The Tower and the Towns Puzzle

[Javed]

Originally Posted by GeoffC

pffftt ...

I'm finding this one difficult.

[GeoffC]

I'm not finding it logical ....

[Javed]

Doesn't seem mathematical either to me. There must be some trick. Like a talking parrot that flies from one of the prisoners' windows to the other (outside the castle) and repeats what each wise man says.

[GeoffC]

Originally Posted by Javed

Doesn't seem mathematical either to me. There must be some trick. Like a talking parrot that flies from one of the prisoners' windows to the other (outside the castle) and repeats what each wise man says.

ah !


you have it .....

[omk3]

Originally Posted by pdurrant

Yes, you have the right answer for the right reasons. There's one typo in your explanation,

Spoiler:

2nd day for 3rd day

, but I'll overlook that.

Oops

Originally Posted by pdurrant

An excellent and correct answer.

Interestingly, it's expressed in a completely different way to omk3's answer. I think it's probably easier to understand this way.

I just love how each one of us is approaching the puzzles from a different perspective yet we can all find a valid solution.
And LazyScot's answer is certainly more concise than mine...

Originally Posted by GeoffC

But you are missing my point.

In order to progress from evening one to the next they have to know that the answer they have come up with is wrong.

How do they know it is wrong without telling the gaoler ?

Have you looked at the spoilers? There's no guesswork involved, as long as we are sure that the other is thinking logically, which is a given in this puzzle. I found the colours on the forehead puzzle much more confusing than this... More evidence of how differently each of us thinks, I guess.

[pdurrant]

Originally Posted by GeoffC

But you are missing my point.

In order to progress from evening one to the next they have to know that the answer they have come up with is wrong.

How do they know it is wrong without telling the gaoler ?

I will give an example. This is definitely in spoiler territory.

Spoiler:

OK, suppose I'm one of the wise men. I'm in my first day in the tower. I can see 2 towns. I cannot know from this whether there are 10 or 13 towns, so I will say nothing to the Gaoler tonight, the first night.

Since I see 2 towns, I know that the other wise man must be seeing either 8 or 11 towns.
Now, if he sees 8 towns, he too cannot tell whether there are 10 or 13 towns, and so he will say nothing to the Gaoler tonight, the first night, and I'll still be in the tower tomorrow morning
But if he sees 11 towns, he will know that there must be 13 towns, he'll tell this to the Gaoler tonight, the first night, and we'll both be free before the morning.

It's the next day. We weren't freed yesterday, so I now know that the other wise man didn't see 11 towns. So he must have seen 8 towns. I can see 2 towns. So there are 10 towns altogether, I'll tell this to the Gaoler tonight, and we'll both be free.

No guesses that are incorrect. But on the second evening I know how many towns there are — not a guess.

[GeoffC]

Originally Posted by omk3

Have you looked at the spoilers? There's no guesswork involved, as long as we are sure that the other is thinking logically, which is a given in this puzzle. I found the colours on the forehead puzzle much more confusing than this... More evidence of how differently each of us thinks, I guess.

Of course and none of them stipulate how the first evening guess is negated.

[pdurrant]

Originally Posted by Javed

Doesn't seem mathematical either to me. There must be some trick. Like a talking parrot that flies from one of the prisoners' windows to the other (outside the castle) and repeats what each wise man says.

No, there's no trick. No talking parrots, or any other communication through pets or wildlife.

Spoiler:

There a kind of slow communication, when they are not released from the tower each evening.

[pdurrant]

Originally Posted by GeoffC

Of course and none of them stipulate how the first evening guess is negated.

See my example.

Also:

Spoiler:

There is no first evening guess. Neither of them knows how many towns there are on the first evening, so neither of them makes a guess to the Gaoler.

One of them might imagine that if a certain thing was true then the other wise man might know (not guess) the number of towns.

[GeoffC]

Originally Posted by pdurrant

No, there's no trick. No talking parrots, or any other communication through pets or wildlife.

Spoiler:

There a kind of slow communication, when they are not released from the tower each evening.

Spoiler:

I'm going to have to walk on by on this one, there is a level of logic (or even guess work) going on that makes it difficult to understand how each wise one knows what the other is thinking, and why it takes 5 days before one of them gives an answer to the gaoler ..... and upon what that answer is based.....

[pdurrant]

Originally Posted by GeoffC

Spoiler:

I'm going to have to walk on by on this one, there is a level of logic (or even guess work) going on that makes it difficult to understand how each wise one knows what the other is thinking, and why it takes 5 days before one of them gives an answer to the gaoler ..... and upon what that answer is based.....

I'm not quite sure where your sticking point is. In a day or three I'll put up my version of a full answer, and hopefully that will do it. If it's still unclear then, I'll happily do my best to explain further.

[omk3]

Originally Posted by GeoffC

Spoiler:

I'm going to have to walk on by on this one, there is a level of logic (or even guess work) going on that makes it difficult to understand how each wise one knows what the other is thinking, and why it takes 5 days before one of them gives an answer to the gaoler ..... and upon what that answer is based.....

Spoiler:

Let's take the simplest case:
One of them sees 11,12 or 13 towns. He is sure the towns are 13, and asks to be freed.

Now the simplest-but-one case:
One of them can see 0,1 or 2 towns. He knows that if the other one could see more than 10 (simplest case above) they would be out on the 1st day. Since they are still in on the 2nd day, he knows the other one sees 10 towns or less.

You know the number of towns you see, and you know the 2 possible numbers the other one can see. If you can discard one of them, you can be sure of the other.
So if you see 2 towns for example, the other one sees either 11 or 8.
But if you haven't been freed on the 1st day, you know he can't see 11. So he must see 8.
And so it goes on.

I hope this helps...

[GeoffC]

Originally Posted by omk3

Spoiler:

Let's take the simplest case:
One of them sees 11,12 or 13 towns. He is sure the towns are 13, and asks to be freed.

Now the simplest-but-one case:
One of them can see 0,1 or 2 towns. He knows that if the other one could see more than 10 (simplest case above) they would be out on the 1st day. Since they are still in on the 2nd day, he knows the other one sees 10 towns or less.

You know the number of towns you see, and you know the 2 possible numbers the other one can see. If you can discard one of them, you can be sure of the other.
So if you see 2 towns for example, the other one sees either 11 or 8.
But if you haven't been freed on the 1st day, you know he can't see 11. So he must see 8.
And so it goes on.

I hope this helps...

Spoiler:

Why ?

He sees 8 - he doesn't know whether it is 10 or 13
He sees 11, therefore it must be 13. They are both out.

However on the next day, he still only sees 8 - and still he doesn't know whether it is 10 or 13.

and the same for each of the remaining days until old age creeps in or one of them takes the plunge and guesses 50:50 at being right.....

[=X=]

Okay I'm simple minded but the only way I can see them 'knowing' the answer without guessing is if one side sees all 13 towns. Since only one side has to give the right answer the person that sees 0. Would not provide an answer

[GeoffC]

Originally Posted by =X=

Okay I'm simple minded but the only way I can see them 'knowing' the answer without guessing is if one side sees all 13 towns. Since only one side has to give the right answer the person that sees 0. Would not provide an answer

You are as simple minded as I am ....