The Tower and the Towns Puzzle

[pdurrant]

I found this one at the very good Smart Kit website: http://www.smart-kit.com/


Two wise men (who really were wise, and knew each other well) were captured by the king of a small kingdom and brought in secret to a lonely tower in the middle of his kingdom, arriving early one morning.

The king told them that as a test of their wisdom they were going to be kept in solitary confinement in the tower, one in a room that looked to the East and one in a room that looked to the West. They would each see half his kingdom (but not necessarily half the number of towns in his kingdom), and be able to count the towns in the half that that they could see. Between them they would be able to see all the towns in his kingdom, but neither would see any of the towns that the other could see.

Every evening a gaoler would visit them in turn, and if either of them could tell the gaoler the total number of towns in the kingdom, they would both be released that same evening. But if either of them gave the wrong answer, they would both be executed.

The last bit of information the king gave them before they were taken to their separate rooms was that there were either 10 or 13 towns in his kingdom.

On the fifth evening after they arrived at the tower, the wise men were freed.

1. How did the wise men know how many towns there were in the small kingdom?
2. How many towns were there in the small kingdom?

I think you might work the general idea of a solution out quite quickly, but finding a way to rigourously state the solution may be harder. I do now have a way to give the answer in language I hope everyone will be able to follow, although it took me a while. But perhaps I'm slow... and I didn't want to look at the answer given until I'd worked one out myself.

Answers in

Spoiler:

spoiler tags

as usual please.

Have fun!

[GeoffC]

Spoiler:

If each of the kings could only see half of the small kingdom, then the number of towns must be 10 (the only answer divisible by 2) ....

Unless it's a Wednesday and I'm really sleepy ..... and thus walking slowly aimlessly around ....

[pdurrant]

Originally Posted by GeoffC

Unless it's a Wednesday and I'm really sleepy ..... and thus walking slowly aimlessly around ....

Sorry, not the answer.

Seeing half the Kingdom just means half the area, not half the number of towns in the kingdom. The towns are not necessarily equally distributed between the East and West halves of the kingdom.

I've added some text to the question to remove this ambiguity.

[GeoffC]

Originally Posted by pdurrant

Sorry, not the answer.

Seeing half the Kingdom just means half the area, not half the number of towns in the kingdom. The towns are not necessarily equally distributed between the East and West halves of the kingdom.

I've added some text to the question to remove this ambiguity.

I did say it were a Wednesday !!!!!

[omk3]

Spoiler:

Let's say I am one of the two men, and see 4. I know he either sees 6 or 9.

If he saw 9, he would know I see either 1 or 4.
If I saw 1, he knows I would know he saw either 9 or 12.
If he saw 12, we'd be out on the 1st day.

We're not, so he knows I know he doesn't see 12.
If I saw 1, I would now know he sees 9 and the towns are 10. We'd be out on the 2nd day.

We're not, so he knows I don't see 1.
If he saw 9, we would be out, because he would now know I see 4, and the towns would be 13. We would be out on the 3rd day.

We are still not out, so I know he doesn't see 9.
Therefore he must see 6. I tell the gaoler it's 10 towns and we're free on the 4th day.

I tried this thought process with me seeing 5 and 3, but I'm still out on the 4th day, not the 5th. If they get out on the 4th day, it's always 10 towns.

[pdurrant]

Originally Posted by omk3

Spoiler:

Let's say I am one of the two men, and see 4. I know he either sees 6 or 9.

If he saw 9, he would know I see either 1 or 4.
If I saw 1, he knows I would know he saw either 9 or 12.
If he saw 12, we'd be out on the 1st day.

We're not, so he knows I know he doesn't see 12.
If I saw 1, I would now know he sees 9 and the towns are 10. We'd be out on the 2nd day.

We're not, so he knows I don't see 1.
If he saw 9, we would be out, because he would now know I see 4, and the towns would be 13. We would be out on the 3rd day.

We are still not out, so I know he doesn't see 9.
Therefore he must see 6. I tell the gaoler it's 10 towns and we're free on the 4th day.

I tried this thought process with me seeing 5 and 3, but I'm still out on the 4th day, not the 5th. If they get out on the 4th day, it's always 10 towns.

You are certainly thinking along the right lines, even if you do not have the answer yet. They do indeed get out on the 5th evening.

[omk3]

Originally Posted by pdurrant

You are certainly thinking along the right lines, even if you do not have the answer yet. They do indeed get out on the 5th evening.

Spoiler:

Heh. I thought I had examined all possibilities, but I hadn't. A number being a possibility and it being a certainty changes everything. I had conveniently avoided 6 or 7 in my previous thinking, but these are the toughest numbers to see.

So let's say I see 6 towns.

I see 6.
I know he sees either 7 or 4.

If he saw 4 he would think I see either 9 or 6.

If I saw 9, I would think he would either see 1 or 4.

If he saw 1, he would think I saw either 9 or 12.

If I saw 12, we'd be out on the 1st day, with me stating there are 13 towns.

We are not out on the 1st day, so if he saw 1, he would know I saw 9, and we would be free on the 2nd day, with him stating there are 10 towns.

We are not free on the 2nd day, so he doesn't see 1. So he must be seeing 4. We would be out on the 3rd day, with me stating there are 13 towns.

We are not free on the 2nd day, so he knows I don't see 9. If he sees 4, he knows I must see 6, and we would be out on the 4th day, with him stating there are 10 towns.

