## Task 26/63

In the year 1604, for the first time since the introduction of the Gregorian calendar, February had five Sundays. In which years does this property recur?
Apparently, the desired property can only occur in leap years, and then only if February 1st falls on a Sunday. After the expiration of a year of 365 days, the weekday for a specific date shifts by 1, since $365=7k+1$. In leap years, the shift is 2, so after four years (one of which is a leap year), the total shift is 5.

For the recurrence to occur after $4n$ years, $n \cdot 5$ must be divisible by 7. The smallest number $n$ for which $n \cdot 5=7m$ with $m=1; 2; 3; \ldots$ is $n=7$.

The first repetition thus occurs after 28 years, the next after another 28 years, and so on. This initially results in the years 1632, 1660, and 1688. However, if a full century (namely one of the years 1700, 1800, or 1900) is exceeded, which are not leap years according to the Gregorian calendar, the shift is 1 smaller. Then $n \cdot 5-1$ must be divisible by 7. This is the case for $n=3; 10; 17; \ldots$.

However, the value $n=3$ is not feasible, as it does not exceed the year 1700. Therefore, $n=10$ must be used, and the interim period is 40 years. Thus, the following years are obtained:

| 1604 | 1632 | 1660 | 1688 |
| :--- | :--- | :--- | :--- |
|      | 1728 | 1756 | 1784 |
|      | 1824 | 1852 | 1880 |
|      | 1920 | 1948 | 1976 |

Since the year 2000 (like the year 1600) is a leap year, a February with five Sundays follows after 28 years, in the year 2004. This then begins a new cycle with the same last digits

$2004 \quad 2032 \quad 2060 \quad 2088 \quad \ldots$