20. Kanga labelled the vertices of a square-based pyramid using $1,2,3,4$ and 5 once each. For each face Kanga calculated the sum of the numbers on its vertices. Four of these sums equalled $7,8,9$ and 10 . What is the sum for the fifth face?
A 11
B 12
C 13
D 14
E 15
SOLUTION
$\mathbf{C}$

One face has total 7 which can only be obtained from the given numbers by adding 1,2 and 4 . Therefore it is a triangular face, and so the label of the top vertex, $x$ say, is one of these three values. Hence the square face has 5 at one vertex. So the smallest possible face total for the square is $5+1+2+3=11$. Therefore the four face totals given, $7,8,9$ and 10, must be the face totals of the triangular faces and their sum is 34 . Note that each vertex except the top belongs to two triangles; and the top belongs to all four. So the sum 34 is twice the sum of all the labels plus an extra $2 x$; that is $34=2(1+2+3+4+5)+2 x=30+2 x$. Hence $x=2$ and the face total for the square face is $1+3+4+5=13$.