Wise Ones in a Tower

A tyrant locks a wise man and his equally wise wife in a tower from which they can see his entire, not very big, capital. He tells them that there are either 11 or 15 towers along the walls of the capital. All of them are visible to either the northern window to wise man Arnold, or the southern window to his Sylvestra, and no tower is visible from both windows. Each morning a guard brings them food and asks how many towers there are. If one of them answers correctly, both are immediately freed; if one answers incorrectly, both are executed. If they don't answer the guard leaves and returns the next morning to ask again. If they do not answer for a week, they are executed anyway.

Arnold and Sylvestra have no means to communicate - no notes, shouts, bribes etc. How do they survive?

The one that sees the towers on the second day answers correctly by saying how many towers are visible outside the window.

The towers are all on one side, so all of the towers are seen by one of them.

The other sees no towers but doesn't answer on the first day as the question implies that there are towers.

The second day the answer is given as the full number of visible towers, correctly presuming that no towers was likely visible from the other window.

It still comes down to a guess, but the best guess is to answer with the number of towers visible. It shortens the horror of waiting to be executed and gives the benefit of immediate release if right, which is most likely.

I'm not hiding this as it's a pure guess and probably wrong.

If A sees 0:
He thinks, Hmm, B is loking at 11 or 15, if B sees 15 he knows that's the answer.
If B doesn't answer on the first day, A knows B has 11 and answers correctly on day 2.

If A sees 1:
Hmm, B would be looking at 10, or 14, if B sees 14 B knows its 15.
If B doesn't answer on the first day, A knows B can see 10, and so there must be 11, A answers on day 2.

If A sees 2:
B would be looking at 9 or 13, logic is essentially the same as seeing 1

If A sees 3:
B is looking at 8 or 12, same logic gets them out.

If A sees 4:
B is looking at 7 or 11
If B is looking at 11, he thinks, hmm, A has 0 or 4, if A has 0 then he will know the answer by day 2. When A doesn't answer on day 2, then on day 3 B knows A has 4 and can answer. A logics all that out; when B doesn't answer on day 3 then A knows B has 7 and can answer on day 4.

If A sees 5:
B is looking at 6 or 10.
if B is looking at 10 he knows A has 1 or 5. When A does't answer on day 2 B knows A has 5 on day 3. If B doesn't answer on day 3 then A knows B has 6 on day 4.

If A sees 6:
B is looking at 5 or 9. If A has 5 he works it out by day 4, so if it gets to day 5 B knows A has 9.

If A sees 7:
B is looking at 4 or 8, If A has 4 he will answer by day 4 so on day 5 B knows A has 8.

There is no way of dividing the towers such neither A nor B can see 0,1,2,3,4,5,6 or 7 towers, so they can always escape by day 5.

Ps, I assume " All of them are visible to either the northern window to wise man Arnold, or the southern window to his Sylvestra, and no tower is visible from both windows." does not mean that either Arnold or Silvestra can definitely see "all of them"; it means that there are no towers that can't be seen from any windows.

Originally posted by iamatiger
If A sees 0:
He thinks, Hmm, B is loking at 11 or 15, if B sees 15 he knows that's the answer.
If B doesn't answer on the first day, A knows B has 11 and answers correctly on day 2.

If A sees 1:
Hmm, B would be looking at 10, or 14, if B sees 14 B knows its 15.
If B doesn't answer on the first day, A knows B can see 10, and so there must be 11, A an ...[text shortened]... all of them"; it means that there are no towers that can't be seen from any windows.

yeah, i figured i was missing something

Apologies for the inexact wording there. It should be, EACH tower is visible from one window or the other, but none are visible from both. So if there are 15 towers, it could be for example that eight visible to the northern window and seven to the southern. So the trick is that if the wise man sees eight, how does he tell whether his wife sees seven (for a total of 15) or three (for a total of 11) ?

And yes, the solution by iamatiger is what I was looking for. Here's my version of the solution.


On day 1:

if there are 15 towers: 0-15 1-14 2-13 3-12 4-11 5-10 6-9 7-8 ; one can see 0 .. 15 towers
if there are 11 towers: 0-11 1-10 2-9 3-8 4-7 5-6 ; one can see 0 .. 11 towers

So if one or the other sees at least 12 towers, they say "there are 15" and walk free. If neither does, the 0-15, 1-14, 2-13 and 3-12 combinations are eliminated.


On day 2:

15 towers: 4-11 5-10 6-9 7-8 ; one can see 4 .. 11 towers
11 towers: 0-11 1-10 2-9 3-8 4-7 5-6 ; one can see 0 .. 11 towers

So if one sees no more than 3 towers, they say "there are 11" and walk free. If neither does, more combinations are eliminated.


On day 3:

15 towers: 4-11 5-10 6-9 7-8 ; one can see 4 .. 11 towers
11 towers: 4-7 5-6 ; one can see 4 .. 7 towers

As above. Eight towers or more seen through one window means there are a total of 15 towers. If neither sees that many, the only remaining combinations lead to 11 towers total.


On day 4:

11 towers: 4-7 5-6 ; one can see 4 .. 7 towers

If they haven't been released by now, they say "there are 11 towers" and leave.