8.2. On 14 cards, different natural numbers are written, their sum is 112. Masha laid out the cards in ascending order. What can the sum of the numbers on the two middle cards be? Find all possible answers.
Answer: 15,16

Solution. If the smallest possible natural numbers, i.e., the card numbers (from 1 to 14), were written on the cards, then their sum would be $1+2+\ldots+14=105$. But the actual sum is 7 more. This means that some numbers are greater than their numbers. If the number on the seventh card is greater than 7, then the numbers on cards 8-14 are also greater, and at least 8 will be added to the sum of 105. Therefore, the number 7 is written on the seventh card. The number on the eighth card can be 8 (example: $1,2,3, \ldots 12,13,21$), or it can be 9 (example: $1,2,3,4,5,6,7,9,10,11,12,13,14,15$). The number 10 cannot be written, because then on each subsequent card, the number is at least 2 more than its number, and at least 14 will be added to the total sum.