Seven friends P, Q, R, S, T, U, and V are sitting in a straight line facing north. Q sits fifth to the right of T. Q does not sit at any of the extreme ends. Two people sit between U and R. S sits third to the left of V. S sits exactly in the middle. U is not an immediate neighbour of P. How many people sit between R and V?
Four
Understanding the Seating Arrangement Puzzle
We are given clues about the seating arrangement of seven friends P, Q, R, S, T, U, and V sitting in a straight line facing North. We need to deduce their exact positions to answer the question.
Key Clues Analysed
- Seven friends: P, Q, R, S, T, U, V.
- Sitting in a straight line, facing North.
- Q sits fifth to the right of T.
- Q does not sit at any of the extreme ends.
- Two people sit between U and R.
- S sits third to the left of V.
- S sits exactly in the middle.
- U is not an immediate neighbour of P.
Step-by-Step Deduction of the Arrangement
Let's use the clues one by one to build the seating arrangement of the seven friends.
Using the Middle Position Clue
There are 7 positions in the line. The middle position is the 4th position.
Clue: S sits exactly in the middle.
| Position 1 | Position 2 | Position 3 | Position 4 | Position 5 | Position 6 | Position 7 |
|---|---|---|---|---|---|---|
| S |
Placing V using S's position
Clue: S sits third to the left of V.
If S is at position 4, and S is to the left of V, V must be to the right of S. Being third to the left of V means V is third to the right of S.
So, V is at position ${4 + 3 = 7}$.
| Position 1 | Position 2 | Position 3 | Position 4 | Position 5 | Position 6 | Position 7 |
|---|---|---|---|---|---|---|
| S | V |
Placing T and Q
Clue: Q sits fifth to the right of T. This implies there are 4 people between T and Q (T _ _ _ _ Q).
Clue: Q does not sit at any of the extreme ends. The extreme ends are Position 1 and Position 7. Position 7 is already taken by V.
So Q cannot be at Position 1 or Position 7. The available positions are 1, 2, 3, 5, 6.
Let's try placing T and Q satisfying "Q is fifth to the right of T" and "Q is not at an end". If Q is at position 6 (not an end), then T would be at position ${6-5=1}$.
| Position 1 | Position 2 | Position 3 | Position 4 | Position 5 | Position 6 | Position 7 |
|---|---|---|---|---|---|---|
| T | S | Q | V |
This arrangement satisfies T _ _ _ _ Q (T at 1, Q at 6) and Q is not at an end.
Placing U and R
Clue: Two people sit between U and R.
The remaining empty positions are 2, 3, and 5.
For two people to sit between U and R, U and R must be separated by two positions. The only pair of available positions that can have two people between them are positions 2 and 5 (with positions 3 and 4 in between). Since position 4 is taken by S, positions 2 and 5 must be where U and R sit.
| Position 1 | Position 2 | Position 3 | Position 4 | Position 5 | Position 6 | Position 7 |
|---|---|---|---|---|---|---|
| T | U/R | S | R/U | Q | V |
So, U and R are at positions 2 and 5 in some order.
Placing P and Finalizing the Arrangement
The only person left to place is P, and the only empty position is 3.
| Position 1 | Position 2 | Position 3 | Position 4 | Position 5 | Position 6 | Position 7 |
|---|---|---|---|---|---|---|
| T | U/R | P | S | R/U | Q | V |
Now, let's use the last clue:
Clue: U is not an immediate neighbour of P.
P is at position 3. The immediate neighbours of P are at positions 2 and 4.
Position 4 is occupied by S, so S is a neighbour of P.
The person at position 2 is also a neighbour of P. According to the clue, U cannot be this neighbour.
Therefore, U cannot be at position 2. This means U must be at position 5, and R must be at position 2.
The Final Seating Arrangement
Replacing U and R based on the last clue, we get the final arrangement:
| Position 1 | Position 2 | Position 3 | Position 4 | Position 5 | Position 6 | Position 7 |
|---|---|---|---|---|---|---|
| T | R | P | S | U | Q | V |
Let's quickly verify all the clues with this final arrangement. All clues are satisfied by the arrangement T R P S U Q V.
Finding the Number of People Between R and V
The question asks: How many people sit between R and V?
In our final arrangement, R is at Position 2 and V is at Position 7.
The people sitting between Position 2 and Position 7 are those at positions 3, 4, 5, and 6.
Looking at the final arrangement (T R P S U Q V), the people between R and V are P, S, U, and Q.
Counting these individuals, we find there are 4 people between R and V.
Conclusion
Based on the step-by-step deduction and the final seating arrangement T R P S U Q V, there are four people sitting between R and V.
Revision Table: Seating Arrangement Summary
| Clue Used | Deduction/Placement | Partial Arrangement |
|---|---|---|
| S in the middle (Position 4) | S is at Pos 4 | _ _ _ S _ _ _ |
| S is 3rd left of V | V is at Pos ${4+3=7}$ | _ _ _ S _ _ V |
| Q is 5th right of T, Q not at ends | T at Pos 1, Q at Pos 6 | T _ _ S _ Q V |
| 2 people between U and R | U & R are at Pos 2 and Pos 5 | T (U/R) _ S (R/U) Q V |
| P takes remaining spot | P is at Pos 3 | T (U/R) P S (R/U) Q V |
| U is not neighbour of P | P neighbours are at 2 & 4. S is at 4. U cannot be at 2. So U is at 5, R is at 2. | T R P S U Q V |
Additional Information: Linear Arrangement Puzzle Solving
Linear arrangement puzzles require careful reading and logical deduction. Here are some tips for solving them:
- Draw the line and mark the positions (e.g., 1 to 7).
- Start with the strongest clues, usually those fixing a position or placing someone relative to an end.
- Absolute positions (like 'middle') are very helpful starting points.
- Relative positions ('left of', 'right of') connect people. Remember directions are based on the person's facing direction (here, North).
- Clues about the number of people 'between' two individuals define a fixed block size.
- Negative clues ('not') are used to eliminate possibilities after some placements are made.
- Always verify your final arrangement against all the original clues to ensure consistency.
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?
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?
5 boys A, B, C, D and E are sitting in a row. E is sitting between D and A, B is to the right of A, C is at one of the extreme ends and C and D are sitting next to each other. Which of the following statement is false?
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?
J, K, M, and N are sitting on a bench facing the same direction. J is sitting next to K but not next to M. N, who is sitting at the extreme right, is not sitting next to either K or M. Which of the following statements is not true?
