Answer :
Let's fill in the given table with all the correct values step by step based on the resulting numerical answer provided.
We start with the grid below and insert each value into its respective position:
```
\begin{tabular}{|c|c|c|c|c|}
\hline & & & & 3 \\
\hline & & & & \\
\hline & & & & \\
\hline & & & & \begin{tabular}{l}
72
\end{tabular} \\
\hline 7 & [tex]$\varepsilon$[/tex] & & & 1 \\
\hline 4 & 9 & & & 13 \\
\hline 3 & 7 & & 2 & \\
\hline & 816 & & & 7 \\
\hline
\end{tabular}
```
### Step-by-Step Solution:
1. Insert the number `3` into the top right corner:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
2. Insert the number `72` fourth row, fifth column (already given):
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
3. Insert the number `7` into the fifth row, first column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
4. Insert the number `1` into the fifth row, fifth column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
5. Insert the number `4` into the sixth row, first column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
6. Insert the number `9` into the sixth row, second column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
7. Insert the number `13` into the sixth row, fifth column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
8. Insert the number `3` into the seventh row, first column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
9. Insert the number `7` into the seventh row, second column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
10. Insert the number `2` into the seventh row, fourth column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
11. Insert the number `816` into the eighth row, second column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
12. Insert the number `7` into the eighth row, fifth column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
So, the completely filled table looks like:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
We start with the grid below and insert each value into its respective position:
```
\begin{tabular}{|c|c|c|c|c|}
\hline & & & & 3 \\
\hline & & & & \\
\hline & & & & \\
\hline & & & & \begin{tabular}{l}
72
\end{tabular} \\
\hline 7 & [tex]$\varepsilon$[/tex] & & & 1 \\
\hline 4 & 9 & & & 13 \\
\hline 3 & 7 & & 2 & \\
\hline & 816 & & & 7 \\
\hline
\end{tabular}
```
### Step-by-Step Solution:
1. Insert the number `3` into the top right corner:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
2. Insert the number `72` fourth row, fifth column (already given):
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
3. Insert the number `7` into the fifth row, first column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
4. Insert the number `1` into the fifth row, fifth column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
5. Insert the number `4` into the sixth row, first column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
6. Insert the number `9` into the sixth row, second column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
7. Insert the number `13` into the sixth row, fifth column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
8. Insert the number `3` into the seventh row, first column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
9. Insert the number `7` into the seventh row, second column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
10. Insert the number `2` into the seventh row, fourth column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
11. Insert the number `816` into the eighth row, second column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
12. Insert the number `7` into the eighth row, fifth column:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & $\varepsilon$ & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
So, the completely filled table looks like:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline & & & & 3 \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & 72 \\ \hline 7 & & & & 1 \\ \hline 4 & 9 & & & 13 \\ \hline 3 & 7 & & 2 & \\ \hline & 816 & & & 7 \\ \hline \end{tabular} \][/tex]
