An evil mathematician catches Annie and Ben, putting them in two separate cells. Annie can see 12 trees outside her cell, and Ben can see 8. They are told that together, they can see all the trees, and no tree is seen by both. Every morning, evil comes to each of them and asks if they think there are 18 or 20 trees in total. They can answer or pass. If the answer to either is wrong, they both will be in prison forever. On the other hand, if one of them answers, both will be freed. The questions are asked separately, and Annie and Ben can’t communicate. Can they escape?
Day 1: Had Annie seen 19 or 20 trees, she would have answered 20 to the evil. Since Annie sees only 12 trees, she passes. She also knows Ben sees 6 or 8.
Since the question comes to Ben, who sees 8 trees, he knows that Annie sees 10 or 12 trees. He passes.
Day 2: Annie knows Ben did not answer yesterday, so the question returned on day 2. She also knows that Ben knows she has seen 10 or 12. She thinks if Ben had seen 6, he would have added 12 + 6 = 18 and given the number 18 as the answer. This is because 10 + 6 = 16 is not among the choices. So, she concludes, Ben sees 8 trees. She answers 20 to the evil.
Impossible Escape Logic Puzzle: MindYourDecisions
