Math Kangaroo, Level 3-4 Problem

Robert had a certain number of identical cubes. He glued a tunnel using half of his blocks (see Fig. 1). With some of the remaining cubes he formed a pyramid (see Fig. 2). How many blocks were not used to build those structures?

Fig. 1	Fig. 2

Solution

The tunnel consists of 4 layers, and each layer has 16 blocks. Therefore, the number of blocks used for the tunnel is $16 \times 4 = 64$. Since the tunnel took one half of all blocks to build, $64$ blocks remained unused.

The pyramid consists of three parts: the foundation, the mid-section and the tower. The foundation is $5 \times 5 = 25$ cubes. The each of the 2 stories in the mid-section is $3 \times 3 = 9$ blocks, or 18 blocks in total. Finally, the tower is build with $3$ blocks. This bring the total number of blocks used to build the pyramid to $25 + 18 + 3 = 46$.

So the number of blocks still left after building both structures is $64 - 46 = 18$.