The figures show the nets of solids with numbered faces. After these solids are assembled, we say that the value of a vertex is the sum of the numbers written on the faces that contain that vertex. For example, the figure on the side shows the net of a pyramid; when this pyramid is assembled, the value of the vertex corresponding to the point indicated in the figure is $1+3+4=8$.

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-14.jpg?height=177&width=391&top_left_y=1942&top_left_x=730)

a) What is the maximum value of a vertex of the pyramid above?

b) The figure below shows the net of a cube. What is the value of the vertex corresponding to the point indicated?

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-14.jpg?height=241&width=450&top_left_y=2321&top_left_x=700)

c) The figure below shows the net of a solid called an octahedron. What is the value of the vertex corresponding to point $\mathbf{A}$?

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-15.jpg?height=226&width=605&top_left_y=247&top_left_x=834)

d) What is the value of the vertex corresponding to point B in the net from the previous item?
Solution

a) In the pyramid, each vertex belongs to three faces. The marked point will become the vertex of the faces with the numbers 2, 3, and 4; since these are the three largest numbers that appear on the faces, this vertex will have the highest sum, which is $2+3+4=9$.

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-39.jpg?height=165&width=192&top_left_y=477&top_left_x=835)

b) In a cube, each vertex belongs to three faces. When assembling the cube, the dotted edges in the figure below will coincide, as will the points $A$ and $B$. We see that the faces that meet at the vertex corresponding to point $A$ are the faces with the numbers 3, 6, and 2; thus, the value of this vertex is $3+6+2=11$.

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-39.jpg?height=277&width=319&top_left_y=895&top_left_x=777)

c) In an octahedron, each vertex belongs to four faces. The figure below shows that, when forming the octahedron, point $A$ will be the common vertex of the faces with the numbers 4, 5, 6, and 7; thus, its value will be $4+5+6+7=22$.

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-39.jpg?height=303&width=414&top_left_y=1376&top_left_x=730)

d) When assembling the octahedron, the two segments indicated by the letter $a$ will form an edge and the points $C$ and $D$ will coincide. Therefore, the segments indicated by $b$ will also coincide, and point $B$ will be brought to point $E$. Thus, the faces that have the vertex corresponding to $B$ in common are the faces with the numbers 1, 2, 4, and 5; the value of this vertex is then $1+2+4+5=12$.

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-39.jpg?height=326&width=437&top_left_y=1936&top_left_x=707)