Facial Sums
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.
Therefore $1+v+u=1+5+u$ and so $v=5$.
Similarly, $1+v+5 = 1+v+u$ and so $u =5$.
Hence the sum for each face is $1+5+5=11$, and so the number at the bottom vertex must be $1$.
The total of all the vertices is $1+5+5+5+1=17$.
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