Comments on: The Tower

By: bambi_76258

bambi_76258 — Sat, 27 Feb 2010 03:41:58 +0000

I think no matter how you try to rationalize the number of cities, it comes down to the meals brought to them each night. Well, in my opinion anyway. Bambi

By: bambi_76258

bambi_76258 — Sat, 27 Feb 2010 03:19:08 +0000

I am not as smart as some of the people who answered this but I am going to give it a try. I think they saw 5 cities each and the way they got out was the dinners brought to them. Each day a dinner came from a different city, from the others vantage point. When the meals were brought in they were from a city that could not be seen by that prisioner. Knowing that there was not 12 cities and each could see half.
He knew the number was 10. Each knew none of the meals were from cities they could see. Bambi

By: hex

hex — Wed, 20 Jan 2010 21:05:25 +0000

But what if the area of the kingdom is not an even number?

By: bilbao

bilbao — Wed, 20 Jan 2010 06:35:38 +0000

Hi Tsopi, not that easy
The expression ‘half of the kingdom’ is literal, and refers to the area/expanse of land. It doesn´t refer to ‘half of the cities’.
Besides, none of the cities is on the border between the two views and visible to both men.

By: Tsopi

Tsopi — Tue, 19 Jan 2010 19:30:19 +0000

I suppose it is as easy as that.As you said each man could see the half of the kingdom witch mean that there were 10 cities ’cause 13 cannot be devided by 2!Lol is that it?

By: Mashplum

Mashplum — Wed, 18 Nov 2009 01:46:54 +0000

Thanks bizarette. I had fun writing it.

By: bizarette18

bizarette18 — Tue, 17 Nov 2009 15:45:03 +0000

Hey, Mashplum, I like your scenario. I had Saruman doing the numerics in mine, and Gandalf patiently waiting to be rescued by eagles, but I like yours better.

By: bilbao

bilbao — Mon, 16 Nov 2009 08:39:03 +0000

Many of you got the correct answer: 7/6 for a total of 13. Some good explanations above.
It is a pleasure to welcome some new visitors. It encourages me to post ever better puzzles…

By: bilbao

bilbao — Mon, 16 Nov 2009 07:07:21 +0000

Hi BlindCupid, the answer to your question is yes, the fact that they escaped on the 5th day and not earlier is important.

Hi Karbon, when any of the wise men tells the jailer the correct solution, both wise men are released simultaneously, no matter who gave the correct solution. So they both know when they are released.

By: venkatg

venkatg — Sat, 14 Nov 2009 20:05:37 +0000

Btw, the reasoning is based on induction. In fact this combination of cities is the only one that takes maximum 5 days . Other combination such as 10,3 would yield 3rd day.

By: Karbon

Karbon — Sat, 14 Nov 2009 19:50:32 +0000

Do any of the wise men have a way of telling when the other one was released?

By: kitchen

kitchen — Sat, 14 Nov 2009 19:15:02 +0000

There are only 13 possible combinations, (0,13)-(6,7) and (0,10)-(5,5).
If any man sees a # of cities that only occurs in one combination, he knows how many cities there are and both are freed. As each night goes by that they’re _not_ freed, both know that the other doesn’t see any of the distinct numbers, eliminating possibilities… it’s a lot like the island and eye color puzzle.

Any combination with 11-13 is eliminated after the first night, 0-2 2nd night, 8-10 3rd night, 3-5 4th night, and on the fifth night, (6,7) is the only remaining possibility.

By: MarkS

MarkS — Sat, 14 Nov 2009 15:15:51 +0000

I’m not particularly good at these things, but here goes-

No one sees 11, 12 or 13 cities since then the solution (there are 13 cities) would be given the first night.
No one sees 0 cities since this means the other sees 13 or 10
cities but since there was no solution the first night the one
seeing 0 cities now knows that there must be 10 cities and gives this answer the second night. Likewise, no one sees
1 or 2 cities, else the solution would have been given the second night.
No one sees 10 cities since that means the other sees 0 or 3 cities but we know that 0 cities is not possible, the answer of 10+3 cities would be given the third night. Likewise no one sees 9 or 8 since 1 and 2 have been ruled out so the answer 9+4 or 8+5 would be given the third night.

No one sees 3 cities since the 10+3 answer was not
given the third night and so 3+7 would have been given on the
fourth night. The same reasoning shows that no one sees 4 or 5 cities.

The only possiblity left for the fifth night is 7+6.

By: BlindCupid

BlindCupid — Fri, 13 Nov 2009 23:01:05 +0000

My best guess would be if one man saw more than ten cities, theremore the answer must be 13.
Though I don’t think the tyrant king would give a question to something so simple..
Does the fact that they escaped on the 5th day have anything to do with the answer?

By: jamesbuc

jamesbuc — Fri, 13 Nov 2009 22:09:49 +0000

Hi Everyone, this is my first post. Lets hope it a good one…

I make it that there 13 cities. One prisoner sees 7, the other 6:

Let prisoner A be the prisoner seeing the (hypothetical) greater number of cities, let prisoner B be the prisoner seeing the fewer number.

On Day 1 prisoner A can go free if he can see 13, 12 or 11 cities (he would know there must be 13 cities in total).

On Day 2 prisoner B can go free if he can see 0, 1 or 2 cities (he would know there must be 10 cities in total or else he would have been freed by virtue of prisoner A giving the correct answer on day 1).

The following combinations remain:

Prisoner A sees 10 cities, Prisoner B sees 3 cities
A sees 9, B sees 4
8 5
7 6
7 3
6 4
5 5

Therefore on Day 3 any prisoner seeing 8, 9 or 10 cities can work out the answer since there is only one combination where each value occurs. Since the prisoners aren’t freed on day 3 the set reduces to the following for Day 4:

7 6
7 3
6 4
5 5

By the same logic on Day 4 any prisoner seeing 3, 4 or 5 cities can win both their freedom. Only one set remains for Day 5:

7 6

Therefore there are 13 villages, one prisoner seeing 7 the other 6.