Facial Sums
The diagram shows a solid with six triangular faces.
At each vertex there is a number and two of the numbers are 1 and 5, as shown.
The sum of the numbers at the three vertices of each face is calculated, and all the sums are the same.
What is the sum of all five numbers at the vertices?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
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 $1+5+5+5+1=17$.