We are not free on the 4th day, so he doesn't see 4. It follows that he sees 7, and we are out on the 5th day, with me stating there are 13 towns.

[GeoffC]

I can't help thinking that the solutions thus far proposed are ignoring this ....

" Every evening a evening a gaoler would visit them in turn, and if either of them could tell the gaoler the total number of towns in the kingdom, they would both be released that same evening. But if either of them gave the wrong answer, they would both be executed. "

How can they proceed to the next possible solution without knowing their previous one was wrong ? Presumably, also, they cannot communicate with each other through locked doors ?

[LazyScot]

Spoiler:

On the first night, if either can see 11, 12 or 13, they can report the answer as 13

On the second night, if either is seeing 0, 1 or 2, they can report the answer as 10, since if the other person could see 11, 12, or 13 they would have already answered.

At this point each must be seeing between 3 and 10 towns.

On the third night, if either is seeing 8, 9 or 10 towns, they can report the answer as 13.

At this point each must be seeing between 3 and 7 towns.

On the fourth night, if either is seeing 3, 4 or 5 towns, they can report the answer as 10 (since there is no way that the other can be seeing 10, 9 or 8 towns to make 13 towns as they would have already answered).

At this point, each must be seeing between 6 and 7 towns.

So if they are still there on the fifth night, either can answer it as being 13 towns.

I'm sure this could be expressed mathematically....

[pdurrant]

Originally Posted by omk3

Spoiler:

Heh. I thought I had examined all possibilities, but I hadn't. A number being a possibility and it being a certainty changes everything. I had conveniently avoided 6 or 7 in my previous thinking, but these are the toughest numbers to see.

So let's say I see 6 towns.

I see 6.
I know he sees either 7 or 4.

If he saw 4 he would think I see either 9 or 6.

If I saw 9, I would think he would either see 1 or 4.

If he saw 1, he would think I saw either 9 or 12.

If I saw 12, we'd be out on the 1st day, with me stating there are 13 towns.

We are not out on the 1st day, so if he saw 1, he would know I saw 9, and we would be free on the 2nd day, with him stating there are 10 towns.

We are not free on the 2nd day, so he doesn't see 1. So he must be seeing 4. We would be out on the 3rd day, with me stating there are 13 towns.

We are not free on the 2nd day, so he knows I don't see 9. If he sees 4, he knows I must see 6, and we would be out on the 4th day, with him stating there are 10 towns.

We are not free on the 4th day, so he doesn't see 4. It follows that he sees 7, and we are out on the 5th day, with me stating there are 13 towns.

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.

Congratulations! I'm most impressed. It took me a lot longer to get to an answer that coherent.

[pdurrant]

Originally Posted by GeoffC

I can't help thinking that the solutions thus far proposed are ignoring this ....

" Every evening a evening a gaoler would visit them in turn, and if either of them could tell the gaoler the total number of towns in the kingdom, they would both be released that same evening. But if either of them gave the wrong answer, they would both be executed. "

How can they proceed to the next possible solution without knowing their previous one was wrong ? Presumably, also, they cannot communicate with each other through locked doors ?

No, they don't have any way of talking to each other, through locked doors or otherwise. Do bear in mind if either of them could tell the gaoler the total number of towns in the kingdom, they would both be released that same evening.

[pdurrant]

Originally Posted by LazyScot

Spoiler:

On the first night, if either can see 11, 12 or 13, they can report the answer as 13

On the second night, if either is seeing 0, 1 or 2, they can report the answer as 10, since if the other person could see 11, 12, or 13 they would have already answered.

At this point each must be seeing between 3 and 10 towns.

On the third night, if either is seeing 8, 9 or 10 towns, they can report the answer as 13.

At this point each must be seeing between 3 and 7 towns.

On the fourth night, if either is seeing 3, 4 or 5 towns, they can report the answer as 10 (since there is no way that the other can be seeing 10, 9 or 8 towns to make 13 towns as they would have already answered).

At this point, each must be seeing between 6 and 7 towns.

So if they are still there on the fifth night, either can answer it as being 13 towns.

I'm sure this could be expressed mathematically....

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.

Congratulations to you too. Goodness, I thought this one would keep you all occupied a lot longer!

[GeoffC]

Originally Posted by pdurrant

No, they don't have any way of talking to each other, through locked doors or otherwise. Do bear in mind if either of them could tell the gaoler the total number of towns in the kingdom, they would both be released that same evening.

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 ?

[LazyScot]

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 ?

Probably somewhere between a hint and a spoiler...

Spoiler:

They are wise, and so they know that the other wise man (woman?) would only answer if they knew the number of towns. So if a night passes without them being released, they know that the other wise did not have enough information to deduce the answer.

And a little bit more of a hint/spoiler

Spoiler:

This represents and "Information flow" between the wise people, and they can make deductions about what the other must know after each night.

[GeoffC]

Originally Posted by LazyScot

Probably somewhere between a hint and a spoiler...

Spoiler:

They are wise, and so they know that the other wise man (woman?) would only answer if they knew the number of towns. So if a night passes without them being released, they know that the other wise did not have enough information to deduce the answer.

And a little bit more of a hint/spoiler

Spoiler:

This represents and "Information flow" between the wise people, and they can make deductions about what the other must know after each night.

pffftt ...