# Mathematics riddle

The question is as follows:

You are taking part in a treasure hunt, where the directions to finding the treasure are given using cryptic clues. You start at a cross-roads, with roads heading out east, west, north, and south. You receive the following instructions about heading towards the treasure from your current location:

(a) If you travel north or travel east, then you will not find the treasure

(b) If the treasure is to the south, then you can also get to the treasure by traveling east

Assuming that all the instruction given to you are true, which direction should you head to find the treasure? Justify your answer.

I think the treasure cannot be in north or east, can it? Perhaps the treasure can only be in south or west? Whereas we can take both the south and east road to get to the treasure in the south, I am not sure which way we can take to get to west. Is it just west alone? It must not, must it?

• The way it is phrased, the obvious conclusion is that the treasure can only be toward the west. But this is not very difficult, since $A: \neg N \wedge \neg E$, $B: S \to E$ giving $B': \neg E \to \neg S$ so $A+B': \neg N \wedge \neg E \wedge \neg S$ Sep 11 '14 at 10:41

Let $N,E,S,W$ denote the proposition that the treasure can be reached by going north (east, south, west resp.). Then we can formulate \begin{align*} A:& \neg N \wedge \neg E\\ B:& S \to E \\ B':& \neg E \to \neg S \\ B'':& \text{true} \to \neg S \\ C:& \neg N \wedge \neg E \wedge \neg S \end{align*} Where $B'$ is obtained by contraposition, $B''$ by using a part of $A$ ($\neg E$) and $C$ by combining $A$ and $B''$ into one. With the hypothesis that $$D: N \vee E \vee S \vee W$$ i.e. that at least one direction is correct, we obtain $$D\wedge C: (N \vee E \vee S \vee W) \wedge (\neg N \wedge \neg E \wedge \neg S) \Rightarrow W \wedge \neg N \wedge \neg S \wedge \neg E$$ in words: west is the only possible direction, assuming at least one direction leads to the treasure and (a) and (b) are true.
$(a)$ implies that you will not find the treasure when travelling north or east. So you can ONLY find the treasure when traveling south or west.
$(b)$ says: IF the treasure is in the south, you also get it when travelling east. This is equivalent to: If cannot get the treasure when traveling to east the treasure will NOT be in the south. (Contraposition) As you know by $(a)$ that the trasure is NOT in the east you now know that it also is NOT in the south. The only place left now is west.