Dices, Cubes and cuboids

Table of Contents

1. Syllabus Overview
2. 1: Basic Concepts of Dice
3. 2: Standard Dice (Opposite Faces)
4. 3: Construction of Cubes (Painting and Cutting)
5. 4: Cuboids (Rectangular Boxes)
6. 5: Advanced Puzzles & Data Sufficiency

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Syllabus Overview Dices {#syllabus-overview-dices}

1. Basic Concepts of Dice: Understanding the structure of a cube, identifying adjacent vs. opposite faces, and visualizing dice nets.
2. Standard Dice (Opposite Faces): Mastering the "Sum-to-7" rule and deducing opposite pairs from multiple views of a single dice.
3. Construction of Cubes (Painting and Cutting): Calculating painted faces (0, 1, 2, 3) for $n \times n \times n$ cubes and $m \times n \times p$ cuboids.
4. Cuboids (Rectangular Boxes): Mastery of dimension-based counting for surface and interior unit cubes.
5. Advanced Puzzles & Data Sufficiency: Methodology for multi-color painting, partial painting adjustments, and DS uniqueness testing.
6. Folding & Unfolding (Nets): (Upcoming) Analyzing various dice nets and predicting face positions after folding into a cube.

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1: Basic Concepts of Dice {#1-basic-concepts-of-dice}

1.1 What is a Dice?

A dice (or die) is a small cube with numbers (or symbols) on each of its six faces. In a standard dice, the numbers from 1 to 6 appear exactly once.

1.2 Adjacent vs. Opposite Faces

Understanding the spatial relationship between faces is the key to solving dice problems:

• Adjacent Faces: Faces that share a common edge. Every face in a cube has 4 adjacent faces.
• Opposite Faces: Faces that do not share any edge or vertex. Every face in a cube has exactly 1 opposite face.

Key Logic:

• In a dice net, opposite faces are always separated by exactly one face.
• Adjacent faces can never be opposite to each other.

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2: Standard Dice (Opposite Faces) {#2-standard-dice}

2.1 The Sum-to-7 Rule

In a standard dice, the sum of any two opposite faces is always 7. This provides a fixed mapping:

2.2 Deducing Opposites from Multiple Views

When given multiple views of the same dice (standard or non-standard), use these rules:

1. Common Face Rule: If two views of a dice show one common face, the faces adjacent to it in both views cannot be opposite to it. The "missing" number must be the opposite.
2. Two Common Faces Rule: If two views show two common faces (e.g., 2 and 3), the third faces in both views (e.g., 1 and 4) must be opposite to each other.

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3: Construction of Cubes (Painting and Cutting) {#3-construction-of-cubes}

3.1 Core Concepts

A large cube of side length $n$ units (made up of $n \times n \times n$ small cubes) is painted on its outer surfaces. Then it is cut into $n^3$ unit cubes. The small cubes are classified by how many of their faces are painted.

Key Components:

• Faces: 6
• Edges: 12
• Corners: 8
• Cubes per edge: $n$

3.2 Formulas for an $n \times n \times n$ Cube

3.3 Worked Examples

Example 1: Cube ($n = 5$)
Question: A cube of side 5 units is painted on all faces. How many small cubes have exactly 2 faces painted?

• $n = 5$
• $2$-painted = $12 \times (5 - 2) = 12 \times 3 = 36$.
• Answer: 36

Example 2: Reverse Problem
Question: A cube is cut into unit cubes. The number of cubes with 2 faces painted is 48. Find $n$.

• $12(n - 2) = 48 \implies n - 2 = 4 \implies n = 6$.
• Answer: 6 ($6 \times 6 \times 6$ cube)

Example 3: Partial Painting
Question: A cube of side 6 is painted on all faces except the bottom. How many cubes have exactly 2 painted faces?

• Standard with full paint: $12(6-2) = 48$.
• Bottom edges affected: 4 edges at the bottom.
• Bottom edges lose 1 paint: $4 \times (6-2) = 16$ cubes drop from "2 faces" to "1 face".
• Bottom corners lose 1 paint: 4 corners drop from "3 faces" to "2 faces".
• Net Result: $48 - 16 + 4 = 36$.
• Answer: 36

3.4 Practice Set 3

1. A cube of side 7 has all faces painted. Find cubes with exactly 1 face painted. ($150$)
2. A cube of side 10 is cut. Find unpainted cubes. ($512$)
3. Number of 1-painted cubes is 96. Find $n$. ($6$)
4. A cube of side 8 is painted only on top and bottom. How many have 1 face painted? ($72$)

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4: Cuboids (Rectangular Boxes) {#4-cuboids}

4.1 Core Concepts

A cuboid is a 3D rectangular box with dimensions $m, n, p$. After painting and cutting, the unit cubes are distributed based on their position.

