A table starts with every entry equal to 0 and is modified using the following steps: (i) adding 1 to all three numbers in any row; (ii) adding 2 to all three numbers in any column. After step (i) has been used a total of times and step (ii) has been used a total of times, the table appears as \begin{tabular}{|l|l|l|} \hline 7 & 1 & 5 \\ \hline 9 & 3 & 7 \\ \hline 8 & 2 & 6 \\ \hline \end{tabular} shown. What is the value of ?
Solution
Since the second column includes the number 1, then step (ii) was never used on the second column, otherwise each entry would be at least 2 . To generate the 1,3 and 2 in the second column, we thus need to have used step (i) 1 time on row 1,3 times on row 2 , and 2 times on row 3 . This gives: \begin{tabular}{|l|l|l|} \hline 1 & 1 & 1 \\ \hline 3 & 3 & 3 \\ \hline 2 & 2 & 2 \\ \hline \end{tabular} We cannot use step (i) any more times, otherwise the entries in column 2 will increase. Thus, . To obtain the final grid from this current grid using only step (ii), we must increase each entry in column 1 by 6 (which means using step (ii) 3 times) and increase each entry in column 3 by 4 (which means using step (ii) 2 times). Thus, . Therefore, .
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