8.6. A triangle is divided into triangular cells as shown in the figure. A natural number is written in each cell. For each side of the triangle, there are four layers parallel to this side, containing seven, five, three, and one cell, respectively. It turns out that the sum of the numbers in each of these twelve layers is a prime number. What is the smallest possible sum of all the written numbers?

![](https://cdn.mathpix.com/cropped/2024_05_06_6ab074edf53f8bb32518g-2.jpg?height=334&width=391&top_left_y=1843&top_left_x=1546)
Answer: 22.

Solution. Example. In each of the three corner cells, we write the number 3, and in each of the others, we write the number 1. Then the sum of the written numbers is $3 \cdot 3 + 13 \cdot 1 = 22$, and the sums of the numbers in the layers are: 11, 5, 3, and 3, respectively.

Estimation. Any corner cell is a separate layer, so each such cell must contain at least the number 2. Consider, for example, the other horizontal layers. In the two layers below the corner cell, we can place numbers with the minimum sums: 1-1-1 and 1-1-1-1-1. Then, in the bottom layer, the arrangement with the minimum sum is: $2+1$ $+1+1+1+1+2=9$, but this is not a prime number. The nearest prime number greater than nine is 11. Therefore, the sum of all numbers is no less than $2+3+5+11=21$.

However, this sum is not achievable, as when considering four layers along the other sides of the original triangle, we find that 2 must be added to each layer of seven cells. Consequently, 22 is the smallest possible sum.

Grading Criteria.

“+” A complete and well-reasoned solution is provided

“土” A generally correct reasoning is provided, containing minor gaps or inaccuracies

“干” The correct answer and example are provided, but the estimation is missing

“-” Only the answer is provided

“-” The problem is not solved or is solved incorrectly