P, Q, R and S are sitting on a bench and facing in the same direction. P sits next to Q but not next to R. S, who is sitting on the extreme right, does not sit next to either Q or R. How many of them each have two neighbors?
P and Q
Solving the Seating Arrangement Puzzle
Let's break down the clues provided in the question to determine the seating arrangement of P, Q, R, and S on the bench. They are all facing in the same direction.
Analyzing the Clues
- Clue 1: P sits next to Q. This means P and Q are adjacent to each other. Possible pairs are PQ or QP.
- Clue 2: P does not sit next to R. R is not adjacent to P.
- Clue 3: S is sitting on the extreme right. The arrangement ends with S: _ _ _ S
- Clue 4: S does not sit next to either Q or R. This is a crucial clue for the position next to S.
Step-by-Step Deduction
- From Clue 3, we know S is on the extreme right: _ _ _ S
- From Clue 4, the person sitting next to S cannot be Q or R. The only person left is P. So, P must be sitting next to S. The arrangement becomes: _ _ P S
- From Clue 1, P sits next to Q. Since S is on one side of P, Q must be on the other side of P (to the left). The arrangement is now: _ Q P S
- We have one spot left and one person left (R). The remaining spot is on the extreme left, so R must sit there. The final arrangement is: R Q P S
- Let's double-check this arrangement against all clues:
- P next to Q? Yes, Q and P are together.
- P not next to R? Yes, Q is between R and P.
- S on the extreme right? Yes.
- S not next to Q? Yes, P is between S and Q.
- S not next to R? Yes, P and Q are between S and R.
Identifying Neighbors
Now that we have the seating order R Q P S, let's see how many neighbors each person has. People sitting at the ends of the bench have only one neighbor, while people in the middle have two neighbors.
- R: Is at the extreme left. Has only one neighbor (Q).
- Q: Is in the middle. Has two neighbors (R and P).
- P: Is in the middle. Has two neighbors (Q and S).
- S: Is at the extreme right. Has only one neighbor (P).
The people with two neighbors are Q and P.
Therefore, two people among them each have two neighbors.
Final Answer Breakdown
Based on the deduced arrangement and neighbor analysis:
| Person | Neighbors | Number of Neighbors |
| R | Q | 1 |
| Q | R, P | 2 |
| P | Q, S | 2 |
| S | P | 1 |
The question asks how many of them each have two neighbors. From the table, P and Q each have two neighbors.
The specific individuals who have two neighbors are P and Q.
Seating Arrangement Revision Table
| Clue | Deduction | Partial Arrangement |
| S on extreme right | S is last | _ _ _ S |
| S not next to Q or R | Person next to S must be P | _ _ P S |
| P next to Q | Q must be left of P | _ Q P S |
| Remaining person R | R fills the last spot | R Q P S |
Additional Information on Linear Arrangements
Linear arrangement puzzles involve arranging people or objects in a straight line or row based on given conditions. Key concepts include:
- Extreme Ends: The positions at the very beginning and very end of the line. People at the ends have only one neighbor.
- Middle Positions: Any position that is not at an extreme end. People in the middle positions have two neighbors (one on each side).
- Adjacent/Next to: Indicates that two items are directly beside each other.
- Solving Strategy: Start with the most concrete clues, especially those fixing positions (like extreme ends) or directly linking/excluding items. Build the arrangement step-by-step, checking against all clues as you go.
Important Questions from Linear Arrangement
Seven people, A, B, C, D, E, F and G, are sitting in a straight row facing the north. Only two people sit to the left of C. Only two people sit between A and B. B sits to the left of A. E is an immediate neighbour of A to the right. Only one person sits to the right of D. F is not an immediate neighbour of B. Who sits at the centre?
Five friends Ram, Priya, Raunak, Aditya and Shyam are sitting in a row facing north. Raunak is to the right of Shyam and Priya is to the left of Shyam but to the right of Rama. Raunak is to the left of Aditya. Who is fourth from the left end?
Six people P, Q, R, S, T and U sit in a row, not necessarily in the same order. If two persons sit in between P and Q, and two persons sit in between T and U, then how many persons sit in between R and S?
Six persons P, Q, R, S, T and U are sitting in a row facing north (not necessarily in the same order). Q sits fourth to the right of S. One person sits between P and Q. Three persons sit between R and T. R sits immediate left of P. How many persons sit between T and U?
Five chairs C01, C02, C03, C04 and C05 are placed one above the other but not necessarily in the same order. If C03 is above C02, C01 is above C05 but below C04 and C05 is above C03, which chair is in the middle?
