Question

7. Докажите, что для любого числа 1, 2, 2n можно записать по кругу так, чтобы любые два соседних отличались не более, чемна 2

Answer 1

Можно это сделать, например, так:
3, ..., 2n − 3, 2n − 1, 2n, 2n − 2, 2n − 4, ..., 2, 1
1 замыкает круг.

Answer 2

Circular Arrangement of Numbers

To prove that for any number 1, 2, 2n, it is possible to arrange them in a circle such that any two adjacent numbers differ by no more than 2, we can use the following approach:

Let's consider the numbers 1, 2, 2n and arrange them in a circle. We need to ensure that the difference between any two adjacent numbers is at most 2.

We can start by placing the number 1 at the top of the circle. Then, we can place the number 2n at the bottom of the circle. This leaves us with the number 2 to place in between 1 and 2n.

To ensure that the difference between any two adjacent numbers is at most 2, we can place the number 2 in such a way that it is equidistant from 1 and 2n. This can be achieved by placing 2 at the midpoint between 1 and 2n on the circle.

This arrangement satisfies the condition that any two adjacent numbers differ by no more than 2, as the difference between 1 and 2 is 1, and the difference between 2 and 2n is also at most 2n-2.

Therefore, it is possible to arrange the numbers 1, 2, 2n in a circle such that any two adjacent numbers differ by no more than 2.