13. $A B C D E F G H$ is a cube in which $A B C D$ is the top face, with vertices $H, G, F$ and $E$ directly below the vertices $A, B$, $C$ and $D$ respectively. A real number is assigned to each vertex. At each vertex, the average of the numbers in the three adjacent vertices is then computed. The averages obtained at $A, B, C, D, E, F, G, H$ are 1, 2, 3, 4, 5, 6, 7, 8 respectively. Find the number assigned to vertex $F$.
(2 marks)
13. $A B C D E F G H$ is a cube, where $A B C D$ is the top face, and $H, G, F, E$ are directly below $A, B, C, D$ respectively. A real number is assigned to each vertex, and then at each vertex, the average of the numbers in the three adjacent vertices is computed. The averages obtained at $A, B, C, D, E, F, G, H$ are 1, 2, 3, 4, 5, 6, 7, 8 respectively. Find the number assigned to vertex $F$.
13. 13
13. Let \(a, b, c, d, e, f, g, h\) denote the numbers assigned, and \(A, B, C, D, E, F, G, H\) the averages obtained, at the vertices \(A\) to \(H\) respectively. First we note that
\[
a+b+c+d+e+f+g+h=1+2+3+4+5+6+7+8=36 \text{.}
\]

Also,
\[
c+f=36-(b+d+h)-(a+e+g)=36-3(A+H).
\]

Similarly, we have
\[
e+f=36-3(A+B) \text{ and } f+g=36-3(A+D) \text{.}
\]

Therefore
\[
\begin{aligned}
3 f & =(c+f)+(e+f)+(f+g)-(c+e+g) \\
& =[36-3(1+8)]+[36-3(1+2)]+[36-3(1+4)]-6 \times 3 \\
& =39
\end{aligned}
\]
from which we get \(f=13\).