answer: true/false boxes

Answer to old chestnut: true/false boxes:

The intended answer to this old brainteaser is that the gold is in box B.

This is arrived at as follows:

Assume the sign on A is true. Then the gold is in box A and the sign on box B is true. But the sign on box B says the sign on box A is false. We've arrived at a contradiction, so we take a step back and scrap our assumption; the sign on A must be false.

So now assume the sign on B is true. The first part is already true; we've found that the sign on A is false. The second part says that the gold is in box A. However, we've reached another contradiction. If the sign on B is true and the gold is in box A, then the sign on A is true!

So assume both signs are false. Then the first part of the sign on B is true, so the second part must be false -- the gold is in box B. Then the sign on A is false as well -- doubly false, in fact.

However, there's a general problem with such problems: we assumed that each statement was either true or false. It is possible to write a statement that cannot be called true or false without creating an inconsistency; the classic example is the Liar's Paradox: "This statement is false." Note that this is exactly what the signs fall into is you put the gold into box A. With those parts being true, the signs reduce to A:"B is true" and B:"A is false". This is a general problem with statements that make claims of truth or falsity of other statements in the same group, and should generally be looked out for in such problems.

This reminds me of a funny joke I played on my little (8 yr. old) brother the other day. I took two paper cups and wrote on cup A "The writing on cup B is FALSE and there is not a dollar under cup B," and on cup B, "The writing on cup A is TRUE and there is not a dollar under cup B," (TRUE and FALSE in all-caps to impress the credulous, just like GOD in the Bible) and told little Freddy that I had placed a dollar under just one of the cups and if he picked the right cup he could keep it but if not he would be taken out of his gifted-and-talented math class and put back with the normals. He got excited and went off talking to himself and drawing diagrams and generally acting like a cute little gifted kid when you give 'em a chance to show off, and five minutes later ran back and pointed to cup B. Whereupon I took my dollar from where I had left it under cup A, and bought an ice cream cone and ate it in front of him as the tears rolled down his face. "But... I saw the dollar sticking out under cup A," he started yammering, as soon as his sobbing subsided enough to allow speech, "I just picked B because it had to be B." I just shook my head sadly and then looked at him and said here's a related problem you might find interesting and I gave him a piece of paper, on one side it said "How do you keep a normal busy? (over)" and on the other side it said "How do you keep a normal busy? (over)" he looked at it and flung it at me and ran to his room.

This is a work of fiction, intended only to dramatize the the "general problem" described by /dev/joe at the end of the last writeup.