8.1 On 12 cards, different natural numbers are written, their sum is 84. 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: 13,14

Solution. If the smallest possible natural numbers, i.e., the card numbers (from 1 to 12), were written on the cards, then their sum would be $1+2+...+12=78$. But the actual sum is 6 more. This means that some numbers are greater than their numbers. If the number on the sixth card is greater than 6, then the numbers on cards 7-12 are also greater, and at least 7 will be added to the sum of 78. Therefore, the number on the sixth card is 6. The number on the seventh card can be 7 (example: $1,2,3, \ldots 10,11,18$), or it can be 8 (example: $1,2,3,4,5,6,8,9,10,11,12,13$). The number 9 cannot be written, because then on each subsequent card, the number is at least 2 more than its number, and at least 12 will be added to the total sum.