Puzzle 46 | Rich or Poor

A place has two kinds of residents, Poor, who always tell the truth, and their opposites, Rich, who always lie. You encounter two people A and B. What are A and B if A says “B is a Poor” and B says “The two of us are opposite types” ?

Answer: Both A and B are Rich

Solution: Let p and q be the statements that A is a Poor and B is a Poor, respectively, so that ¬p and ¬q are the statements that A is a Rich and B is a Rich, respectively. Let us consider the possibility that A is a Poor, this is the statement that p is true. If A is a Poor, then he is telling the truth when he says that B is a Poor, so that q is true, and A and B are the same type. However, if B is a Poor, then B’s statement that A and B are of opposite types, the statement  (p ? ¬q) ? (¬p ? q), would have to be true, which it is not, because A and B are both Poors. Consequently, we can conclude that A is not a Poor, that is, that p is false. If A is a Rich, then because everything a Rich says is false, A’s statement that B is a Poor, that is, that q is true, is a lie. This means that q is false and B is also a Rich. Furthermore, if B is a Rich, then B’s statement that A and B are opposite types is a lie, which is consistent with both A and B being Rich. We can conclude that both A and B are Rich.

This puzzle is contributed by Feroz Baig.