Comprehensive CSAT Guide: Cubes

1. Introduction and Importance

In the UPSC and MPSC CSAT, spatial reasoning and visual-spatial intelligence are tested extensively. Questions based on Cubes—specifically painted cubes that are cut into smaller identical pieces—are a staple of the reasoning section. These questions appear complex because they require visualizing a 3D object, but they can be solved accurately and swiftly using simple mathematical formulas and logical deductions.

Mastering this topic guarantees easy marks because the underlying structure of the problems rarely changes. Once you memorize the core formulas and understand how faces, edges, and corners work, you can solve any painted cube problem in under a minute.

2. Core Concepts, Formulas, and Tricks

A standard cube has:

• 6 Faces (Flat surfaces)
• 12 Edges (Lines where two faces meet)
• 8 Corners/Vertices (Points where three edges meet)

When a large cube (say of side ‘L’) is painted on all its outside faces and then cut into smaller, identical cubes (each of side ‘l’), we use a fundamental ratio denoted by ‘n’:

n = (Length of the side of the larger cube) / (Length of the side of the smaller cube)

Total number of smaller cubes formed = n³

The Core Formulas

Based on the position of the smaller cubes within the larger painted cube, they will have a specific number of painted faces:

• 1. Three faces painted (Corner cubes): These are always located at the corners of the large cube. Regardless of the size of ‘n’ (as long as n > 1), the number is always fixed.
  Formula: 8
• 2. Two faces painted (Middle cubes): These are located along the edges of the large cube, between the corners.
  Formula: 12 × (n – 2)
• 3. One face painted (Central cubes): These are located in the center of the faces of the large cube.
  Formula: 6 × (n – 2)²
• 4. Zero faces painted (Inner/Core cubes): These are completely hidden inside the large cube.
  Formula: (n – 2)³

3. Solved Examples with Step-by-Step Explanations

Example 1: The Basic Painted Cube

Question: A large cube of side 12 cm is painted red on all its faces. It is then cut into smaller cubes, each of side 3 cm. How many smaller cubes have exactly one face painted red?

Step-by-Step Explanation:
1. Calculate ‘n’: n = (Side of large cube) / (Side of small cube) = 12 / 3 = 4.
2. Identify the formula for “exactly one face painted”: 6 × (n – 2)².
3. Substitute n = 4 into the formula: 6 × (4 – 2)².
4. Calculate: 6 × (2)² = 6 × 4 = 24.
Answer: 24 cubes have exactly one face painted.

Example 2: Finding the Inner Core

Question: A solid wooden cube is painted blue on all sides and then cut into 125 smaller, identical cubes. How many of these smaller cubes have no paint on them?

Step-by-Step Explanation:
1. First, find ‘n’ from the total number of cubes. We know Total cubes = n³.
2. Therefore, n³ = 125, which means n = 5.
3. We need the number of cubes with “zero faces painted” (inner cubes).
4. Formula: (n – 2)³.
5. Substitute n = 5: (5 – 2)³ = 3³ = 27.
Answer: 27 cubes have no paint on them.

Example 3: Different Colors on Opposite Faces

Question: A large cube is painted Red on two opposite faces, Blue on two opposite faces, and Green on the remaining two opposite faces. It is then cut into 64 smaller cubes. How many smaller cubes have at least one face painted Blue and no other color?

Step-by-Step Explanation:
1. Find ‘n’: n³ = 64, so n = 4.
2. The question asks for cubes with only Blue paint. This means they must have exactly ONE face painted, and that face must be Blue.
3. Cubes with exactly one face painted are located on the faces of the large cube. The formula for one face of the large cube is (n – 2)².
4. How many faces are painted Blue? Exactly two faces.
5. Calculate for these two Blue faces: 2 × (n – 2)² = 2 × (4 – 2)² = 2 × 2² = 2 × 4 = 8.
Answer: 8 cubes have only Blue paint on them.

4. Pro-Tips to Avoid Common Mistakes

• “At least” vs. “Exactly”: Pay very close attention to these words. “At least two faces painted” means you must add the cubes with 2 faces painted AND the cubes with 3 faces painted. “Exactly two faces” means ONLY the cubes with 2 faces painted.
• Cutting vs. Pieces: A cube cut ‘x’ times along one axis produces ‘x+1’ pieces. If a cube requires 3 cuts along length, width, and height, the total pieces are 4 × 4 × 4 = 64. The value of ‘n’ in this case is 4.
• The Corner Trap: Remember that cubes with 3 painted faces are always 8, regardless of the size of the cube (unless the cube is simply 1x1x1 or cut in a non-standard way, which is rarely asked).
• Cross-check your math: The sum of cubes with 3 faces + 2 faces + 1 face + 0 faces must always equal n³!

5. Practice Questions

Q1. A cube is painted yellow on all sides and cut into 216 smaller cubes of equal size. How many cubes will have exactly two faces painted?

1. 48
2. 36
3. 24
4. 64

(Hint: n³ = 216, so n = 6. Use the formula for 2 faces: 12 × (n – 2). Answer: 1)

Q2. A cube of side 5 cm is painted on all its surfaces. If it is sliced into 1 cubic centimeter cubes, how many cubes will have at least one face painted?

1. 125
2. 98
3. 27
4. 116

(Hint: It’s easier to subtract the unpainted cubes from the total cubes. Total = 125. Unpainted = (5-2)³ = 27. Answer: 125 – 27 = 98. Option 2)

Q3. A cube is painted red on three adjacent faces and black on the remaining three faces. It is then cut into 64 smaller cubes. How many cubes have only one face painted red?

1. 12
2. 8
3. 16
4. 24

(Hint: Find ‘n’. Only one face painted happens in the center of the faces. How many faces are red? 3 faces. Use 3 × (n-2)². Answer: 1)