Question 19
We place the digits $1$ through $9$ one by one in a $3 \times 3$ grid. The digit $1$ may be placed in an arbitrarily chosen box; each subsequent digit comes in a box that is horizontally or vertically adjacent to the box that contains the previous digit. See, for example, the picture below. We call such a grid a *snake grid*. The score of a box in a snake grid is the sum of the digits in all boxes with one side adjacent to the box. The total score of a snake grid is the sum of the scores of all its boxes. For example, the snake grid in the example (row by row, adding up from left to right) has total score $6+13+10+9+20+19+10+15+14 = 116$.
$$
\begin{array}{|c|c|c|}
\hline
1 & 2 & 9 \\
\hline
4 & 3 & 8 \\
\hline
5 & 6 & 7 \\
\hline
\end{array}
$$
How many possible total scores can such a snake grid have?
$$
\begin{array}{|c|c|c|}
\hline
1 & 2 & 9 \\
\hline
4 & 3 & 8 \\
\hline
5 & 6 & 7 \\
\hline
\end{array}
$$
How many possible total scores can such a snake grid have?