Question

Which of the following is the correct position of H?

• A. Left of C
• B. Exactly Opposite of B
• C. Exactly Opposite of E
• D. Cannot be Determined
• A. B is to the immediate right of F.
• B. G is sitting two places to the left of F.
• C. There are only six persons whose sitting positions can be exactly determined.
• D. All of these.

Answer 1

The Correct Option is B

Step 1: Fix anchors using symmetry.
Place \(C\) at the top (without loss of generality). Since \(F\) is opposite \(C\), put \(F\) opposite \(C\).
Condition (iii) “\(H\) is the only one sitting between \(G\) and \(C\)” - the three consecutive seats are either \(G\!-\!H\!-\!C\) or \(C\!-\!H\!-\!G\).
Step 2: Try both orders; reject the impossible one.
If \(G\!-\!H\!-\!C\) (clockwise), then using (iv) \(E\) two to the right of \(B\) and “\(E\) not next to \(F\)” leads to a contradiction (no place for \(E\)).
Hence the only feasible order is \(C\!-\!H\!-\!G\) (clockwise).
Step 3: Place \(E\) and \(B\).
\(E\) is two to the right of \(B\) and not next to \(F\). The only way to satisfy both is \(E\) on the seat just before \(C\) (clockwise), which forces \(B\) two seats to its left (clockwise).
Step 4: Fill the remaining seats.
The last two seats go to \(A\) and \(D\), with \(D\) not next to \(F\) - \(D\) sits two to the right of \(F\).
Final circular order (clockwise): \(C,\;H,\;G,\;A,\;F,\;B,\;D,\;E\).
Here, \(H\) is directly opposite \(B\). \(\Rightarrow\) \(\boxed{\text{Exactly Opposite of B}}\).

Answer 2

From the finalized order \(C,\;H,\;G,\;A,\;F,\;B,\;D,\;E\) (clockwise):
\(\bullet\) Immediate right of \(F\) is \(B\) \(\Rightarrow\) (a) is true.
\(\bullet\) Two places to the left of \(F\) is \(G\)\( \Rightarrow\) (b) is true.
\(\bullet\) All eight positions are uniquely fixed, not just six\( \Rightarrow \)(c) is false.
Hence the only statement that is not correct is \(\boxed{(c)}\).