## Task 1 - 240821

Klaus reports on all the days of his stay at the summer camp:

(1) Every morning the weather was either consistently sunny or consistently rainy.

(2) Every afternoon the weather was either consistently sunny or consistently rainy.

(3) On exactly seven days, there was rainy weather.

(4) If it rained in the afternoon, it was sunny in the morning.

(5) On exactly five afternoons, it was sunny.

(6) On exactly six mornings, it was sunny.

Determine whether the number of days Klaus spent at the summer camp can be uniquely determined from these statements! If this is the case, give this number! Also, provide a possible distribution of sunny and rainy mornings and afternoons according to the statements!
Let $x$ be the number of days when it was sunny both in the morning and in the afternoon, $y$ the number of days when it was sunny in the morning and rainy in the afternoon, and $z$ the number of days when it was rainy in the morning and sunny in the afternoon.

Since, according to (4), there was no day when it was rainy both in the morning and in the afternoon, it follows from (1) and (2) that each day had exactly one of the three weather distributions mentioned for $x, y$, and $z$. The total number of days Klaus spent at the summer camp is thus $x+y+z$. From (3), (5), and (6), we have

$$
y+z=7, \quad x+z=5, \quad x+y=6
$$

Adding these three equations, we get $2 \cdot(x+y+z)=18$, so $x+y+z=9$. The sought number can thus be uniquely determined; it is 9.

Further, it follows that $x=9-7=2, y=9-5=4, z=9-6=3$, and thus for the weather distribution:

On exactly 2 days it was sunny both in the morning and in the afternoon,

on exactly 4 days it was sunny in the morning and rainy in the afternoon, and on exactly 3 days it was rainy in the morning and sunny in the afternoon.

As a review of statements (1) to (6) shows, these numbers lead to a possible weather distribution (according to the statements), for example, in the following table:

$$
\begin{array}{llllllllll}
\text { Day } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\text { Morning } & \mathrm{s} & \mathrm{s} & \mathrm{s} & \mathrm{s} & \mathrm{s} & \mathrm{s} & \mathrm{r} & \mathrm{r} & \mathrm{r} \\
\text { Afternoon } & \mathrm{s} & \mathrm{s} & \mathrm{r} & \mathrm{r} & \mathrm{r} & \mathrm{r} & \mathrm{s} & \mathrm{s} & \mathrm{s}
\end{array}
$$