VisionIAS - Video Classroom Lecture

Reasoning Class 01

CSAT Reasoning: Cubes and Dice

Overview of CSAT and Its Importance

Civil Services Aptitude Test (CSAT) is a crucial part of the UPSC Civil Services Preliminary Examination.
While it is qualifying in nature, it plays a pivotal role in a candidate`s journey towards becoming an IAS officer.
Despite being a qualifying paper, many deserving candidates face challenges in clearing CSAT, particularly the reasoning section.
Over the past few years, there has been a noticeable increase in the difficulty level of CSAT, making it imperative for aspirants to give due attention to it.

Syllabus Coverage

The CSAT paper broadly covers three areas:
Basic Numeracy, Data Interpretation (DI), and Data Sufficiency (DS)
Logical Reasoning - Within the logical reasoning section, several topics are crucial, including calendars, ranking and comparison, cubes and dice, clocks, seating and complex arrangements, direction sense, series, blood relations, visual reasoning, coding-decoding, Venn diagrams, syllogism, selections, and miscellaneous topics.
Reading Comprehension

Focus on Cubes and Dice

Understanding Dice

A die is a cube-shaped object commonly used in games like Ludo. In CSAT, understanding the properties and types of dice is essential for solving related problems.
Types of Dice
Standard Die: In a standard die, the sum of the numbers on opposite faces is always seven. The opposite face pairs are as follows:
1 opposite to 6
2 opposite to 5
3 opposite to 4
Non-Standard Die: Any die that does not follow the properties of a standard die is considered non-standard. In non-standard dice, the numbers or symbols on the faces can be arranged in any manner, and the sums of opposite faces do not necessarily equal seven.

Visible Faces on a Die

At any given time, a maximum of three faces of a die are visible. Importantly, opposite faces are never visible simultaneously.
Determining Possible Numbers on a Face - When analyzing a die, if certain numbers are visible, one can deduce which numbers cannot be on the opposite unseen faces based on the properties of a standard die.

Rules to Find Opposite Faces on a Die

To solve problems involving the determination of opposite faces on dice, two primary rules are applied.
Rule 1: One Common Face
If in two different views of a die, only one face is common, then:
Start from the common face and write the adjacent faces in a clockwise order for both views.
The corresponding adjacent faces in the two views are opposite to each other.
The face opposite to the common face is the one not present in either of the views (the missing number).
Example: Given two views of a die with the number 2 common in both:
First view: 2, 1, 6
Second view: 2, 5, 3

Starting from 2 and moving clockwise:

First view sequence: 2-1-6
Second view sequence: 2-5-3

Opposite pairs:

1 opposite to 5
6 opposite to 3
The missing number (4) is opposite to the common face (2)

Rule 2: Two Common Faces

If in two different views of a die, two faces are common, then:
The non-common faces in the two views are opposite to each other.
The common faces cannot be opposite to each other.
Example: Given two views with faces 2 and 5 common in both:
The first view has 6 as the non-common face.
The second view has 3 as the non-common face.
Therefore, 6 and 3 are opposite faces.

Practice Problems Applying the Rules

Determining the Opposite Face of a Number:
- Given multiple views of a die, apply the appropriate rule to deduce which number is opposite to the given number.
- For instance, if asked which number is opposite to 3, and based on the views and common faces, use Rule 1 or Rule 2 to find the answer.
Identifying Standard and Non-Standard Dice:
- Analyze the visible numbers to determine if a die is standard.
- If opposite pairs (that sum to seven) are both visible, it cannot be a standard die.
Solving Previous Year Questions:
- Practice questions from past CSAT papers, such as determining the symbol opposite a certain face when given multiple views.

Cubes and Cuboids

Definitions and Properties

Cube: A three-dimensional figure with all sides equal (length = breadth = height).
Cuboid: A three-dimensional figure where length, breadth, and height may differ.

Volume Formulas

Cube: Volume = side³
Cuboid: Volume = length × breadth × height

Components of Cubes and Cuboids

Corners (Vertices): Points where three faces meet. Both cubes and cuboids have eight corners.
Edges: Line segments where two faces meet. Both shapes have twelve edges.
Faces: Flat surfaces. Both shapes have six faces.

Cutting Cubes and Cuboids

When a larger cube or cuboid is cut into smaller identical cubes:
Total Number of Smaller Cubes:
- Calculated by dividing the dimensions of the larger shape by the side length of the smaller cube in each dimension and multiplying the results.
- Formula: Number of Smaller Cubes = (Length / Side of Small Cube) × (Breadth / Side) × (Height / Side)
Example: A cube of side 5 cm is cut into smaller cubes of side 1 cm:
Number of Smaller Cubes: (5/1)³ = 125

Calculating the Minimum Number of Cuts

Minimum Cuts Required: To obtain and pieces, minimum cuts required = n - 1 when all cuts are in the same plane.
However, to maximize the number of pieces with a given number of cuts, distribute the cuts equally among all three dimensions.

