# Searching for gold

You are a treasure hunter in search of the legendary gold of the Isle of Goon. According to your research the gold is buried either behind the old church, or under the giant palm tree. You travel to Goon with the intention of asking about the gold. Inhabitants of this island are not known for their truthfulness, so you should be wary about believing what they tell you.

You see two islanders, let’s call them person A and person B, and you ask them where the gold is buried. Person A tells you “The next statement person B makes will be false.” Person B says “If the statement person A just made was false, then the gold is buried behind the church.”

From this information, can you determine where the gold is buried?

Possibilities:

A truth, B truth - impossible, as this contradicts what A is saying

A truth, B lie - impossible. Any logical implication in the form $false \implies x$ is true for all x (see, e.g. https://math.stackexchange.com/questions/137890/why-is-it-sensical-for-a-proposition-with-a-false-antecedent-to-validate-to-true ). So since "the statement person A just made was false" is itself false, the entire statement can't be false, since it's in the form $false \implies (gold.buried.behind.church)$

A lie, B truth - possible, and the gold is buried behind the church

Both lie - impossible, as A would be telling the truth if B was lying

So the gold is behind the church.

Obviously, a big note to the above is it assumes that A knows what B is going to say in advance, and that the two will avoid logical paradoxes. Because, on a common-sense level, it's certainly physically possible for them to both say those statements even when there's no gold behind the church.

Likewise, if you say "Either this statement is a lie or I have a million dollars in my bank", the only non-paradoxical solution is that you have a million dollars... but that doesn't make it true!

To understand the false implies all thing, imagine somebody saying "if (thing that isn't true) then (some other statement)". Like, "if pigs can fly, then I can read minds". You can never call the person a liar for not being able to read minds, because pigs can't fly, so the overall statement isn't false.

• The second part of your answer, about these statements being physically possible with no gold behind the church, is actually the solution I had in mind. In the world I created for this puzzle, the gold is under the giant palm tree. Oct 6 '14 at 22:11
• Note that the "false implies anything is true" idea is true because we defined it that way, not because it must be that way. As the linked answer states, "if a then b" is actually defined as "b or not a". We could have easily defined it to be false whenever a is false, but that's the same thing as "a and b"! To put it differently, we choose "You can never call the person a liar" to mean "the person is telling the truth" because of how we treat vacuous answers. Oct 8 '14 at 19:28
• @EnvisionAndDevelop I agree, though I think this definition is much more intuitive. Oct 8 '14 at 21:13

Short answer: Gold is behind church

A's sentence illustrate that one of them tells truth and the other lies. why?

if A=truth => B=lie

if A=lie => B=truth

Hence,

if A=truth => B=lie ... B's condition is true and is saying that gold is not behind church. (because: the statement person A just made was true, then the gold is not buried behind the church.) B is a liar, So gold is behind church

if A=lie => B=truth ... B's condition is false and is saying that gold is behind church. B is a truth-teller gold is behind church

• "B's condition is true and is saying that gold is not behind church" B isn't saying the gold is not behind the church Oct 6 '14 at 21:59
• @BenAaronson, Yeah that is probably a bug in my answer :D - But I have edited answer. Thanks
– Rafe
Oct 7 '14 at 6:06

If B's statement is true, then A told a lie. Then, by B's statement, the gold is behind the church.

On the other hand, if B lied, B's statement can only be false if A lied. But A truthfully said B would lie. So it is not possible.

• "A told the truth but the gold is behind the church anyway." that wouldn't make B's statement false. Oct 6 '14 at 21:37
• Oops... corrected. The point was that p=>q can be false only if q is false. Oct 6 '14 at 21:41
• But that's not what you need. You need that p=>q can only be false if p isn't false. Oct 6 '14 at 21:44
• Yep. I completely messed it up again. Now I hope it is right. Oct 6 '14 at 21:45
• Looks right to me Oct 6 '14 at 21:45