Four identical cubes are stacked vertically one on top of another to form a tower. The edge length of each...

1. TRANSLATE the problem information

• Given information:
  • 4 identical cubes stacked vertically
  • Edge length of each cube = 13 cm
  • Need to find total exposed surface area

• What this tells us: We need to find the total area of all faces that are visible/not covered by other cubes

2. INFER the approach needed

• Key insight: When cubes are stacked, some faces get covered while others remain exposed
• Strategy: Count the exposed faces for each cube, then multiply by the area per face
• What to calculate first: Area of one face, since all faces are identical squares

3. Calculate the area of one cube face

• Area of square face = \(\mathrm{side^2 = 13^2 = 169\ cm^2}\)

4. VISUALIZE the tower and systematically count exposed faces

• Picture the tower from bottom to top:

Bottom cube: Sits on ground with 3 cubes above it

• Bottom face: exposed (touching ground/table)
• 4 side faces: all exposed
• Top face: covered by the cube above
• Total exposed faces: \(\mathrm{1 + 4 = 5}\)

Second cube: Sandwiched between other cubes

• Top face: covered by cube above
• Bottom face: covered by cube below
• 4 side faces: all exposed
• Total exposed faces: \(\mathrm{0 + 4 = 4}\)

Third cube: Also sandwiched between other cubes

• Same logic as second cube
• Total exposed faces: 4

Top cube: Has no cube above it

• Top face: exposed (open to air)
• Bottom face: covered by cube below
• 4 side faces: all exposed
• Total exposed faces: \(\mathrm{1 + 4 = 5}\)

5. SIMPLIFY to find total exposed surface area

• Total exposed faces = \(\mathrm{5 + 4 + 4 + 5 = 18}\) faces
• Total exposed surface area = \(\mathrm{18 \times 169 = 3042\ cm^2}\)

Answer: C) 3042

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak VISUALIZE skill: Students struggle to mentally picture which faces are covered versus exposed when cubes are stacked.

Many students think about surface area of individual cubes (\(\mathrm{6 \times 169 = 1014}\) per cube) and multiply by 4 cubes to get 4056 cm². They fail to realize that stacking covers some faces, reducing the total exposed area. This leads them to select Choice E (4056).

Second Most Common Error:

Poor INFER reasoning about the stacking pattern: Students may correctly understand that some faces get covered but make systematic counting errors.

For example, they might think each cube loses 2 faces (top and bottom) when stacked, giving 4 exposed faces per cube, leading to \(\mathrm{4 \times 4 \times 169 = 2704\ cm^2}\). However, this ignores that the bottom cube's bottom face and top cube's top face remain exposed. This causes them to select Choice B (2704).

The Bottom Line:

This problem requires strong spatial visualization skills to mentally "see" a 3D tower and systematically account for hidden versus visible faces. Students who jump to calculations without careful visualization typically overcount or undercount the exposed surfaces.