Four couples M-N, O-P, R-L, S-E meet to have a dinner together on an eight-seater rectangular dining table with four seats each on opposite sides. The men are named as M, O, R, S. Two of the men, O, R, sit on the two corners of one side of the table and M sits beside R. Further, none of the couples occupy seats opposite to each other. However, a person sits adjacent to the seat opposite to his/her spouse. Based on the information, which of the following conclusions can be drawn?
Table Setup and Constraints
The problem describes a seating arrangement puzzle around an eight-seater rectangular dining table. The table has two opposite sides, each seating four people. We can label the seats as follows:
Side 1: $[T1]--[T2]--[T3]--[T4]$ Side 2: $[B1]--[B2]--[B3]--[B4]$ Where T1 is opposite B1, T2 opposite B2, T3 opposite B3, and T4 opposite B4.
Key constraints:
- Four couples are involved: M-N, O-P, R-L, S-E. The men are M, O, R, S.
- O and R sit at the corners of one side (e.g., Side 1).
- M sits adjacent to R on the same side.
- No couple sits directly opposite each other.
- Crucially, a person sits adjacent to the seat that is opposite their spouse's seat. This implies spouses are seated diagonally across from each other.
Deducing the Seating Arrangement
We analyze the seating possibilities based on the constraints. Let's assume O and R sit on Side 1.
Scenario 1: O at T1, R at T4
- Given O is at a corner (T1) and R is at the other corner (T4) on Side 1.
- M sits beside R (T4), so M must be at T3. The arrangement starts: $[O]--[T2]--[M]--[R]$.
- Apply the spouse rule (Person A at seat X, spouse B, opposite seat Y, adjacent to Y are Z1, Z2 => B is at Z1 or Z2):
- R (at T4), spouse L: Seat opposite T4 is B4. Seat adjacent to B4 is B3. Thus, L must be at B3.
- O (at T1), spouse P: Seat opposite T1 is B1. Seat adjacent to B1 is B2. Thus, P must be at B2.
- M (at T3), spouse N: Seat opposite T3 is B3. Seats adjacent to B3 are B2 and B4. Since P is at B2, N must be at B4.
- The arrangement on Side 2 is now determined for P, L, N: $[B1]--[P]--[L]--[N]$.
- The remaining unassigned seats are T2 (Side 1) and B1 (Side 2). The remaining people are S and E, who are spouses.
- Checking the spouse rule for S-E: If S is at T2 and E is at B1 (or vice-versa), the rule holds. For example, S(T2) and spouse E(B1). Opposite T2 is B2. Seats adjacent to B2 are B1 and B3. E is at B1, which is adjacent to B2. This configuration is valid.
- A possible valid seating arrangement is: Top side: $[O]-[S/E]-[M]-[R]$, Bottom side: $[E/S]-[P]-[L]-[N]$.
Scenario 2: R at T1, O at T4
- Given R is at a corner (T1) and O is at the other corner (T4) on Side 1.
- M sits beside R (T1), so M must be at T2. The arrangement starts: $[R]--[M]--[T3]--[O]$.
- Apply the spouse rule:
- R (at T1), spouse L: Seat opposite T1 is B1. Seat adjacent to B1 is B2. Thus, L must be at B2.
- O (at T4), spouse P: Seat opposite T4 is B4. Seat adjacent to B4 is B3. Thus, P must be at B3.
- M (at T2), spouse N: Seat opposite T2 is B2. Seats adjacent to B2 are B1 and B3. Since L is at B2 and P is at B3, N must be at B1.
- The arrangement on Side 2 is now determined for N, L, P: $[N]--[L]--[P]--[B4]$.
- The remaining unassigned seats are T3 (Side 1) and B4 (Side 2). The remaining people are S and E, who are spouses.
- Checking the spouse rule for S-E: If S is at T3 and E is at B4 (or vice-versa), the rule holds. For example, S(T3) and spouse E(B4). Opposite T3 is B3. Seats adjacent to B3 are B2 and B4. E is at B4, which is adjacent to B3. This configuration is valid.
- A possible valid seating arrangement is: Top side: $[R]-[M]-[S/E]-[O]$, Bottom side: $[N]-[L]-[P]-[E/S]$.
Evaluating the Conclusions
Both deduced seating arrangements are valid and satisfy all conditions. We now check the given options based on these arrangements.
- Option A: R sits opposite to N and S sits opposite to P.
- Scenario 1: R(T4) is opposite N(B4). S/E(T2) is opposite P(B2). This matches.
- Scenario 2: R(T1) is opposite N(B1). S/E(T3) is opposite P(B3). This matches.
- Conclusion: Option A is correct.
- Option B: R sits opposite to P and O sits opposite to N.
- In both scenarios, R is opposite N (not P), and O is opposite E/S (not N).
- Conclusion: Option B is incorrect.
- Option C: S sits opposite to N and O sits opposite to L.
- In both scenarios, S is opposite P (not N), and O is opposite E/S (not L).
- Conclusion: C is incorrect.
- Option D: M sits opposite to L and O sits opposite to E.
- Scenario 1: M(T3) is opposite L(B3). O(T1) is opposite E/S(B1). This matches.
- Scenario 2: M(T2) is opposite L(B2). O(T4) is opposite E/S(B4). This matches.
- Conclusion: Option D is correct.
The conclusions that can be drawn from the given information are A and D.
Important Questions from Seating Arrangement
- P and Q have been allotted a hostel room with two beds, a study table, and an almirah. P is an avid bird-watcher and wants to sit at the table and watch birds outside the window. Q does not mind that as long as his bed is close to the ceiling fan.
Which one of the following arrangements suits them the most? Six persons P, Q, R, S, T and U are sitting around a circular table facing the center not necessarily in the same order. Consider the following statements:
• P sits next to S and T.
• Q sits diametrically opposite to P.
• The shortest distance between S and R is equal to the shortest distance between T and U.
Based on the above statements, Q is a neighbor ofSeven cars P, Q, R, S, T, U and V are parked in a row not necessarily in that order. The cars T and U should be parked next to each other. The cars S and V also should be parked next to each other, whereas P and Q cannot be parked next to each other. Q and S must be parked next to each other. R is parked to the immediate right of V. T is parked to the left of U.
Based on the above statements, the only INCORRECT option given below is:- Four persons P, Q, R and S are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and R cannot sit adjacent to each other. S should be seated to the right of Q. The number of distinct seating arrangements possible is:
- P, Q, R, S, T and U are seated around a circular table. R is seated two places to the right of Q. P is seated three places to the left of R. S is seated opposite U. If P and U now switch seats, which of the following must necessarily be true?
