22. Cleo builds a pyramid with identical metal spheres. Its square base is a $4 \times 4$ array of spheres, as shown in the diagram. The upper layers are a $3 \times 3$ array of spheres, a $2 \times 2$ array of spheres and a single sphere at the top. At each point of contact between two spheres, a blob of glue is placed. How many blobs of glue will Cleo place?
A 72
B 85
C 88
D 92
E 96
SOLUTION
$\mathbf{E}$

Consider first how to join each individual layer. In the $4 \times 4$ layer there are $4 \times 3 \times 2=24$ points of contact. Similarly, in the $3 \times 3$ layer there are $3 \times 2 \times 2=12$ points of contact and in the $2 \times 2$ layer there are $2 \times 1 \times 2=4$ points of contact. Now consider how to join the layers together. To join the $4 \times 4$ layer to the $3 \times 3$ layer, each sphere in the $3 \times 3$ layer has 4 points of contact with the lower layer, making $3 \times 3 \times 4=36$ points of contact. Similarly to join the $3 \times 3$ layer to the $2 \times 2$ layer, each sphere in the $2 \times 2$ layer has 4 points of contact with the lower layer making $2 \times 2 \times 4=16$ points of contact and the single sphere in the top layer has 4 points of contact with the $2 \times 2$ layer. Hence the total number of points of contact is $24+12+4+36+16+4=96$ and therefore 96 blobs of glue are used.