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Pensolve Christmas Riddle

A great old Emperor has finally decided that he must appoint who will succeed him between his two sons and one daughter. He is concerned that if if gives the empire to just one son then the other will be jealous and cause a rebellion. So he devises a challenge for his two sons where they must demonstrate that they can trust each other and then they can share the kingdom, if they cannot then the entire empire will go to his daughter.

He gives his sons the following instructions (who previously had never seen outside the castle):

  1. You must not talk to each other for the remainder of the challenge
  2. You must climb to the top of opposite towers in my mighty castle
  3. From your vantage points you can each see exactly half of the empire and collectively you can see all of the cities in the empire
  4. Each day at noon I will send a squire to deliver water to each of you and will get one opportunity to guess whether there are 10 or 13 cities in the empire
  5. You can decide not to guess because if one of you says the wrong answer then you have both failed the challenge and the empire will go to your sister

On the fifth day one son spoke and correctly picked how many cities were in the empire. How many were there and why?

Cheers and have a Merry Christmas!

Hint 1: Think about what would happen if one of the brothers could see 12 cities, would he know how many cities there are?

Answer: The riddle is solved by deduction, so if someone doesn’t do something then you can deduce information from that. If one brother was looking at 12 cities, then he would know that there can not be 10 cities, so there must be 13 cities. He should be able to work this out on day 1. Since no one says anything then we can deduce that no one was looking at 12 cities. The same applies to 11 and 13 cities. If a brother was looking at 0, 1 or 2 cities, and after midday on day 1 he knows that his brother can not see more than 10 cities, then on day 2 he can safely declare there are only 10 cities. But no one says anything on day 2, so no one is looking at 0, 1, 2, 11, 12 or 13 cities. On day 3, If someone is looking at 8, 9 or 10 cities and they know their brother is looking at more than 2 cities, then they know that there are 13 cities. No one says anything so no one is looking at 0, 1, 2, 8, 9, 10, 11, 12, 13. On day 4, the same logic applies, which eliminates cities 3, 4, 5. So on day 5 both brothers should know that there are 13 cities and one is looking at 6 cities and one is looking at 7.