Question

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 ________.

• A. H1
• B. H2
• C. H3
• D. H4

Hint:

First place all four hills in a west-to-east order using clues (i), (ii) and (iv); this pins down H2 and H3 to the two middle spots and fixes H2 at position 3.
Then use clues (iii) and (v) to test which of the two middle hills can be the southernmost one.

Answer 1

The Correct Option is C

Step 1: List what is known and unknown separately.
There are two separate orderings buried in the clues: an east-west order of all four hills, and a single southernmost label that applies to just one of them. Handle the east-west order first, since most clues (i, ii, iv) are about it.

Step 2: Eliminate positions for H2 and H3 using clues (i) and (ii).
Clue (i) removes H2 and H3 from the easternmost spot; clue (ii) removes them from the westernmost spot. With 4 hills and 4 east-west spots, removing 2 hills from 2 of the 4 spots leaves only the middle two spots open to them, so $\{H2, H3\}$ occupies $\{$position 2, position 3$\}$ and $\{H1, H4\}$ occupies $\{$position 1, position 4$\}$.

Step 3: Pin down H2 using the count in clue (iv).
Clue (iv) gives a number: exactly 2 hills west of H2. Testing position 2 for H2 gives only 1 hill to the west, which does not match. Testing position 3 gives exactly 2 hills to the west, which matches. So H2 must sit at position 3, and H3 takes position 2 by elimination.

Step 4: Build the southernmost candidate list from clue (iii).
Clue (iii) removes both position 1 and position 4 from being the southernmost hill, whatever hills sit there. Since H1 and H4 occupy those end positions, the southernmost hill must be whichever hill sits in the middle, that is, either H3 (position 2) or H2 (position 3).

Step 5: Apply the east-count test from clue (v) to both candidates.
For H3 at position 2, hills to its east are positions 3 and 4, a count of 2, meeting the at-least-two bar exactly. For H2 at position 3, only position 4 lies east of it, a count of 1, which fails the bar. So H2 is eliminated, leaving H3 as the only hill that can be the southernmost one.

Final Answer:
Working through the east-west order first and then applying the east-count rule shows H3 is the southernmost hill, so option (C) is correct. \[ \boxed{\text{H3}} \]