Let the numbers at two of the other vertices be $ u$ and $v$, as shown in the diagram. The three faces sharing the vertex labelled with the number 1 all have the same sum. Then $1+v+u=1+5+u$ and so $v=5$.
Similarly, $1+v+5 = 1+v+u$ so $u =5$.
Hence the sum for each faces is $1+5+5=11$, and we see that the number at the bottom vertex is $1$.
The total of all the vertices is $1+5+5+5+1=17$.
This problem is taken from the UKMT Mathematical Challenges.