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Posted on 02 March 2023, by Kevin Mora

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One of my favorite puzzles in discrete math is the "Queen's Treasure".

There are three chests, and only one of them contains the queen's treasure. Chest 1 and Chest 2 claim to be empty, while Chest 3 claims that "chest 2 has the treasure." You are allowed to open only one chest. How do you find the treasure?

Let $p_i$ denote "treasure is in chest $i$" for $i=1,2,3$. We will refer to these chest inscriptions as following: $\neg p_1, \neg p_2, \ p_2 $.

$$ (\neg p_1 \land \neg\neg p_2 \land p_2) \ \lor \newline (\neg\neg p_1 \land\neg p_2 \land\neg p_2) \ \lor \newline (\neg\neg p_1 \land\neg\neg p_2 \land p_2)$$ $$ (\neg p_1 \land p_2 \land \neg p_2) \ \lor \newline (p_1 \land\neg p_2 \land\neg p_2) \ \lor \newline (p1 \land p_2 \land p_2) $$

At this point, we will find a contradiction: $p_2 \land\neg p_2$. This expression will evaluate to false, so we don't need to worry about it anymore; we will thus refer to it as $F$.

$$ F \lor(p_1 \land\neg p_2) \lor (p_1 \land p_2) $$ $$ (p_1 \land\neg p_2) \lor (p_1 \land p_2) $$ $$ p_1 \land (\neg p_2 \lor p_2) $$ $$ p_1 \land T $$ $$ ∴ p_1 \land T = p_1$$

We just proved that the treasure is in Chest 1 ($p_1$).