A Consider the $4 \times 4$ "multiplication table" below. The numbers in the first column multiplied by the numbers in the first row give the remaining numbers in the table. For example, the 3 in the first column times the 4 in the first row give the $12(=3 \cdot 4)$ in the cell that is in the 3 rd row and 4 th column.
\begin{tabular}{|c|c|c|c|}
\hline 1 & 2 & 3 & 4 \\
\hline 2 & 4 & 6 & 8 \\
\hline 3 & 6 & 9 & 12 \\
\hline 4 & 8 & 12 & 16 \\
\hline
\end{tabular}

We create a path from the upper-left square to the lower-right square by always moving one cell either to the right or down. For example, here is one such possible path, with all the numbers along the path circled:
\begin{tabular}{|c|c|c|c|}
\hline 1 & 2 & 3 & 4 \\
\hline 2 & 4 & 6 & 8 \\
\hline 3 & 6 & 9 & 12 \\
\hline 4 & 8 & 12 & 16 \\
\hline
\end{tabular}

If we add up the circled numbers in the example above (including the start and end squares), we get 48 . Considering all such possible paths:
(a) What is the smallest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.
(b) What is the largest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.
A The minimum is 46 and the maximum is 50 . To see this more easily, tilt the grid 45 degrees:
$$
\begin{array}{l}
1 \\
2 \quad 2 \\
3 \quad 4 \quad 3 \\
\begin{array}{llllll}
4 & & 6 & & 6 & \\
& 8 & & 9 & & 8
\end{array} \\
12 \quad 12 \\
16 \\
\end{array}
$$

Now every path must include exactly one number from each row. The smallest and largest numbers in each row are respectively at the edge and in the middle, so the smallest and largest totals are achieved by the paths below:

These totals are 46 and 50 .