Four hills H1, H2, H3, and H4 are present in an area. The following observations are made about them: (i) Neither H2 nor H3 is the easternmost hill. (ii) Neither H2 nor H3 is the westernmost hill. (iii) Neither the easternmost hill nor the westernmost hill is the southernmost hill. (iv) Two hills are located to the west of H2. (v) The southernmost hill has at least two hills to its east. The southernmost hill is ________.
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First pin down the west-to-east order using clues (i), (ii) and (iv); this fixes H2 and H3 in the two middle positions. Then use clue (v) to pick between them for the southernmost hill.
Step 1: Assign a numeric west-to-east rank to every hill. Let $r(H)$ denote the position of hill $H$ counted from the west, so $r(H) \in \{1,2,3,4\}$, with $r=1$ the westernmost hill and $r=4$ the easternmost hill. Clues (i) and (ii) tell us $r(H2) \notin \{1,4\}$ and $r(H3) \notin \{1,4\}$, so both $r(H2)$ and $r(H3)$ lie in $\{2,3\}$, which forces $\{r(H1), r(H4)\} = \{1,4\}$. Step 2: Turn clue (iv) into an equation. Clue (iv) states that exactly two hills have a smaller rank than H2, that is $r(H2) - 1 = 2$, giving $r(H2) = 3$. Since $r(H2)$ and $r(H3)$ together occupy $\{2,3\}$, this leaves $r(H3) = 2$. Step 3: Turn clue (v) into an inequality and test each remaining candidate. From clue (iii), the southernmost hill has rank in $\{2,3\}$, so its candidates are H3 ($r=2$) and H2 ($r=3$). Clue (v) requires the southernmost hill to have at least two hills to its east, which in rank terms means $4 - r \geq 2$, i.e. $r \leq 2$. Step 4: Apply the inequality. For H3, $r = 2$, so $4 - 2 = 2 \geq 2$: the condition holds. For H2, $r = 3$, so $4 - 3 = 1 < 2$: the condition fails. Only H3 satisfies $r \leq 2$ among the two candidates. Step 5: Conclude. The hill with rank 2, which is H3, is the only hill consistent with every one of the five clues, so it must be the southernmost hill. \[ \boxed{\text{Southernmost hill} = H3} \]
