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 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.

Answer 1

The Correct Option is C

Step 1: Fix the extreme (east-west) positions.
Line up the four hills from west to east in positions 1, 2, 3, 4, where position 1 is the westernmost hill and position 4 is the easternmost hill. Clues (i) and (ii) say H2 and H3 are neither the easternmost nor the westernmost hill, so neither of them can sit in position 1 or position 4. That forces positions 1 and 4 to be occupied by H1 and H4 (in some order), while H2 and H3 must occupy the two middle positions, 2 and 3.
Step 2: Use clue (iv) to fix H2's exact position.
Clue (iv) says exactly two hills lie to the west of H2. In a west-to-east line of 4 hills, a hill has exactly two hills west of it only if it sits in position 3 (positions 1 and 2 are west of it, position 4 is east of it). If H2 were in position 2, only one hill would be west of it, which contradicts clue (iv). So H2 must be in position 3, and consequently H3 (the only hill left for the middle slots) is in position 2.
Step 3: Write out the full west-to-east order.
The order is: position 1 (H1 or H4), position 2 = H3, position 3 = H2, position 4 (H4 or H1, whichever was not used in position 1).
Step 4: Apply clue (iii) to shortlist the southernmost hill.
Clue (iii) says the southernmost hill is neither the easternmost hill (position 4) nor the westernmost hill (position 1). So the southernmost hill must be one of the two middle hills, H3 (position 2) or H2 (position 3), and options (A) H1 and (D) H4 are ruled out.
Step 5: Apply clue (v) to choose between H2 and H3.
Clue (v) says the southernmost hill has at least two hills to its east. Counting hills strictly east of each position: position 2 (H3) has positions 3 and 4 to its east, i.e. 2 hills, so H3 satisfies clue (v). Position 3 (H2) has only position 4 to its east, i.e. 1 hill, so H2 fails clue (v) and is ruled out as well, eliminating option (B).
Step 6: Conclude.
Only H3 satisfies every clue simultaneously, so H3 is the southernmost hill.
\[ \boxed{\text{Southernmost hill} = H3} \]