Problem 7.8. Tanya and Vера are playing a game. Tanya has cards with numbers from 1 to 30. She arranges them in some order in a circle. For each pair of adjacent numbers, Vера calculates their difference, subtracting the smaller number from the larger one, and writes down the resulting 30 numbers in her notebook. After that, Vера gives Tanya a number of candies equal to the smallest number written in the notebook. Tanya arranges the cards so as to get as many candies as possible. What is the maximum number of candies she can get?
Answer: 14.

Solution. We will prove that Tanya cannot get 15 candies. Let's look at the card with the number 15. The number 15 differs from all the remaining numbers, except for the number 30, by no more than 14. Thus, at least one of the differences involving the number 15 will be no more than 14.

Let's provide an example of an arrangement where Tanya can get 14 candies. Arrange the numbers in a circle in the following order: $1,16,2,17,3,18, \ldots, 14,29,15,30$.