Question:
Four hills \(H_1, H_2, H_3\), and \(H_4\) are present in an area. The following observations are made about them:
(i) Neither \(H_2\) nor \(H_3\) is the easternmost hill.
(ii) Neither \(H_2\) nor \(H_3\) 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 \(H_2\).
(v) The southernmost hill has at least two hills to its east.
The southernmost hill is .
Four hills \(H_1, H_2, H_3\), and \(H_4\) are present in an area. The following observations are made about them:
(i) Neither \(H_2\) nor \(H_3\) is the easternmost hill.
(ii) Neither \(H_2\) nor \(H_3\) 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 \(H_2\).
(v) The southernmost hill has at least two hills to its east.
The southernmost hill is .
(i) Neither \(H_2\) nor \(H_3\) is the easternmost hill.
(ii) Neither \(H_2\) nor \(H_3\) 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 \(H_2\).
(v) The southernmost hill has at least two hills to its east.
The southernmost hill is .
Show Hint
First use clues (i) and (ii) to see that only \(H_1\) and \(H_4\) can be the east/west extremes; then use clue (iv) to fix \(H_2\)'s exact position, and finally check clue (v) against the two remaining candidates for southernmost.
Updated On: Aug 10, 2026
- \(H_1\)
- \(H_2\)
- \(H_3\)
- \(H_4\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Fix the east-west extremes.
From clue (i), neither \(H_2\) nor \(H_3\) is the easternmost hill. From clue (ii), neither \(H_2\) nor \(H_3\) is the westernmost hill. So both the easternmost and the westernmost hill must come from \(\{H_1, H_4\}\). That leaves \(H_2\) and \(H_3\) to occupy the two middle positions in the east-west order.
Step 2: Place \(H_2\) using clue (iv).
Rank the four hills from west to east as position 1 (westernmost) to position 4 (easternmost). Clue (iv) says two hills lie west of \(H_2\), so \(H_2\) sits at position 3, with positions 1 and 2 to its west and position 4 to its east. This is consistent with Step 1, since position 3 is one of the two middle slots.
Step 3: Use clues (iii) and (v) to find the southernmost hill.
Clue (iii) says the southernmost hill is not the easternmost or westernmost hill, so the southernmost hill is \(H_2\) or \(H_3\).
Check \(H_2\): from Step 2, only position 4 lies east of \(H_2\), so only one hill is to its east. Clue (v) needs at least two hills east of the southernmost hill, so \(H_2\) fails this test and cannot be the southernmost hill.
So the southernmost hill is \(H_3\).
Step 4: Fix \(H_3\)'s east-west position and check clue (v).
Position 3 is taken by \(H_2\), and \(H_3\) cannot be at position 1 (westernmost, ruled out by clue (ii)) or position 4 (easternmost, ruled out by clue (i)). So \(H_3\) must sit at position 2. That leaves 2 hills (positions 3 and 4) to its east, which satisfies clue (v) exactly.
Every clue is now satisfied with the west-to-east order: one of \(H_1/H_4\), then \(H_3\), then \(H_2\), then the other of \(H_4/H_1\).
Why the other options are wrong:
\(H_1\) is wrong because it is one of the two extreme (easternmost or westernmost) hills, and clue (iii) rules out an extreme hill being southernmost.
\(H_2\) is wrong because it has only one hill to its east, which fails clue (v).
\(H_4\) is wrong for the same reason as \(H_1\), it is the other extreme hill.
Final Answer:
The southernmost hill is \(H_3\). \[ \boxed{H_3} \]
From clue (i), neither \(H_2\) nor \(H_3\) is the easternmost hill. From clue (ii), neither \(H_2\) nor \(H_3\) is the westernmost hill. So both the easternmost and the westernmost hill must come from \(\{H_1, H_4\}\). That leaves \(H_2\) and \(H_3\) to occupy the two middle positions in the east-west order.
Step 2: Place \(H_2\) using clue (iv).
Rank the four hills from west to east as position 1 (westernmost) to position 4 (easternmost). Clue (iv) says two hills lie west of \(H_2\), so \(H_2\) sits at position 3, with positions 1 and 2 to its west and position 4 to its east. This is consistent with Step 1, since position 3 is one of the two middle slots.
Step 3: Use clues (iii) and (v) to find the southernmost hill.
Clue (iii) says the southernmost hill is not the easternmost or westernmost hill, so the southernmost hill is \(H_2\) or \(H_3\).
Check \(H_2\): from Step 2, only position 4 lies east of \(H_2\), so only one hill is to its east. Clue (v) needs at least two hills east of the southernmost hill, so \(H_2\) fails this test and cannot be the southernmost hill.
So the southernmost hill is \(H_3\).
Step 4: Fix \(H_3\)'s east-west position and check clue (v).
Position 3 is taken by \(H_2\), and \(H_3\) cannot be at position 1 (westernmost, ruled out by clue (ii)) or position 4 (easternmost, ruled out by clue (i)). So \(H_3\) must sit at position 2. That leaves 2 hills (positions 3 and 4) to its east, which satisfies clue (v) exactly.
Every clue is now satisfied with the west-to-east order: one of \(H_1/H_4\), then \(H_3\), then \(H_2\), then the other of \(H_4/H_1\).
Why the other options are wrong:
\(H_1\) is wrong because it is one of the two extreme (easternmost or westernmost) hills, and clue (iii) rules out an extreme hill being southernmost.
\(H_2\) is wrong because it has only one hill to its east, which fails clue (v).
\(H_4\) is wrong for the same reason as \(H_1\), it is the other extreme hill.
Final Answer:
The southernmost hill is \(H_3\). \[ \boxed{H_3} \]
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