4.2 General Formulas ($m, n, p \ge 2$)

• Total small cubes: $m \times n \times p$
• 3 faces painted: 8 (Corner cubes)
• 2 faces painted: $4 \times [(m - 2) + (n - 2) + (p - 2)]$
• 1 face painted: $2 \times [(m-2)(n-2) + (n-2)(p-2) + (p-2)(m-2)]$
• 0 faces painted: $(m-2)(n-2)(p-2)$

4.3 Worked Examples

Example 1: Standard Cuboid ($5 \times 6 \times 7$)
Question: How many cubes have exactly 2 faces painted?

• $m=5, n=6, p=7$
• $4[(5-2) + (6-2) + (7-2)] = 4[3 + 4 + 5] = 48$.
• Answer: 48

Example 2: Inner Cubes
Question: How many cubes share no face with the exterior ($5 \times 6 \times 7$)?

• $(5-2)(6-2)(7-2) = 3 \times 4 \times 5 = 60$.
• Answer: 60

4.4 Practice Set 4

1. A cuboid $6 \times 8 \times 10$ is painted. Find 2-painted cubes. ($72$)
2. Find 1-painted cubes for $6 \times 8 \times 10$. ($208$)
3. Find total "at least one face painted" for $4 \times 5 \times 6$. ($96$)
4. Find integer pairs $(a, b)$ for cuboid $a \times a \times b$ if 2-painted count is 92. ($2a+b=29$)

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Summary 3.5

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5: Advanced Puzzles & Data Sufficiency {#5-advanced-puzzles}

5.1 Core Concepts

Advanced puzzles in geometry and logic move beyond simple formulas to require situational adjustments. The primary challenges include:

• Multi-View Dice: Using adjacency lists to find hidden faces in non-standard dice.
• Multi-Color Painting: Managing separate counts for different face colors (e.g., Red/Blue/Green).
• Partial Painting: Adjusting standard formula counts based on unpainted or removed faces.
• Data Sufficiency (DS): Deciding if given statements uniquely determine a geometric property.

5.2 Methodology: The Adjacency Table

For complex dice views, list all faces that appear together. Since an opposite face never shares an edge, any face appearing in the same view is adjacent.

• Step 1: Identify the "Common Face" across views.
• Step 2: List all faces adjacent to it.
• Step 3: The missing face from the set of six is the opposite.

5.3 Worked Examples: Multi-Color & Partial Logic

Example 1: Non-Standard Dice (Letters)
Question: Three views of a dice are: (A, B, C), (A, D, E), and (B, D, F). Which face is opposite A?

• From View 1: A is adjacent to {B, C}
• From View 2: A is adjacent to {D, E}
• Combining: A is adjacent to {B, C, D, E}.
• Answer: F is opposite A.

Example 2: Two-Color Cube Painting
Question: A cube ($n=4$) has Top/Bottom painted RED and sides painted BLUE. How many cubes have exactly 2 faces painted of the same color?

• Edges between Blue faces: 4 vertical edges. Each has $n-2 = 2$ cubes. $4 \times 2 = 8$.
• Edges between Red faces: None (Top/Bottom are opposite).
• Mixed Edges: Top/Bottom horizontal edges meet sides (Red-Blue edges). Not counted.
• Answer: 8

Example 3: Partial Painting Adjustment
Question: A cuboid ($5 \times 6 \times 7$) is painted on all sides except the front ($5 \times 7$). Find the count of 2-painted cubes.

• Base Count (Full Paint): $4[(5-2) + (6-2) + (7-2)] = 48$.
• Adjustment (Edge Loss): Horizontal edges (3 cubes each) and Vertical edges (5 cubes each) of the front face lose 1 layer. $48 - (2 \times 3 + 2 \times 5) = 48 - 16 = 32$.
• Adjustment (Corner Gain): The 4 corners that were 3-painted now become 2-painted. $32 + 4 = 36$.
• Answer: 36

5.4 Worked Examples: Data Sufficiency (DS)

Example 4: Cube Dimension Finding
Question: How many cubes in an $n \times n \times n$ cube have exactly two faces painted?

• I: The total number of small cubes is 125. ($n=5$, Sufficient)
• II: The corner cubes count is 8. (Always true for any $n$, Insufficient)
• Answer: Statement I alone is sufficient.

Example 5: Dice Logic
Question: What number is opposite 2 on the dice?

• I: The dice is standard. (Opposite is 5, Sufficient)
• II: Face 1 is adjacent to 3, 4, 5, 6. (Means 1 opposite 2, Sufficient)
• Answer: Either I or II alone is sufficient.

5.5 Practice Set 5

1. A dice has faces (A, B, C, D, E, F). View 1: (A, B, C), View 2: (A, D, E), View 3: (B, D, F). Find opposite of C. (D)
2. A cube ($n=6$) is cut into 216 units. Find 1-painted cubes. (96)
3. A cuboid ($4 \times 5 \times 6$) has faces of different colors. Find cubes with 2 different colors. (36)
4. DS: Find $n$ for a cube. I: Total surface area is 150. II: $n$ faces painted. (I alone)

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Summary 5.6