He imprisons them in his prison and decides to test their cleverness. They are kept in two different cells, which are located on opposite sides of the prison, so that they cannot com- municate in any way. Max's cell's window has twelve steel bars, while Jenny's cell's window has eight. The first day of their imprisonment, Mr. Giovanni tells first Max and then Jenny that he has decided to give them a riddle to solve. The rules are simple, and solving the riddle is the only hope the two have to escape.
● In the prison there are no bars on any window, door or passage, except for windows in the to logicians' cells, which are the only barred ones (this implies that each cell has at least one bar on its window).
● Mr. Giovanni will ask the same question to Max every morning: "Are there eighteen or twenty bars in my castle?"
● If Max doesn't answer, the same question will then be asked to Jenny the night of the same day.
● If either of them answers correctly, and is able to explain the logical reasoning behind their answer, Mr. Giovanni will immediately free both of them and never bother them again.
● If either of them answers wrong, Mr. Giovanni will throw away the keys of the cells and hold Max and Jenny prisoners for the rest of their lives.
● Both Max and Jenny know these rules.
As soon as either Max or Jenny answer, the decision is made. Only if they don't give an answer does the question pass on to the other person.What’s the minimum number of days they will need to escape?
They soon find the answer. Impressed that they could escape, Mr. Giovanni makes them another offer. Mr. Giovanni is an avid fan of chess, and hence gives them two problems. If they solve both the problems, Mr. Giovanni offers to not only free all the Pokémon he has captured till date, but also to give Max 5 of his strongest Pokémon, which can help Max easily defeat all the gym leaders.

Explanation

The explanation: Let's start on the time of release and follow the thought process. Max learns that Jenny has 8 bars on the morning of the 4th day. To avoid long lists of "he knows she knows" let's use some shorthand. M represents Max and R represents Jenny "M12: R=6,8" means "In the case that Max has 12 bars, Max knows that Jenny has 6 or 8 bars". On the next indent level, "R6" and "R8" describes both possible cases. Here's the thought process of Max on the morning of the 4th day, just before he announces that there are 20 bars in total. The logic starts like this: Max knows Jenny has 6 or 8 bars. If Jenny had 6 bars, she would think Max had 12 or 14 bars. If Jenny had 6 bars and assumed Max had 12, Max would think... etc.
`M12: R=6,8 - R6: M=12,14 - - M14: R=4,6 - - - R4: M=14,16 - - - - M16: R=2,4 - - - - - R2: M=16,18 - - - - - - M18`: ​If M had 18 bars, he would have answered "20" on the first morning - - - - - - M16: ​Since M didn't answer day 1, R would answer "18" on the first evening. - - - - - R4: ​That didn't happen, so M would answer "20" on second morning - - - - M14: ​That didn't happen, so R would answer "18" on second evening - - R6: ​That didn't happen, so M would answer "20" on the third morning - - M12: ​That didn't happen, so R would answer "18" on the third evening - R8: ​None of the above happened, so this is the only remaining choice Max announces that 20 is the answer on the fourth morning.