We call a calendar week even or odd depending on whether the sum of the dates of the days within that week is even or odd. Starting from the first Monday in January, out of 52 consecutive weeks, how many can be even?
Let's take a calendar in our hands. After a brief examination, we can see that within a month, odd and even weeks alternate. Among 7 consecutive numbers starting with an odd number, there are always 4 odd numbers (including the first and the last), so their sum is even. However, if the first number is even, then 3 odd numbers appear during the week, meaning their sum is odd.

If this regularity continued, it would be easy to count the odd and even weeks: out of 52 consecutive weeks, 26 would be even and 26 would be odd.

However, this is not entirely the case. When a month has 31 days, the week of the last day does not follow with an even number after 31, but rather an odd number again: 1.

For example, if January 1 falls on a Monday, then the first 4 weeks are: even, odd, even, odd. The 5th week: 29, 30, 31, 1, 2, 3, 4 is even, and the 6th week: 5, 6, 7, 8, 9, 10, 11 is also even.

Now, two even weeks follow each other, increasing the number of even weeks by 1.

There are 5 more cases where we have a 31-day month: March, May, July, August, October, assuming that December 31 falls on the 53rd week.

Let's mark the even weeks with $p$ and the odd weeks with $t$ in the calendar. The sequence starts regularly, then two even weeks follow, and then it continues regularly until the next 31-day month.

Let's write down the sequence:

20

1st row: $p$,

, $p$,

, $p$, $p$,

, $p$,

, $p$, $t$, $p$, $p$, $t$, $p$, $\ldots$ 2nd row: $\quad t$, $p$, $t$, $p$, $t$, $p$, $t$, $p$, $p$, $t$, $\ldots$

If we omit one of the two adjacent $p$s in the first row and shift the row 1 position to the left (2nd row), as if it were regular, the order is restored, meaning that the second irregular period (31-day month) does not increase the number of even weeks. Out of the 6 31-day months, only 3 increase the number of even weeks by 1 each, so in total we have $26 + 3 = 29$ even and 23 odd weeks.

If January 1 does not fall on a Monday but, for example, on a Sunday, then the 364th day, 52 weeks later, is December 31, which also marks a leap. In a leap year, the regularity is disrupted again due to February 29. This totals 8 occasions that can increase the number of even weeks by 4. Therefore, in a year, a maximum of 30 even weeks is possible.

1996, for example, was a leap year, January 1 fell on a Monday, and there were 29 even weeks.

Kormos Márton (Debrecen, KLTE Gyak. Gimn., 12. o.t.)