Solving the Seating Arrangement Puzzle

This puzzle involves arranging six persons P, Q, R, S, T, and U in a straight row facing north. We are given several clues about their relative positions. To solve this, we need to combine the clues step by step to find the unique arrangement.

Analyzing the Clues

Let's list the given conditions:

• 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.

Building the Arrangement

We should start with the most restrictive clues or those connecting multiple persons. The clue "R sits immediate left of P" gives us a block of two people: RP. This block will always sit together in that order.

Next, consider "One person sits between P and Q". Since we have the RP block, P is followed or preceded by one person before Q. Given R P is a block, the sequence must be Q _ R P or R P _ Q.

Now, let's use "Q sits fourth to the right of S". This means there are three people between S and Q: S _ _ _ Q.

Let's try to combine the sequences. If we use R P _ Q, and Q is fourth to the right of S (S _ _ _ Q). To combine these, S must be placed before the R P block. Let's see if the number of people fits. If the sequence is S _ _ _ Q, and then R P _ Q, we might get something like S _ _ _ R P _ Q. This involves 8 positions, but we only have 6 people. So, R P _ Q combined with S _ _ _ Q in this manner is unlikely in a 6-person row.

Let's try the other sequence involving R, P, and Q: Q _ R P. Now combine this with S _ _ _ Q. The Q in S _ _ _ Q is the same Q as in Q _ R P. So, S must be placed to the left of Q with three people in between. If we place S at the beginning, S is at position 1, then the 4th position to its right is position 5. So, Q must be at position 5. The sequence S _ _ _ Q takes up positions 1 through 5.

Now, let's fit Q _ R P starting from Q at position 5. Q is at 5, then one person (position 6), then R, then P. Since there are only 6 positions total, this doesn't fit (Q=5, \_=6, R=7, P=8).

Let's reconsider placing the RP block first and seeing how the others fit.

Assume the six seats are numbered 1 to 6 from left to right.

The clue "Three persons sit between R and T" implies R and T are at positions like (1 and 5) or (2 and 6).

We know RP is a block.

Case 1: R is at position 1. Then P is at position 2 (R immediate left of P). Three persons between R (1) and T. So T is at position 1 + 4 = 5. The arrangement so far is R P _ _ T _. Positions: R=1, P=2, T=5.

• One person between P (2) and Q. Q must be at position 2+2=4 (P _ Q). Arrangement: R P _ Q T _. Positions: R=1, P=2, Q=4, T=5.
• Q (4) sits fourth to the right of S. S must be at position 4-4=0. This is impossible as position 1 is the first seat. So this case doesn't work.

Case 2: R is at position 2. Then P is at position 3 (R immediate left of P). Three persons between R (2) and T. T must be at position 2+4 = 6. The arrangement so far is _ R P _ _ T. Positions: R=2, P=3, T=6.

• One person between P (3) and Q. Q must be at position 3+2=5 (P _ Q). Arrangement: _ R P _ Q T. Positions: R=2, P=3, Q=5, T=6.
• Q (5) sits fourth to the right of S. S must be at position 5-4=1. Arrangement: S R P _ Q T. Positions: S=1, R=2, P=3, Q=5, T=6.

Let's check if this partial arrangement S R P _ Q T (positions 1, 2, 3, 5, 6 filled) fits all conditions for 6 persons. The persons placed are S, R, P, Q, T. The missing person is U. There is one empty spot at position 4. So U must be at position 4.

The complete arrangement is S R P U Q T.

Verifying the Final Arrangement (S R P U Q T)

• R sits immediate left of P: R is at 2, P is at 3. Yes.
• One person sits between P and Q: P is at 3, Q is at 5. U (at 4) sits between them. Yes.
• Three persons sit between R and T: R is at 2, T is at 6. P (3), U (4), Q (5) sit between them. Yes.
• Q sits fourth to the right of S: S is at 1, Q is at 5. Positions 2, 3, 4 are between them. Q is at the 4th position to the right of S (1→2→3→4→5). Yes.

All conditions are satisfied by the arrangement S R P U Q T.

Finding Persons Between T and U

In the final arrangement S R P U Q T:

• U is at position 4.
• T is at position 6.

The persons sitting between U and T are those at positions 5. The person at position 5 is Q.

Therefore, only one person (Q) sits between T and U.

Conclusion

Based on the given clues, the final arrangement of the six persons is S R P U Q T. In this arrangement, only one person sits between T and U.