The Roman senator Meritorius, his brother, his son, and his daughter have varying oratory skill levels. They are seated in rows and columns as shown in the figure with exactly one person sitting in each box. It is known that
(i) Meritorius' daughter and his brother are seated in the same column.
(ii) His son is seated diagonally across the sibling of the worst orator.
(iii) The best and worst orators are seated in the same row.
Who is the best orator?
Column 1
Column 2
Row 1
Row 2
Show Hint
Fix the daughter and brother in one column using clue (i), then use the diagonal-seat clue (ii) to find who is the worst orator before applying the same-row clue (iii).
Step 1: Label the four seats.
Call the seats R1C1, R1C2, R2C1 and R2C2, one person per seat. Diagonal pairs are (R1C1, R2C2) and (R1C2, R2C1).
Step 2: Fix the daughter and brother using clue (i).
D and Br share a column. Take that to be Column 1, without losing anything, so $\{D,Br\}=\{R1C1,R2C1\}$ in some order. Then $\{M,S\}=\{R1C2,R2C2\}$ in some order in Column 2.
Step 3: Enumerate all four placements.
Case (D=R1C1, Br=R2C1, S=R2C2, M=R1C2): diagonal of S is R1C1=D, so D's sibling S is the worst orator; S's row (Row 2) also holds Br, so Br is the best orator.
Case (D=R1C1, Br=R2C1, S=R1C2, M=R2C2): diagonal of S is R2C1=Br, so Br's sibling M is the worst orator; M's row (Row 2) also holds Br, so Br is the best orator.
Case (D=R2C1, Br=R1C1, S=R2C2, M=R1C2): diagonal of S is R1C1=Br, so Br's sibling M is the worst orator; M's row (Row 1) also holds Br, so Br is the best orator.
Case (D=R2C1, Br=R1C1, S=R1C2, M=R2C2): diagonal of S is R2C1=D, so D's sibling S is the worst orator; S's row (Row 1) also holds Br, so Br is the best orator.
Step 4: Compare all four outcomes.
Every one of the four valid placements gives the exact same best orator, no matter how the seats are actually filled in.
