The King and the Two Wizards

The riddle below is a fun challenge! So far, no AI has been able to solve it (I just tested it with o3-mini, and it failed). But some humans can! That’s my challenge to you: try to solve it and let me know your answer

A small kingdom had a king who regularly received counsel from two very smart wizards. Though the two wizards were truly wise, they didn't show much interest in the daily affairs of the country.

One day, the king grew fed up with the wizards and locked them in a tower in the middle of his kingdom. He informed them that, as a test of their wisdom, they would be kept in solitary confinement in the tower—one in a room facing east and the other in a room facing west.

Each would see half the kingdom (though not necessarily half the number of cities in the kingdom) and would be able to count the cities in the half that they could see. Between them, they would be able to see all the cities in the kingdom, but neither would see any of the cities that the other could see. Additionally, they could not communicate with each other.

Also, neither of them could partially see a city; any city they could see, they could see in its entirety.

Every evening, a jailer would visit them in turn, and ask them about the total number of cities in the kingdom. If either of them could tell the jailer the total number of cities in the kingdom, they would both be released that same evening.

However, if either of them provided the wrong answer, they would both be executed. The poor wizards had no idea about the number of cities in the kindgom.

The last piece of information the king gave them before they were taken to their separate rooms was that there were either 12 or 13 cities in his kingdom.

The first night, the wizards didn't say anything when the jailer asked. The next night, they stayed silent again. Days went by, and they just kept quiet.

However, on the eighth evening after they were locked in the tower, one of wizards finally spoke. His answer was correct, and as a result, they were both set free.

So I have these questions for you:

1. How many cities were there in the small kingdom?
2. How many cities did each see?
3. How did the wizards know how many cities there were in the small kingdom?