Question 175
A $3 \times 3$ square is given with numbers written in each $1 \times 1$ cell. The allowed moves are to choose a row, a column or a diagonal and add or subtract 1 from the three numbers in it. Prove that it is not possible to start with the numbers in the configuration on the left and arrive at the numbers in the configuration on the right after some operations.
| 0 | 1 | 0 |
| :--- | :--- | :--- |
| 1 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 1 |
| :--- | :--- | :--- |
| 0 | 1 | 0 |
| 1 | 0 | 1 |
| 0 | 1 | 0 |
| :--- | :--- | :--- |
| 1 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 1 |
| :--- | :--- | :--- |
| 0 | 1 | 0 |
| 1 | 0 | 1 |