Solution
Da or Ne, and the Gold
Show the problem again
On an island of knights (truthful) and knaves (liars), the natives answer yes/no questions with one of two words, "Da" or "Ne", and you do not know which one means yes. You meet a native whose type is unknown and ask him: "If I asked you whether there is gold on this island, would you answer 'Da'?" He replies "Da." Is there gold on the island? Answer yes or no.
Worked solution
Write $m$ for the truth value of "Da means yes". A speaker who answers a question with yes/no value $y$ says "Da" exactly when $y = m$. Let $g$ be the truth about gold and take a knight first. Asked directly, he would say Da exactly when $g = m$. The nested question therefore has true answer $u = (g = m)$, and he reports Da exactly when $u = m$, that is, when $(g = m) = m$, which simplifies to $g$. For a knave, the direct answer has value $\neg g$, so he would say Da exactly when $\neg g = m$; asked about that fact he lies, and the second lie cancels the first, again giving Da exactly when $g$. So in every one of the eight worlds (gold or not, Da meaning yes or no, knight or knave) the reply is Da if and only if there is gold. We do not learn what Da means or who the speaker is, but we do learn the answer. Answer: yes, there is gold.
Source: In the spirit of Raymond Smullyan's knights-and-knaves puzzles and George Boolos's 'Hardest Logic Puzzle Ever' (1996), whose answerers reply 'da' or 'ja'; the embedded-question trick is folklore. Statement written for AxiomIQ.