19. In the final of a dancing competition, each of the three members of the jury gives each of the five competitors 0 points, 1 point, 2 points, 3 points or 4 points. No two competitors get the same mark from any individual judge.
\begin{tabular}{|c|c|c|c|c|c|}
\cline { 2 - 6 } \multicolumn{1}{c|}{} & Adam & Berta & Clara & David & Emil \\
\hline I & 2 & 0 & & & \\
\hline II & & 2 & 0 & & \\
\hline III & & & & & \\
\hline Sum & 7 & 5 & 3 & 4 & 11 \\
\hline
\end{tabular}
Adam knows all the sums of the marks and a few single marks, as shown. How many points does Adam get from judge III?
A 0
B 1
C 2
D 3
E 4
SOLUTION
B

The table can be partially completed as follows. Berta scored 5 points in total. Therefore her score from judge III is 3 . Clara's total is 3 . Therefore she cannot have been given 4 by any of the judges. David's total is 4 . He cannot have been given a score of 4 by any judge, since, if so, both the other two judges must have given him 0 . This is impossible, as judges I and II give 0 to Berta and Clara, respectively. Therefore judge I gives 4 to Emil, and judge II gives 4 to either Adam or Emil. Emil's total is 11. So if he gets 4 from judges I and II, he gets 3 from judge III which is not possible as judge III gave Berta 3. Hence judge II gives 4 to Adam. Because Adam's total is 7 , it now follows that Adam gets a score of 1 from judge III. (Note: the final four scores are not uniquely determined.)
\begin{tabular}{|c|c|c|c|c|c|}
\cline { 2 - 6 } \multicolumn{1}{c|}{} & Adam & Berta & Clara & David & Emil \\
\hline I & 2 & 0 & & & 4 \\
\hline II & 4 & 2 & 0 & 1 & 3 \\
\hline III & 1 & 3 & & & 4 \\
\hline Sum & 7 & 5 & 3 & 4 & 11 \\
\hline
\end{tabular}