Problem 6.7. A store received one box each of oranges, apples, cucumbers, and potatoes. The boxes were numbered from 1 to 4, and each had an inscription:

- First box: “Potatoes are in the second box.”
- Second box: “Oranges are not in this box.”
- Third box: “Apples are here.”
- Fourth box: “Cucumbers are in the 1st or 2nd box.”

It turned out that the inscriptions on the fruits (oranges and apples) were false, while those on the vegetables (cucumbers and potatoes) were true. Determine what is in each box.
(a) In the first box
(1) there are oranges
(b) In the second box
(2) there are apples
(c) In the third box
(3) there are cucumbers
(d) In the fourth box
(4) there are potatoes
Answer: $\mathrm{a} 2 \mathrm{~b} 3 \mathrm{c} 1 \mathrm{~d} 4$.

Solution. Let's look at the third box. It cannot contain vegetables (since the inscription would be false) and it cannot contain apples (since the inscription would be true); therefore, it must contain oranges.

Now let's see where the apples can be. It is not the second box (since the inscription on it would be true), and it is not the fourth box (since the cucumbers would be in the 1st or 2nd box, and the inscription on it would be true); therefore, it is the first box. It remains to note that the inscriptions on the 2nd and 4th boxes are true, from which it follows that the cucumbers are in the second, and the potatoes in the fourth.