Dice

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Dice — Key Facts for SSC CGL • A standard dice (cube) has 6 faces numbered 1 to 6. The numbers on opposite faces sum to 7: 1↔6, 2↔5, 3↔4. • In a standard dice view, adjacent faces share an edge. Two faces that share only a corner are NOT adjacent. • Open Dice (unfolded nets): There are 11 valid nets out of possible 6-face arrangements (11 unique ways to fold into a cube). • In a single dice roll: 1 appears on top → opposite face (6) is always at the bottom. The four side faces are all possible except 1 and 6. • Number of dots (pips) on opposite faces: Sum always equals 7.

⚡ Exam Tip: When given two views of a dice that share a common face, the third face can be determined by the rule that opposite faces sum to 7. For dice nets, if two faces are adjacent in the net, they are not opposite. Count the visible faces in each view and use elimination.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Dice — SSC CGL Study Guide

Core Concept: Dice questions test your ability to visualise three-dimensional objects and their different views. There are two main types: (1) standard dice (opposite faces sum to 7), and (2) dice nets (2D layouts that fold into 3D cubes). Both require strong spatial reasoning.

Rule 1 — Opposite Faces: In every standard dice, the pairs of opposite faces are fixed:

• 1 is opposite to 6
• 2 is opposite to 5
• 3 is opposite to 4

Rule 2 — Adjacent Faces: If face A is on top and face B is visible from the side, then A and B are adjacent (share an edge). They cannot be opposite.

Worked Example 1 (Finding Opposite Face): Four views of the same dice are shown. If 2 is adjacent to 1, 3, 4, and 6 — and 1 is adjacent to 2, 3, 5, and 6 — which face is opposite to 2?

• 2 is adjacent to: 1, 3, 4, 6
• Remaining faces: 5 is not adjacent to 2
• Therefore: 2 is opposite to 5

Worked Example 2 (Two Views): View 1: 1 on top, 2, 3, 4 visible on sides. View 2: 5 on top, 1, 2, 6 visible on sides. What is opposite to 3?

• From View 1: 3 is adjacent to 1 (top), 2 (side), 4 (side)
• From View 2: 3 appears on a side face when 5 is top
• Faces adjacent to 3: 1, 2, 4 (from View 1) and… what about 5 and 6?
• Since 6 does not appear in either view as adjacent to 3 (only 1, 2, 4 visible), and 5 is also not adjacent to 3… → 3 is opposite to 5? Wait: If 1 is top in View 1, and 5 is top in View 2, then 1 and 5 are not adjacent → they could be opposite or share a hidden relationship.
• Actually: In View 1, faces 2, 3, 4 are adjacent to 1. In View 2, faces 1, 2, 6 are adjacent to 5. The face NOT listed as adjacent to 3 is 5 or 6 → Since 3 and 5 don’t appear together, and 3 and 6 don’t appear together, both could be candidates… We need more info. Usually the question gives a third view.
• Therefore, 3 is opposite to 6 (from the standard dice rule that 3↔4 are opposite — wait 3↔4, so 3 is opposite to 4! Let me re-examine).

Correction: In standard dice, 3↔4 are opposite. If we’re given multiple views, the answer must match the standard rule. So 3 is opposite to 4.

Worked Example 3 (Dice Net): Which of the following arrangements, when folded, will NOT form a standard dice?

• A valid dice net: Each number’s opposite must be correctly placed. For example, in the net with 1 on top and 6 below (with 2, 3, 4, 5 around), the 1 and 6 must be on opposite faces in the net.
• An invalid net: One where 1 and 6 are on adjacent faces in the unfolded layout — they can never be opposite when folded, violating the 1↔6 rule.
• Answer: The net where 1 and 6 share an edge is invalid.

Common Student Mistakes: Assuming faces on opposite corners of a net are opposite (they are NOT), forgetting that a standard dice always has 1↔6, 2↔5, 3↔4 as opposite pairs, and confusing clockwise vs anticlockwise arrangement of adjacent faces.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Dice — Comprehensive SSC CGL Notes

Theoretical Foundation: Dice are right rectangular prisms (specifically cubes) with numbered faces. The standard dice convention (opposite faces summing to 7) was established to ensure fairness in games. SSC CGL reasoning questions exploit the constraint that only one arrangement of opposite pairs is possible in a standard dice.

All 11 Valid Dice Nets: There are exactly 11 unique nets that fold into a standard dice. Each net has exactly 6 squares arranged in different patterns:

• Cross shape (1 centre + 4 arms)
• T-shape variations
• L-shape variations
• Z-shape variations
• Staircase shapes

Key constraint: No net can have more than 4 squares in a row and certain 6-square arrangements (like a 2×3 rectangle without proper arrangement) cannot fold into a dice.

Dice Numbering Variants: In Indian competitive exams, “standard dice” always means opposite faces sum to 7. However, be aware that some questions use non-standard dice where this rule does not apply — these are always clearly labelled or indicated by a different arrangement.

Important Rules for Adjacent Faces: When dice are placed on a surface:

1. Only 5 faces are visible (4 sides + top OR 4 sides + bottom if upside down)
2. If two dice show the same number on top, their side arrangements must be mirror images
3. When a dice is rolled forward (towards you), the top face becomes the front, the bottom face becomes the top

Two Dice Together: When two dice are placed adjacent:

• The faces touching each other are opposite pairs (each = 7)
• If die A shows 3 on the touching face and die B shows 4 on the touching face, both are valid (since 3+4=7 but wait — that means die B’s 3 is on the opposite face to die B’s touching face, so die A’s touching face is 3 → die A’s 4 must be opposite to 3… This requires careful analysis)

Worked Example — Two Dice: Two standard dice are placed side by side. Die 1 shows 3 on the top and 2 on the front face. Die 2 shows 5 on the top and 2 on the front face. The touching faces show (3 on die 1) and (4 on die 2). Is this arrangement possible?

• Die 1: top=3, front=2. Since 1↔6 and 2↔5 and 3↔4, the adjacent faces of 3 are: 1, 2, 4, 5
• Die 2: top=5, front=2. Adjacent faces of 5: 1, 2, 3, 6 (or 2, 3, 4, 6 depending on orientation — wait, 5↔2 are opposite, so 5’s adjacent faces are 1, 3, 4, 6)
• Die 1’s touching face = 3, Die 2’s touching face = 4 → 3+4=7 ✓
• Die 2’s front face = 2. Since die 1’s front = 2 and die 2’s front = 2, they both have the same front value. This is possible if the two dice are oriented as mirror images. → Arrangement is possible.

SSC CGL PYQ Pattern (2019-2023):

• 2023 Tier-I: 1-2 dice questions — most common type is “which face is opposite to X” from given two views
• 2022 Tier-I: 1 dice net question (identifying valid/invalid net)
• Most difficult: Questions where three different views of the same dice are given and you must find the position of a specific face
• Key rule: From any view showing 3 adjacent faces, the remaining three faces are determined (including the one opposite the corner)

Practice Strategy: Master the opposite-pair rule first (1↔6, 2↔5, 3↔4). Then practice questions where you’re given two views and must find the third. For dice nets, draw all 11 nets and memorize the constraint that opposite numbers cannot be adjacent in a valid net.

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