Terry made a cube using cardboard. Terry then made 9 identical cardboard cubes to form solid P as shown below. (Top view, Front view and Side view are indicated on the drawing.)

1. Determine the maximum height of each column when viewed from the front.

Looking from the front, the solid is divided into 3 columns (left, centre, right).

• The left column has a maximum height of 2 cubes (due to the stack at the left-back position).
• The centre column has a maximum height of 1 cube.
• The right column has a maximum height of 1 cube.

1. Determine the condition for the least possible unpainted faces

To minimise the number of unpainted faces, we must maximise the number of painted faces. The painted faces are the exposed faces of the solid. Thus, we need to arrange the 9 cubes to form a solid with the maximum possible surface area.

2. Find the maximum number of exposed faces for 9 cubes

Each of the 9 cubes has 6 faces, giving a total of $9 \times 6 = 54$ faces.
When cubes are glued together, each glued connection hides 2 faces (one from each cube).
To maximise the exposed faces, we must minimise the number of glued connections.
The minimum number of connections to keep 9 cubes together as a single solid is 8 (forming a straight line or a simple tree structure without loops).
Number of hidden faces = $8 \times 2 = 16$.
Number of exposed (painted) faces = $54 - 16 = 38$.