RRB Group D Reasoning Cube Notes: Introduction, Examples
RRB Group D Reasoning Cube it covers defining 'n' as the ratio of large to small cube side lengths, calculating total cubes as n³, and classifying smaller cubes based on colored faces. Key formulas are provided for cubes with three (8), two (12*(n-2)), one (6*(n-2)²), and zero ((n-2)³) surfaces colored, along with memory aids.
RRB Group D Reasoning Cube: Cube-cutting problems are an important part of RRB Group D Reasoning exams, designed to test a candidate’s spatial reasoning and visualization skills. These questions typically involve a larger cube, often painted on its surfaces, which is then divided into smaller, identical cubes. Aspirants are asked to determine the number of smaller cubes with a specific number of colored faces. Understanding the systematic approach, key formulas, and memory tricks for corner, edge, face, and inner cubes can help candidates solve these problems quickly and accurately in competitive exams.
RRB Group D Reasoning Cube
Cube-cutting problems are a common component of competitive exams, testing spatial reasoning. Check systematic approach to solving these, focusing on scenarios where a larger, painted cube is divided into smaller, identical cubes. We will also explore the classification of these smaller cubes based on their colored surfaces, using straightforward formulas to eliminate the need for complex visual aids.
Also Read:
Understanding Cube Cutting Basics
In cube-cutting problems, a large cube, often painted on all surfaces, is cut into smaller, identical cubes. The core challenge is determining the number of smaller cubes with specific numbers of colored faces. For example, if a 4 cm large cube is painted green on all surfaces and then cut into smaller cubes of 1 cm side length, its dimensions are 4 cm × 4 cm × 4 cm. To cut a 4 cm length into 1 cm pieces, 3 cuts are required along each dimension (length, breadth, and height).
The 'n' Variable and Total Cubes
To systematically solve any cube-cutting problem, the first step is to calculate the variable 'n'.
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Definition of 'n': The variable 'n' represents the ratio of the side length of the larger cube to the side length of the smaller cube.
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n = (Measurement of Large Cube) / (Measurement of Small Cube)
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Example Calculation: If the large cube is 4 cm and the small cube is 1 cm:
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n = 4 / 1 = 4
Once 'n' is determined, the total number of small cubes created can be found.
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Total Small Cubes: The total number of small cubes formed is always n³.
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In our example (n=4), Total Cubes = 4³ = 64.
Classification of Cubes by Colored Surfaces
After cutting, paint only remains on the exterior surfaces of the smaller cubes. The interior cubes remain uncolored. Smaller cubes are categorized based on the number of colored faces they possess, with a maximum of three colored surfaces possible for any single small cube.
1. Corner Cubes (Three Surfaces Colored)
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These cubes are found at the corners of the original large cube.
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The number of Corner Cubes is always fixed at 8, irrespective of the value of 'n'.
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Exam questions asking for cubes with three surfaces colored will always have 8 as the answer.
2. Middle Cubes (Two Surfaces Colored)
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These cubes are located along the edges of the original large cube, excluding the corner positions.
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These are also referred to as cubes with four surfaces colorless.
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Formula: The number of middle cubes is given by 12 * (n - 2).
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Example (n=4): 12 * (4 - 2) = 12 * 2 = 24.
3. Central Cubes (One Surface Colored)
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These cubes are situated at the center of each of the 6 faces of the original large cube.
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They may also be referred to as cubes with five surfaces colorless.
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Formula: The number of central cubes is given by 6 * (n - 2)².
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Example (n=4): 6 * (4 - 2)² = 6 * (2)² = 6 * 4 = 24.
4. Inner Cubes (Zero Surfaces Colored / Colorless)
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These cubes come from the core of the original large cube and were never exposed to the paint. They have no colored faces.
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Formula: The number of inner cubes is given by (n - 2)³.
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Example (n=4): (4 - 2)³ = 2³ = 8.
Memory Aids for Cube Formulas
A simple pattern helps in memorizing the formulas for the number of colored surfaces:
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3 Surfaces Colored: This is always fixed at 8, representing the fixed number of corners in any cube.
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2 Surfaces Colored: Remember the base formula: 12 * (n - 2).
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1 Surface Colored: From the previous formula, reduce the coefficient by 6 (from 12 to 6) and increase the power of (n - 2) by 1 (from ¹ to ²). The formula becomes 6 * (n - 2)².
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0 Surfaces Colored: From the previous formula, remove the coefficient (6 becomes 1) and increase the power of (n - 2) by 1 (from ² to ³). The formula becomes (n - 2)³.
Worked Examples
Let's consider a scenario: A 5 cm cube is cut into 1 cm cubes.
Here, n = 5/1 = 5.
1. How many small cubes have only one surface colored?
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This refers to Central Cubes.
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Formula: 6 * (n - 2)²
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Calculation: 6 * (5 - 2)² = 6 * (3)² = 6 * 9 = 54.
2. How many small cubes have at least two surfaces colored?
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"At least two" means cubes with two surfaces colored OR three surfaces colored.
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Two Surfaces Colored (Middle Cubes): 12 * (n - 2) = 12 * (5 - 2) = 12 * 3 = 36.
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Three Surfaces Colored (Corner Cubes): 8 (always fixed).
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Total = 36 + 8 = 44.
3. How many small cubes have at most one surface colored?
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"At most one" means cubes with one surface colored OR zero surfaces colored.
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One Surface Colored (Central Cubes): We calculated this as 54.
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Zero Surfaces Colored (Inner Cubes): (n - 2)³ = (5 - 2)³ = 3³ = 27.
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Total = 54 + 27 = 81.