Q is to the right of R but to the left of P. S is to the left of R and to the…

In a single-row linear-arrangement puzzle, every clue of the form 'X is to the right/left of Y' is really just an inequality between X and Y's positions. Convert each clue into an inequality, chain together the ones that share a common person, and you get one total order for everybody in the row; the seat exactly halfway along that order is the middle seat.

1. "Q is to the right of R" gives R < Q (R comes before Q in the row).

2. "Q is to the left of P" gives Q < P. Combined with step 1: R < Q < P.

3. "S is to the left of R" gives S < R.

4. "S is to the right of T" gives T < S. Combined with step 3: T < S < R.

5. R is common to both chains, so they link into a single order: T < S < R < Q < P.

6. Reading left to right, the five seats are T, S, R, Q, P — seats 1 to 5.

7. With five seats, the seat exactly in the middle is seat 3, occupied by R.

Cross-check against every original clue:

• Q is to the right of R (seats 4 and 3) — holds.

• Q is to the left of P (seats 4 and 5) — holds.

• S is to the left of R (seats 2 and 3) — holds.

• S is to the right of T (seats 2 and 1) — holds.

• Every original clue is satisfied, and only R has exactly two people on each side of it.