Challenge #40 Result
Wow! So many responses were received to this puzzle....over 300. Many strategies were attempted, and conflicting answers were offered, which makes for excellent dialogue. The reply I selected was from Lauri Kultii, from Finland, which I have included below. Thanks to all Zeno's Patrons who submitted answers, and keep coming back to the Coffeehouse, where mental pleasures are welcomed! The mind is a wonderful tool....Barnette
Here is the original problem, along with Lauri's response:
Challenge: One night at Zeno's a Guinness-loving ale patron placed a bet that nobody in the Coffeehouse could meet this challenge: First, she placed on a table two boxes labeled "A" and "B". A sign on box A says "The sign on box B is true and the bottle of Guinness is in box A". A sign on box B says "The sign on box A is false and the Guinness is in box A". Assuming there is a bottle of Guinness in one of the boxes, which box contains the bottle? Can you figure it out from the information provided? Explain your answer.
Response: city: Helsinki
comments: Let: 'p' stand for 'The sign on box B is true'
'q' stand for 'The bottle of Guinness is in box A'
'r' stand for 'The sign on box A is false'.
Now it can be shown that assuming Guinness to be in box A yields a contradiction in every possible state of affairs.
So, let's assume: the Guinness is in box A.
Now we have just two possibilities concerning the status of sign on box A; either the sign (the proposition the sign states) is true or it is false. Let's consider each possibilities in turn.
1. If the sign on box A is true it states 'p AND q'. So, p is true. That is, the sign on box B is true. It means that 'r AND q' is true. But 'r' means that sign on box A is false which contradicts our assumption at the beginning of this paragarph, namely that sign on box A is true.
2. What about if the sign on box A is false? Then either p or q or both should be false. Let's examine all possibilities in turn.
i) 'p AND NOT-q'. NOT-q straightforwardly contradicts our very first assumption that the Guinness is in box A.
ii) 'NOT-p AND q'. NOT-p means that sign on box B (that is 'r AND q') is false. That can happen in three possible ways.
a) 'NOT-r AND q'. NOT-r contradicts our assumption at the beginning of 2 (i.e. that sign on box A is false).
b) 'r AND NOT-q'. NOT-q contradicts our very first assumption that Guinness is in box A.
c) 'NOT-r AND NOT-q'. Same as above.
iii) 'NOT-p AND NOT-q'. NOT-q again contradicting the very first assumption that Guinness is in box A.
So, under assumption that Guinness is in box A a contradiction is inevitable. So, we deduce that Guinness is in box B (as it was stated that Guinness is in one of the boxes). (By the way, assuming that Guinness is in box B does not yield contradiction in all possible states of affairs. I tried.)