14. [9] You have a $2 \times 3$ grid filled with integers between 1 and 9 . The numbers in each row and column are distinct, the first row sums to 23 , and the columns sum to 14,16 , and 17 respectively.
\begin{tabular}{c|c|c|c|} 
& 14 & 16 & 17 \\
\hline 23 & $a$ & $b$ & $c$ \\
\hline & $x$ & $y$ & $z$ \\
\hline
\end{tabular}

What is $x+2 y+3 z$ ?
Answer: 49 The sum of all 6 numbers is $14+16+17=47$, so $x+y+z=47-23=24$. If three distinct digits sum to 24 , they must be 7,8 , and 9 , because any other triple of digits would have a smaller sum. So, we try placing these digits in for $x, y$, and $z$, and the only arrangement that does not force equal digits in any row or column is $x=8, y=7, z=9$. In this case, $x+2 y+3 z=49$.