Question 70
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.
| | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
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:
| | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
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.
| | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
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:
| | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
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.