SSC CGL Reasoning Dice: Understanding Types problems & Solutions

SSC CGL Reasoning Dice is often considered challenging because questions require quick visualisation, identifying opposite faces, and analysing multiple dice views accurately. Understanding concepts like Closed Dice, Open Dice, Rotation of Dice, and proven solving techniques can help improve speed, reduce confusion, and strengthen performance in CBT-2 and Tier-2 reasoning sections.

Types of Dice Problems

The study of dice problems generally involves three main categories, progressing from fundamental to more complex concepts:

1. Closed Dice: Focuses on dice presented in their typical, closed, cubic form.

2. Open Dice: Deals with dice unfolded into a 2D net, where you must visualise how it folds.

3. Rotation of Dice: Involves analysing the changes in visible faces as a die is rotated.

Anatomy of a Die

Understanding the basic structure of a die is crucial for solving problems. Consider the example of a Ludo die:

• Corners (Vertices): A standard die has 8 corners (4 upper, 4 lower).

• Lines (Edges): It has 12 edges (4 on top, 4 on bottom, 4 connecting top and bottom faces).

• Surfaces (Faces): It has 6 faces (typically 3 are visible, 3 hidden).

Properties of Die Surfaces

Key properties govern how faces relate to each other on a die:

• Visibility: Only three surfaces of a closed die are visible from any single perspective.

• Opposite Surfaces: Each surface possesses only one opposite surface.

• Adjacent Surfaces: Every surface is connected to four adjacent (neighbouring) surfaces. (Total surfaces for any face are: 1 (itself) + 1 (opposite) + 4 (adjacent) = 6 surfaces).

Types of Closed Dice

Closed dice are categorised into two primary types based on the relationship between their adjacent and opposite faces:

1. Ordinary/General/Simple Dice 

• Definition: An ordinary die is characterized by the sum of any two adjacent (neighboring) faces being 7.

• Example: If faces 4 and 3 are visible and adjacent, their sum (4+3=7) indicates an ordinary die.

• Rule for Opposite Faces: For an ordinary die, if the sum of adjacent faces is 7, you cannot determine the opposite face using the 'sum of 7' rule applicable to standard dice.

2. Standard Dice

• Definition: A standard die is identified by the sum of any two adjacent faces being NOT 7.

• Rule for Opposite Faces: In a standard die, the sum of opposite faces is always 7.

• To find the opposite face: Subtract the visible face number from 7.

• Example: Opposite of 1 = 7 - 1 = 6; Opposite of 4 = 7 - 4 = 3; Opposite of 2 = 7 - 2 = 5.

(Memory Tip: Adjacent Surfaces are neighbouring faces. Opposite Surfaces are faces directly across from each other.)

Identifying Dice Types and Opposites

To determine the opposite face, first identify the type of die:

• Example 1: Opposite of 6. If no two visible adjacent faces sum to 7, it's a Standard Die. The opposite of 6 is 1 (7-6=1).

• Example 2: Opposite of 6. If any two visible adjacent faces sum to 7 (e.g., 4+3=7), it's an Ordinary/Normal Die. The opposite face cannot be definitively determined from a single view. The answer is "Cannot Be Determined".

(Memory Tip: Adjacent faces can never be opposite faces.)

Generalizing Face Representation

Dice faces are not limited to numbers. They can feature colors, letters, or symbols. The principle remains the same: adjacent faces cannot be opposite. For instance, if Red, Yellow, and Pink are visible, Yellow and Red cannot be opposite Pink.

Solving Dice Problems with Multiple Views

When multiple views of a die are provided, specific methods help determine opposite faces. These are frequently tested in competitive exams.

Case 1: One Common Face (One Common Trick)

• Condition: Two views of a die share only one common face.

• Method:

1. Identify the common face.

2. From the common face, rotate clockwise in both views and list the faces.

3. Faces aligning vertically in the list are opposite pairs.

4. The common face's opposite is the face that is not visible in either of the given views.

Case 2: Two Common Faces (Two Common Trick)

• Condition: Two views of a die share two common faces.

• Method:

1. Identify the two common faces.

2. The remaining (uncommon) faces in each view will be opposite to each other.

Case 3: No Common Face (No Common Trick)

• Condition: Two views of a die have no common face.

• Method: If it's a standard die, opposite faces sum to 7. Otherwise, specific relationships like top-to-top and side-to-side opposition are often assumed, but this requires context.

More Than Two Dice Views

When more than two views are provided, apply the One Common Trick or Two Common Trick by selecting any two appropriate views. The elimination method also becomes very powerful.

Elimination Method for Multiple Views

• Principle: An adjacent face can never be the opposite face.

• Application: To find the opposite of a specific face, list all faces adjacent to it across all given views. These listed faces cannot be its opposite. The only face remaining from the standard set (e.g., 1-6) must be the opposite.

Identifying Incorrect Statements about Face Adjacency

This advanced question tests your understanding of face relationships. You must identify a statement that is definitely false based on a die's properties.

Core Principle for Adjacency:

• A given face has one opposite face and four adjacent faces.

• A face cannot be adjacent to its own opposite face.

• Thus, any given face must be adjacent to four other faces.