In a year, there is at least and at most how many months where the 13th falls on a Friday?
First, let's examine which day of the year the 13th of each month falls on: January 13 is the 13th day of the year, February 13 is the 13 + 31 = 44th day, March 13 is the 44 + 28 = 72nd day, and so on. In leap years, February has 29 days instead of 28, so from March 13 onwards, each date is shifted by one day. This gives us the following table:

The 13th of the month is the how manyth day of the year

| Month | I. | II. | III. | IV. | V. | VI. | VII. | VIII. | IX. | X. | XI. XII. |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| Common Year | 13 | 44 | 72 | 103 | 133 | 164 | 194 | 225 | 256 | 286 | 317 | 347 |
| Leap Year | 13 | 44 | 73 | 104 | 134 | 165 | 195 | 226 | 257 | 287 | 318 | 348 |

Since a week consists of 7 days, two days of the year fall on the same day of the week if the number of days between them is divisible by 7, or if their sequence numbers give the same remainder when divided by 7. Let's determine the remainder when the numbers in the above table are divided by 7:

| Month | I. II. | III. | IV. V. VI. VII. VIII. IX. X. XI. XII. |  |  |  |  |  |  |  |  |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Common Year | 6 | 2 | 2 | 5 | 0 | 3 | 5 | 1 | 4 | 6 | 2 | 4 |
| Leap Year | 6 | 2 | 3 | 6 | 1 | 4 | 6 | 2 | 5 | 0 | 3 | 5 |

If, for example, the first Friday of a year was January 4, then in that year, the days of the year that give a remainder of 4 when divided by 7 fall on a Friday. If this is a common year (as it was in 1974), then according to the above table, the 13th of the IX. and XII. months fall on a Friday. Based on this, we can easily calculate how many 13ths fall on a Friday in a given year, depending on which day the first Friday of the year falls on:

| The first Friday of the year is January | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Common Year | 1 | 3 | 1 | 2 | 2 | 2 | 1 |
| Leap Year | 1 | 2 | 2 | 1 | 2 | 3 | 1 |

We can see that in every year, the 13th falls on a Friday at least once and at most three times. Both cases do indeed occur: 1971 was a common year and January 2 was a Friday, so the 13th fell on a Friday three times that year; in this year (1975), January 3 was a Friday, so the 13th falls on a Friday only once this year (in June, according to the middle table).