Maximizing the Number of Pieces

Formula: Number of Pieces = (Number of Cuts in Length + 1) × (Number of Cuts in Breadth + 1) × (Number of Cuts in Height + 1)
To maximize pieces, cuts should be as evenly distributed as possible across the three dimensions.
Example: With six cuts, the maximum number of pieces is achieved by making two cuts in each dimension:
Number of Pieces: (2 + 1) × (2 + 1) × (2 + 1) = 27

Painting of Cubes and Cuboids

When a cube or cuboid is painted on all faces and then cut into smaller cubes:

Determining the Number of Smaller Cubes with Painted Faces

Three Faces Painted:
- Only the corner pieces have three faces painted.
- Number: Always 8 (since a cube has 8 corners).
Two Faces Painted:
- Occur along the edges but not at the corners.
- Calculations:
  - For each edge, subtract the corner pieces.
  - Formula for Cuboid: Number = 4(L - 2) + 4(B - 2) + 4(H - 2)
  - The formula for Cube: Number = 12(N - 2), where N is the number of pieces along one edge.
One Face Painted:
- Found on the faces but not along the edges.
- Calculations:
  - Subtract the border layers (edges and corners) from each face.
  - Formula for Cuboid: Number = 2[(L - 2)(B - 2) + (B - 2)(H - 2) + (L - 2)(H - 2)]
  - Formula for Cube: Number = 6(N - 2)²
No Faces Painted:
- The innermost cubes do not touch any painted face.
- Formula for Cuboid: Number = (L - 2)(B - 2)(H - 2)
- Formula for Cube: Number = (N - 2)³

Examples:

Cube Painted on All Faces and Cut into Smaller Cubes:
- For a cube of side 5 cm cut into 125 smaller cubes of 1 cm side:
  - Three Faces Painted: 8
  - Two Faces Painted: 12(N - 2) = 12(5 - 2) = 36
  - One Face Painted: 6(N - 2)² = 6(3)² = 54
  - No Faces Painted: (N - 2)³ = 27

Advanced Problems on Painted Cubes

When cubes are painted with different colors on different faces:
Opposite Faces Painted the Same Color: For a cube painted so that opposite faces have the same color (e.g., red, green, yellow), calculations adjust based on color distribution.
Calculating Cubes with Specific Color Combinations:
- Total Cubes with a Specific Color:
  - Multiply the number of faces painted in that color by the number of smaller cubes on one face.
- Cubes with Only One Color:
  - Consider the inner portions of the painted faces, excluding edges and corners.
- Cubes with Multiple Colors:
  - Identify edges and corners where different colored faces meet.
  - Calculate based on the number of such edges and corners.
Example: A cube is painted with red, green, and yellow colors on opposite faces and cut into 125 smaller cubes:
Cubes with All Three Colors:
Only corner cubes can have three faces painted.
Number: 8
Cubes with Only Red Color:
Calculate the inner face cubes on red-painted faces, excluding edges and corners.
Number: Depends on N; for N = 5, only red cubes = 9 (on one face) × 2 (since there are two red faces) = 18

Folding and Unfolding of Dice

Understanding how two-dimensional nets fold into three-dimensional dice is crucial for certain types of reasoning problems.
Opposite Faces in Unfolded Views:
- In a net of a cube, squares that are opposite in the net will have opposite faces when folded.
- Rule: Alternate squares in the same row or column will be opposite faces.
Practice Problems:
- Given a net, determine which faces will be opposite when folded.
- Given a folded cube, determine which unfolded net corresponds to it.

Class Activities and Problem-Solving

Throughout the lecture, several problems were presented to illustrate the concepts:
Calculation-Based Problems:
- Determining the number of smaller cubes with certain painted faces.
- Calculating cuts required for a given number of pieces.
Conceptual Questions:
- Determining opposite faces in dice.
- Understanding properties of standard and non-standard dice.
Higher Difficulty Level Questions:
- Cubes painted with different colors on adjacent faces.
- Calculating the number of cubes with specific color combinations.

Topics to be Discussed in the Next Class:

Sitting and Complex Arrangements: Strategies to solve seating arrangement problems.
Direction Sense: Techniques to tackle direction-related reasoning questions.
Series Completion: Approaches for number and letter series problems.