feira 13 - What is the maximum number of Friday the 13ths that can occur in a non-leap year? In this case, what is the $10^{\text{th}}$ day of the year?
Feira 13 - Given that the days of the week repeat every 7 days, the difference between the days of the week is given by the remainder when dividing by 7 the number of days that have passed.

$\mathrm{In}$ the following table:

- in the first row, the number of days between the 13th of one month and the 13th of the next month;
- in the second row, the remainder when we divide this number by 7;
- in the third row, the remainder when we divide by 7 the number of days between the 13th of January and the 13th of the corresponding month, that is, it is obtained by summing the results obtained in the previous row from January to the corresponding month and then calculating the remainder when dividing by 7.

| J-F | F-M | M-A | A-M | M-J | J-J | J-A | A-S | S-O | O-N | N-D |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 31 | 28 | 31 | 30 | 31 | 30 | 31 | 31 | 30 | 31 | 30 |
| 3 | 0 | 3 | 2 | 3 | 2 | 3 | 3 | 2 | 3 | 2 |
| 3 | 3 | 6 | 1 | 4 | 6 | 2 | 5 | 0 | 3 | 5 |

The equal values in the last row mean that in these months the 13th fell on the same day of the week. In particular, this last row tells us that the 13th of February, the 13th of March, and the 13th of November correspond to the same day of the week. Therefore, we have at most three Friday the 13ths.

In this case, the 13th of January occurred 3 days before Friday, that is, on a Tuesday, and the 10th of January occurred 3 days before, that is, on a Saturday.

Note: The 6th Friday the 13th occurs only when the 1st day of the month is a Sunday. Thus, another way, perhaps simpler, to solve the problem is to determine the maximum number of times the 1st day of the month is a Sunday in a non-leap year.