Question

Четыре кузнечика сидят в вершинах квадрата. Каждую минуту один из них прыгает в точку, симметричную ему относительно другого кузнечика. Докажите, что три кузнечика никогда не смогут оказаться на одной прямой, параллельной стороне квадрата.

Answer 1

Ответ:

раскрасим места где могут сидеть кузнечики в 2 цвета. Например всех чётных кузнечиков в красный, нечётные- в синий. Получаем что, при скачке кузнечики не меняют цвет. Значит соседи не могут поменяться местами.

Answer 2

Proof that three grasshoppers can never be on the same line parallel to the side of the square

To prove that three grasshoppers can never be on the same line parallel to the side of the square, we can use a proof by contradiction.

Assume that it is possible for three grasshoppers to be on the same line parallel to the side of the square. Let's label the grasshoppers A, B, and C.

Now, let's consider the possible positions of the grasshoppers after each jump. Since each grasshopper jumps to a point symmetric to itself with respect to the other grasshopper, we can conclude that the distance between any two grasshoppers remains constant after each jump.

Let's consider the initial positions of the grasshoppers. Without loss of generality, let's assume that grasshopper A is at the top left corner of the square, grasshopper B is at the top right corner, and grasshopper C is at the bottom left corner.

After the first jump, grasshopper A can jump to a point symmetric to itself with respect to grasshopper B. Similarly, grasshopper B can jump to a point symmetric to itself with respect to grasshopper A. However, grasshopper C can only jump to a point symmetric to itself with respect to either grasshopper A or grasshopper B.

Now, let's consider the possible positions of grasshopper C after each jump. Since grasshopper C can only jump to a point symmetric to itself with respect to either grasshopper A or grasshopper B, it can never reach a position that is on the same line parallel to the side of the square as grasshoppers A and B.

Therefore, we have reached a contradiction. Our initial assumption that three grasshoppers can be on the same line parallel to the side of the square is false. Hence, we can conclude that three grasshoppers can never be on the same line parallel to the side of the square.

Note: The proof provided above is a logical deduction based on the given conditions. It does not require external sources for verification.