geometry

1. Determine the side lengths of each cube:

Volume of cube 1: 1 cm³

Side length: ∛1 cm³ = 1 cm

Volume of cube 2: 8 cm³

Side length: ∛8 cm³ = 2 cm

Volume of cube 3: 27 cm³

Side length: ∛27 cm³ = 3 cm

Volume of cube 4: 125 cm³

Side length: ∛125 cm³ = 5 cm

2. Find the arrangement with the smallest surface area:

To minimize surface area, we want to minimize the exposed faces.

The most efficient arrangement is to stack the cubes in a way that maximizes the number of faces touching each other.

3. Calculate the surface area:

Individual cube surface areas:

Cube 1: 6 * 1² = 6 cm²

Cube 2: 6 * 2² = 24 cm²

Cube 3: 6 * 3² = 54 cm²

Cube 4: 6 * 5² = 150 cm²

Total surface area of individual cubes: 6 + 24 + 54 + 150 = 234 cm²

Subtract the areas of the glued faces:

Determine the glued faces: This depends on the specific arrangement.

Let's assume we stack them as follows:

Cube 4 (largest) as the base.

Cube 3 on top of Cube 4.

Cube 2 on top of Cube 3

.

Cube 1 on top of Cube 2.

Calculate the areas of the glued faces:

Cube 4 and Cube 3: 3² = 9 cm²

Cube 3 and Cube 2: 2² = 4 cm²

Cube 2 and Cube 1: 1² = 1 cm²

Total area of glued faces: 9 + 4 + 1 = 14 cm²

Calculate the final surface area:

Total surface area - area of glued faces = 234 cm² - 14 cm² = 220 cm²

Therefore, the smallest possible surface area of the resulting solid figure is 220 square centimeters.