Question 25
Consider a $99 \times 99$ table with its columns and rows labelled $1$ to $99$ from left to right and bottom to top respectively. Lucía writes the numbers from $1$ to $3160$ in increasing order by steps, in the following way:
| | | | | | | | | | |
|--------|----|----|----|----|----|----|----|----|----|
| Row 4 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | | |
| Row 3 | 26 | 9 | 10 | 11 | 12 | 13 | 18 | | |
| Row 2 | 27 | 8 | 5 | 4 | 3 | 14 | 17 | | |
| Row 1 | 28 | 7 | 6 | 1 | 2 | 15 | 16 | | |
| | 47 | 48 | 49 | 50 | 51 | 52 | 53 | | |
* Number $1$ is in the cell of row $1$ and column $50$.
* In the first step, she fills every cell which is a neighbor of $1$, anti-clockwise.
* In the second step, she fills every cell which is a neighbor of any cell filled in the previous step, clockwise.
* She continues this way in each step, filling the cells that are neighbors of the cells filled in the previous step and changing the orientation.
The figure shows the table after Lucía completed the first three steps.
In which step will Lucía write the number $2022$? In which row and which column is $2022$ written?
| | | | | | | | | | |
|--------|----|----|----|----|----|----|----|----|----|
| Row 4 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | | |
| Row 3 | 26 | 9 | 10 | 11 | 12 | 13 | 18 | | |
| Row 2 | 27 | 8 | 5 | 4 | 3 | 14 | 17 | | |
| Row 1 | 28 | 7 | 6 | 1 | 2 | 15 | 16 | | |
| | 47 | 48 | 49 | 50 | 51 | 52 | 53 | | |
* Number $1$ is in the cell of row $1$ and column $50$.
* In the first step, she fills every cell which is a neighbor of $1$, anti-clockwise.
* In the second step, she fills every cell which is a neighbor of any cell filled in the previous step, clockwise.
* She continues this way in each step, filling the cells that are neighbors of the cells filled in the previous step and changing the orientation.
The figure shows the table after Lucía completed the first three steps.
In which step will Lucía write the number $2022$? In which row and which column is $2022$ written?