Bebras 2022 - Level 6 Question 14

Consider the number grid shown below.



| Row / Column | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 2 | 2 | 1 | 1 |
| 2 | 1 | 2 | 1 | 2 |
| 3 | 2 | 1 | 2 | 1 |
| 4 | 1 | 2 | 2 | 2 |

We may repeatedly swap any two rows or any two columns.

How many of the grids below can be obtained from the initial grid by such transformations?



Grid 1:

| Row / Column | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 2 | 1 | 1 | 2 |
| 2 | 1 | 2 | 1 | 2 |
| 3 | 1 | 2 | 2 | 1 |
| 4 | 2 | 1 | 2 | 1 |

Grid 2:

| Row / Column | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 2 | 1 | 1 | 1 |
| 2 | 2 | 2 | 1 | 2 |
| 3 | 1 | 2 | 2 | 1 |
| 4 | 1 | 2 | 2 | 2 |

Grid 3:

| Row / Column | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 2 | 1 | 2 | 1 |
| 2 | 1 | 2 | 1 | 2 |
| 3 | 1 | 2 | 2 | 1 |
| 4 | 1 | 2 | 2 | 2 |

Grid 4:

| Row / Column | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 2 | 1 | 2 | 2 |
| 2 | 1 | 2 | 2 | 1 |
| 3 | 2 | 1 | 2 | 1 |
| 4 | 1 | 2 | 1 | 2 |

Grid 5:

| Row / Column | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 1 | 2 | 2 | 2 |
| 2 | 2 | 2 | 2 | 1 |
| 3 | 2 | 2 | 1 | 1 |
| 4 | 2 | 1 | 1 | 2 |