues, then you could conclude there are 20 because your friend must see at least four statues. But you only see 12, so you pass. This signals to your friend that you see at most 14 statues.

Now if your friend saw four or five statues, knowing that you see at most 14 statues, she could conclude there would have to be 18 statues. But she sees eight statues, so she has to pass again. This signals to you that your friend sees at least six statues.

Day 4

If you saw 13 or 14 statues, then you could conclude there are 20 because your friend must see at least six statues. But you only see 12, so you pass. This signals to your friend that you see at most 12 statues.

Now if your friend saw six or seven statues, knowing that you see at most 12 statues, she could conclude there would have to be 18 statues. But your friend sees eight statues, so she has to pass again. This signals to you that she sees at least eight statues.

Day 5

Since your friend sees at least eight statues, and you see 12, you know there are 20. You guess 20 and get freed!