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: Set up positions along the west-to-east line.
Since the clues talk about "easternmost", "westernmost", and hills "to the west or east" of each other, place the four hills at 4 positions along a line, numbered 1 (westernmost) to 4 (easternmost).

Step 2: Use clues (i) and (ii) to place H2 and H3.
Clue (i) says neither H2 nor H3 is at position 4, and clue (ii) says neither is at position 1. So H2 and H3 must occupy the two middle positions, 2 and 3, in some order, which forces H1 and H4 to occupy positions 1 and 4 in some order.

Step 3: Use clue (iv) to fix H2's exact position.
Clue (iv) says exactly two hills lie to the west of H2. If H2 were at position 2, only 1 hill (position 1) lies west of it, which is too few. If H2 were at position 3, exactly 2 hills (positions 1 and 2) lie to its west, matching the clue. So H2 is at position 3, which leaves position 2 for H3.

Step 4: Use clue (iii) to rule out the ends.
Clue (iii) says the southernmost hill is not the easternmost or the westernmost hill. So the southernmost hill must be H3 (position 2) or H2 (position 3), since those are the only two positions left.

Step 5: Use clue (v) to choose between H3 and H2.
Clue (v) says the southernmost hill has at least two hills to its east. Position 2 (H3) has positions 3 and 4 to its east, which is 2 hills, so it satisfies the clue. Position 3 (H2) has only position 4 to its east, just 1 hill, which fails the at-least-two requirement. So the southernmost hill cannot be H2.

Final Answer:
Only H3 satisfies every clue, so H3 is the southernmost hill. \[ \boxed{\text{H3}} \]