Question

r	s	t
u	v	w
x	y	z

Each of the letters in the table above represents one of the numbers 1, 2, or 3, and each of these numbers occurs exactly once in each row and exactly once in each column. What is the value of $r$ ?
(1) $v+z=6$
(2) $s+t+u+x=6$

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Both statements together are sufficient.

Answer 1

Answer: (E)

Both statements together are sufficient.

• In the following discussion, “row/column convention” means that each of the numbers $1,2,$ and $3$ appears exactly once in any given row and exactly once in any given column.

1. Given that $v+z=6$, then both $v$ and $z$ are equal to $3$, since no other sum of the possible values is equal to $6$. Applying the row/column convention to row $2$, and then to row $3$, it follows that neither $u$ nor $x$ can be $3$. Since neither $u$ nor $x$ can be $3$, the row/column convention applied to column $1$ forces $r$ to be $3$; SUFFICIENT.
2. If $u=3$, then $s+t+x=3$. Hence, $s=t=x=1$, since the values these variables can have does not permit another possibility. However, this assignment of values would violate the row/column convention for row 1, and thus u cannot be $3$. If $x=3$, then $s+t+u=3$. Hence, $s=t=u=1$, since the values these variables can have does not permit another possibility. However, this assignment of values would violate the row/column convention for row 1, and thus $x$ cannot be $3$. Since neither $u$ nor $x$ can be $3$, the row/column convention applied to column 1 forces $r$ to be $3$; SUFFICIENT.

• The correct answer is D; each statement alone is sufficient.