I heard about a little logic puzzle from Presh Talwalker a few months ago:

Afer coming up with my solution, it seemed a lot simpler than others I read ¯\_(ツ)_/¯.

Try it for yourself before continuing on.

The Problem (if you don’t want to watch the video)

Alice and Bob have been trapped by an evil logician in separate cells of a prison. Alice sees 12 trees and Bob sees 8 trees. The captor tells both Alice and Bob they can together see all the trees, but no tree is seen by both people. Every day Alice is asked ‘Are there 18 or 20 trees in total?’, if she passes then Bob is asked this question. If both pass this process is repeated the next day. If either guess correctly, both are immediately freed. If either guesses incorrectly, both are imprisoned forever. Assuming they both know the rules but can’t communicate, can they escape with certainty?

My Solution (spoilers)

On Day One (Part A) Alice looks out of the window. She sees 12 Trees (A=12). Because she is asked if the total number of trees is 18 or 20, she knows that she is the majority regardless (A>B must hold). Further Alice realizes that if B=6, then from Bob’s POV A=12 or A=14. And if B=8, then from Bob’s POV A=10 or A=12. As of now she has no way of choosing T=18 or T=20, so she passes.

On Day One (Part B) Bob looks out of the window. He sees 8 Trees (B=8). He immediately realizes he is the minority similar to Alice’s initial realization. Bob realizes that if A=10, then from Alice’s POV B=8 or B=10. But we already know that Bob is the minority, so that second option is inconsistent. We can eliminate A=10 because we know A>B. So now Bob knows by elimination that A=12, and thus T=20. Escape with certainty is possible on the first day.

The Heuristic

This is a binary problem, there can only be 18 or 20 trees. The heuristic is we look for aspects of the problem statement that can be reduced to a binary nature. Then if we can make a relationship between these variables and the answer variable, we can use simple logic to find the answer. The critical component is realizing A has strictly more trees than B, and both Alice and Bob know it. Taking all the facts together, in this case we found that one option (T=18 / A=10) led to an inconsistent result when Bob looks at it from Alice’s point of view, because A>B is violated, so the other option must be the case. The other important aspect of this problem is looking at facts shared among Alice and Bob (the problem statement, the T=18 or T=20 question, and A>B).

The Takeaway

If you look back the solution is not complicated. A lot of the apparent difficulty in brainteasers derives from how they’re framed, and that’s the main takeaway here. Many apparently difficult problems can be simply solved with reframes. One of the useful beliefs in my world is that all problems can be solved by some little trick that just needs to be found.