The image represents a top view of a three-layer pyramid composed of 14 identical cubes. Each cube is assigned a natural number such that the numbers corresponding to the cubes in the bottom layer are all different, and the number on each subsequent cube is the sum of the numbers from the four adjacent cubes in the lower layer.

Determine the smallest number divisible by four that can belong to the topmost cube.

(A. Bohiniková)

![](https://cdn.mathpix.com/cropped/2024_04_17_dec1ce4fe3ed83b7cc6eg-2.jpg?height=446&width=456&top_left_y=2090&top_left_x=800)
The number belonging to the topmost cube is determined by four numbers from the second layer, and these are entirely determined by the numbers in the first layer. Each corner cube in the first layer is adjacent to one cube from the second layer, each non-corner cube on the edge of the first layer is adjacent to two cubes from the second layer, and the central cube of the first layer is adjacent to all four cubes of the second layer. Thus, the total sum in the topmost cube is contributed to once by four cubes, twice by four cubes, and four times by one cube.

For mutually different numbers in the first layer, the smallest possible number in the topmost cube is

$$
1 \cdot(9+8+7+6)+2 \cdot(5+4+3+2)+4 \cdot 1=62 .
$$

The nearest larger number divisible by four is 64, which is the smallest number belonging to the topmost cube that meets all the requirements. This result can be achieved, for example, by replacing the number 9 in the previous sum with the number 11.

Evaluation. 2 points for determining the smallest possible number; 2 points for determining the smallest number divisible by four and a specific valid realization; 2 points based on the quality of the commentary.

Note. Let the numbers in the first layer be denoted as follows:

$$
\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array}
$$

Then the numbers in the second layer are:

$$
\begin{array}{cc}
a+b+d+e & b+c+e+f \\
d+e+g+h & e+f+h+i
\end{array}
$$

And the number in the topmost cube is:

$$
a+c+g+i+2(b+d+f+h)+4 e
$$