In our city, there are three cinemas named after the cardinal directions. It is known about their opening hours that:

1. at least one cinema is open every day,
2. if the southern cinema is open, then the northern cinema is not open,
3. the northern and eastern cinemas are never open at the same time,
4. if the eastern cinema is open, then the southern or northern cinema is also open.

We went to the southern cinema and found out that it was closed. Which of the remaining cinemas is certainly open?

(M. Dillingerová)

Hint. Consider all possible scenarios of the cinemas being open or closed without the limiting conditions.
We are interested in the situation where the southern cinema is closed. Without further information about opening hours, the following four cases may arise, which we will compare with the conditions given:

a) If the northern and eastern cinemas were open, we would be in conflict with the third condition. This situation will not occur.

b) If the northern cinema is open and the eastern cinema is closed, we are not in conflict with any of the conditions. This situation is possible.
c) If the northern cinema is closed and the eastern cinema is open, we would be in conflict with the fourth condition. This situation will not occur.

d) If the northern and eastern cinemas are closed, we would be in conflict with the first condition. This situation will not occur.

The only situation that meets all the conditions given is b): when the southern cinema is closed, the northern cinema is certainly open.

Other guidance. What options meet the third condition?

Alternative solution. In view of the third condition, we can replace the fourth condition with the following:

$4^{\prime}$. If the eastern cinema is open, then the southern cinema is also open.

According to the third condition, we can distinguish three cases:

a) The northern cinema is open and the eastern cinema is closed. Then, according to the second condition, the southern cinema is closed.

b) The northern cinema is closed and the eastern cinema is open. Then, according to the fourth condition (or its just mentioned replacement), the southern cinema is open.

c) Both the northern and eastern cinemas are closed. Then, according to the first condition, the southern cinema is open.

In none of these cases are we in conflict with the conditions we did not use in the derivation. For clarity, we list all possible open (1) or closed (0) cinema situations in the table:

| $\mathrm{S}$ | 1 | 0 | 0 |
| :---: | :---: | :---: | :---: |
| $\mathrm{V}$ | 0 | 1 | 0 |
| $\mathrm{J}$ | 0 | 1 | 1 |

If the southern cinema is closed, then the northern cinema is certainly open (and the eastern cinema is certainly closed).

Note. The discussion in the solution of the problem can be conducted in various ways. For these purposes, we will illustrate the options that the conditions in the task allow separately - empty cells can contain either 1 or 0:

The first condition allows the following options

| $\mathrm{S}$ | 1 |  |  |
| :---: | :---: | :---: | :---: |
| $\mathrm{V}$ |  | 1 |  |
| $\mathrm{J}$ |  |  | 1 |

The second condition allows the following options

| $\mathrm{S}$ | 0 |  |
| :---: | :---: | :---: |
| $\mathrm{V}$ |  |  |
| $\mathrm{J}$ | 1 | 0 |

The third condition allows the following options

| $\mathrm{S}$ | 1 | 0 | 0 |
| :---: | :---: | :---: | :---: |
| $\mathrm{V}$ | 0 | 1 | 0 |
| $\mathrm{J}$ |  |  |  |

The fourth condition allows the following options

| $\mathrm{S}$ |  | 1 | 1 |  |
| :---: | :---: | :---: | :---: | :---: |
| $\mathrm{V}$ | 1 | 1 | 1 | 0 |
| $\mathrm{J}$ | 1 |  | 1 |  |

Selecting the options that meet all four conditions simultaneously (i.e., the "intersection" of these four tables) leads to the table at the end of the previous solution.