Question 66
Let $m$, $n$ be integers greater than $1$. A $m \times n$ grid is given. We want to write integers in each square so that
(i) at least one of the entries are nonzero, and
(ii) for each square $S$, $\sigma(S) = 0$, where $\sigma(S)$ denotes the sum of the entry in all the squares which are next to $S$ (i.e., all the squares which share an edge with $S$).
For example, let $m = 3$, $n = 4$. Assume that we write $1$ in all unit squares.
| 1 | 1 | 1 | 1 |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 |
If we write $\sigma(S)$ in each square, it shows
| 2 | 3 | 3 | 2 |
|---|---|---|---|
| 3 | 4 | 4 | 3 |
| 2 | 3 | 3 | 2 |
and so in this way condition (ii) does not hold.
We call $(m, n)$ a *good pair* if we can write integers in the squares with conditions (i) and (ii).
For example, consider the pair $(2, 2)$.
| 0 | 1 |
|---|---|
| -1 | 0 |
By writing integers as above, the desired conditions indeed hold. Therefore $(2, 2)$ is a good pair.
(1) Let $m = 3$. Find all integer $n \le 10$ such that $(m, n)$ is good pair.
(2) Find the number of good pair $(m, n)$ such that $2 \le m, n \le 10$. We consider ordered pairs, i.e., pairs $(m, n)$ and $(n, m)$ are considered different if $n \ne m$.
(i) at least one of the entries are nonzero, and
(ii) for each square $S$, $\sigma(S) = 0$, where $\sigma(S)$ denotes the sum of the entry in all the squares which are next to $S$ (i.e., all the squares which share an edge with $S$).
For example, let $m = 3$, $n = 4$. Assume that we write $1$ in all unit squares.
| 1 | 1 | 1 | 1 |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 |
If we write $\sigma(S)$ in each square, it shows
| 2 | 3 | 3 | 2 |
|---|---|---|---|
| 3 | 4 | 4 | 3 |
| 2 | 3 | 3 | 2 |
and so in this way condition (ii) does not hold.
We call $(m, n)$ a *good pair* if we can write integers in the squares with conditions (i) and (ii).
For example, consider the pair $(2, 2)$.
| 0 | 1 |
|---|---|
| -1 | 0 |
By writing integers as above, the desired conditions indeed hold. Therefore $(2, 2)$ is a good pair.
(1) Let $m = 3$. Find all integer $n \le 10$ such that $(m, n)$ is good pair.
(2) Find the number of good pair $(m, n)$ such that $2 \le m, n \le 10$. We consider ordered pairs, i.e., pairs $(m, n)$ and $(n, m)$ are considered different if $n \ne m$.