Here is a logic puzzle that depends on the game theory concept of common knowledge.
Can you figure it out? Alice and Bob are taken prisoners by an evil logician. They are given one chance to be set free. Alice and Bob are placed in cells that have a view of a courtyard with trees. There are 20 trees in all, of which Alice sees 12 and Bob sees 8. Neither prisoner knows how many trees the other sees. But each prisoner is told the trees are partitioned between them: together they see all the trees, but individually no tree is seen by both of them.
They have to figure out the total number of trees, but they are not allowed to communicate with each other. Each day the logician visits Alice in her cell and asks, “Are there 18 or 20 trees in total?” Alice has two choices: she can guess or pass. If Alice passes, then the logician visits Bob in his cell and asks the same question. Bob also can guess or pass. If Bob passes, then the logician retires for the night asks and repeats asking the questions the next day. Both prisoners know the procedure of how the logician is asking questions. There are consequences to guessing. If either person guesses incorrectly, then they are both trapped forever. If either person guesses correctly, however, then they are both set free immediately. Obviously they could guess and have a 50% chance. But can they do better? Is there a way they can escape with certainty?
Hover to show the answer.
This is a complex one. Answer is coming.