3. On a glade, 25 gnomes gathered. It is known that 1) every gnome who put on a hat also put on shoes; 2) 12 gnomes came without a hat; 3) 5 gnomes came barefoot. Which gnomes are more and by how many: those who came in shoes but without a hat, or those who put on a hat?
Answer. There are 6 more gnomes who put on a cap.

## Solution.

From condition 2, it follows that $25-12=13$ gnomes came in a cap.

From condition 1, we get that exactly 13 gnomes came both in a cap and in shoes.

From condition 3, it follows that a total of $25-5=20$ gnomes came in shoes.

Thus, $20-13=7$ gnomes came in shoes but without a cap.

Therefore, those who put on a cap (13 gnomes) are 6 more than those who came in shoes but without a cap (7 gnomes).

## Grading Criteria.

- Correct solution - 7 points.
- Correct answer with incomplete justification - 3-4 points.
- Only the correct answer without justification - 2 points.