GK 2020 Q22
22. Cleo builds a pyramid with identical metal spheres. Its square base is a 4 × 4 array of spheres, as shown in the diagram. The upper layers are a 3 × 3 array of spheres, a 2 × 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
Grey Kangaroo 2020, Question 22. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 96
Consider first how to join each individual layer. In the 4 × 4 layer there are 4 × 3 × 2 = 24 points of contact. Similarly, in the 3 × 3 layer there are 3 × 2 × 2 = 12 points of contact and in the 2 × 2 layer there are 2 × 1 × 2 = 4 points of contact. Now consider how to join the layers together. To join the 4 × 4 layer to the 3 × 3 layer, each sphere in the 3 × 3 layer has 4 points of contact with the lower layer, making 3 × 3 × 4 = 36 points of contact. Similarly to join the 3 × 3 layer to the 2 × 2 layer, each sphere in the 2 × 2 layer has 4 points of contact with the lower layer making 2 × 2 × 4 = 16 points of contact and the single sphere in the top layer has 4 points of contact with the 2 × 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.
Solution © UK Mathematics Trust, from the GK 2020 solutions.