7.6. Three children: Andrei, Petru, and Mihai, each have a cardboard. On each cardboard, a number is written. They all know that the numbers on the cardboards are distinct non-zero natural numbers, the sum of which is equal to 15. Andrei knows that the number written on his cardboard is the smallest. Petru knows that the number written on his cardboard is the largest, and Mihai knows that the number written on his cardboard is neither the smallest nor the largest. Andrei says: "I cannot determine what numbers are written on your cardboards." Then Petru says: "I, too, cannot determine what numbers are written on your cardboards." Finally, Mihai says: "At first, I could not determine what numbers are written on your cardboards, but after what you said, I now know what numbers you have written on your cardboards." Determine what number is written on each boy's cardboard.
## Solution:

On the three cards, the numbers can be written as: $(1,2,12),(1,3,11),(1,4,10),(1,5,9),(1,6,8)$, $(2,3,10),(2,4,9),(2,5,8),(2,6,7),(3,4,8),(3,5,7),(4,5,6)$. Since Andrei cannot know what numbers are written on the other cards, it follows that Andrei cannot have the number 4 written on his card; otherwise, he could have confidently stated what numbers are written on the other cards. Similarly, Petru cannot have the number 11 or 12 written on his card. Thus, the remaining possibilities are: $(1,4,10),(1,5,9),(1,6,8),(2,3,10),(2,4,9),(2,5,8),(2,6,7),(3,4,8)$, $(3,5,7)$. Since Mihai can now state what numbers are written on the other cards, it follows that the number 3 is written on his card. Therefore, the numbers written on the cards are $(2,3,10)$. So, Andrei has the number 2 written on his card, and Petru has the number 10 written on his card.