11. Johan has a large number of identical cubes. He has made the structure on the right by taking a single cube and then sticking another cube to each face. He wants to make an extended structure in the same way so that each face of the structure on the right will have a cube stuck to it. How many extra cubes will he need to complete his extended structure?

A 10
B 12
C 14
D 16
E 18
Solution
E

In the given structure, there are five exposed faces on each of the six cubes that were added to the original cube and hence there are 30 faces to cover on the structure. Adding a cube so that one of its edges touches an edge of the original cube covers two faces on the structure. The original cube has 12 edges and so 12 cubes are required to cover 24 of the faces of the structure. A further six cubes are required to cover the final six outer faces of the structure. Therefore the number of extra cubes Johan needs to complete his extended structure is $12+6=18$.