# 3. Variant 1

Vasya cut out a triangle from cardboard and numbered its vertices with the digits $1,2,3$. It turned out that if Vasya's triangle is rotated 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya's triangle is rotated 6 times clockwise around its vertex numbered 2 by an angle equal to the angle at this vertex, it will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by an angle equal to the angle at this vertex, it will return to its original position. What is the smallest $n$ that Vasya could have named so that his claim is true for at least one cardboard triangle?
Answer: 4.

Solution: Since after 12 turns around the first vertex of the triangle, it returns to its original position, it makes one or several full rotations of $360^{\circ}$. Therefore, the angle at the first vertex is not less than $30^{\circ}$. Reasoning similarly, we get that the angle at the second vertex is not less than $60^{\circ}$. Thus, the angle at the third vertex, by the sum of the angles in a triangle, is not more than $90^{\circ}$ and $n \geq 4$.

Four turns are sufficient: a triangle with angles $30^{\circ}, 60^{\circ}$, and $90^{\circ}$ will work.

## Variant 2

Vasya cut out a triangle from cardboard and numbered its vertices with the digits $1,2,3$. It turned out that if Vasya's triangle is rotated 24 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya's triangle is rotated 8 times clockwise around its vertex numbered 2 by an angle equal to the angle at this vertex, it will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by an angle equal to the angle at this vertex, it will return to its original position. What is the minimum $n$ that Vasya could have named so that his claim is true for at least some cardboard triangle?

Answer: 3.

## Variant 3

Vasya cut out a triangle from cardboard and numbered its vertices with the digits $1,2,3$. It turned out that if Vasya's triangle is rotated 10 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya's triangle is rotated 5 times clockwise around its vertex numbered 2 by an angle equal to the angle at this vertex, it will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by an angle equal to the angle at this vertex, it will return to its original position. What is the minimum $n$ that Vasya could have named so that his claim is true for at least some cardboard triangle?

Answer: 5.

## Variant 4

Vasya cut out a triangle from cardboard and numbered its vertices with the digits $1,2,3$. It turned out that if Vasya's triangle is rotated 18 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya's triangle is rotated 3 times clockwise around its vertex numbered 2 by an angle equal to the angle at this vertex, it will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by an angle equal to the angle at this vertex, it will return to its original position. What is the minimum $n$ that Vasya could have named so that his claim is true for at least some cardboard triangle?

Answer: 9.

## Variant 5

Vasya cut out a triangle from cardboard and numbered its vertices with the digits $1,2,3$. It turned out that if Vasya's triangle is rotated 15 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya's triangle is rotated 3 times clockwise around its vertex numbered 2 by an angle equal to the angle at this vertex, it will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by an angle equal to the angle at this vertex, it will return to its original position. What is the minimum $n$ that Vasya could have named so that his claim is true for at least some cardboard triangle?

Answer: 10.