A solid block is painted on all its outer faces and then cut into small $2$-cm cubes as shown. (a) How many $2$-cm cubes did Shu Xiu cut from the block? (b) How many of these $2$-cm cubes had none of the faces painted?

1. Find the number of $2$-cm cubes along each dimension of the block.

Number of cubes along length = $32 \div 2 = 16$
Number of cubes along height = $24 \div 2 = 12$
Number of cubes along depth = $12 \div 2 = 6$

2. Calculate the total number of cubes for part (a).

Total cubes = $16 \times 12 \times 6 = 1152$

3. Determine the dimensions of the unpainted interior region in terms of cubes.

The unpainted cubes form an inner block. We remove $1$ layer of cubes from each of the $6$ faces (i.e., $2$ layers per dimension).
Interior length = $16 - 2 = 14$ cubes
Interior height = $12 - 2 = 10$ cubes
Interior depth = $6 - 2 = 4$ cubes

4. Calculate the total number of unpainted cubes for part (b).

Number of unpainted cubes = $14 \times 10 \times 4 = 560$