Aptitude Dice Cheatsheet

Cheatsheet Content

### Introduction to Dice Dice problems are common in aptitude tests, focusing on spatial reasoning, visualization, and basic arithmetic. A standard die is a cube with 6 faces, numbered 1 to 6. Opposite faces always sum to 7 (e.g., 1 opposite 6, 2 opposite 5, 3 opposite 4). #### Key Concepts - **Adjacent Faces:** Faces sharing an edge with a given face. - **Opposite Faces:** Two faces separated by three other faces when moving around the die. - **Rotation:** How a die behaves when rolled or rotated. ### Standard Dice Rules A standard die follows specific rules: 1. **Sum of Opposite Faces:** The sum of numbers on any two opposite faces is always 7. - 1 is opposite 6 - 2 is opposite 5 - 3 is opposite 4 2. **Adjacent Faces Never Opposite:** Any two adjacent faces can never be opposite to each other. 3. **Visible Faces:** In a single view, you can see at most 3 faces. These 3 faces are always adjacent to each other. ### Types of Dice Problems Aptitude questions involving dice generally fall into these categories: #### 1. Open Dice - A 3D die is unfolded into a 2D pattern. - You need to identify which faces are opposite or adjacent when folded. - **Rule:** In an open die, alternate faces are always opposite to each other. - Example: `A B C D E F` (linear arrangement), `A` is opposite `C`, `B` is opposite `D`. - Faces sharing only a corner are adjacent. - Faces sharing an edge are adjacent. #### 2. Closed Dice (Standard/Non-Standard) - Given one or more views of a closed die. - You need to find the number opposite a given face or predict a missing face. - **Standard Die:** Always follows the "sum of opposite faces is 7" rule. - **Non-Standard Die:** Does NOT follow the "sum of opposite faces is 7" rule. The problem will provide enough information to deduce face arrangements. #### 3. Rotated Dice - Given two or more views of the same die after it has been rotated. - You need to determine the orientation of faces. #### 4. Cube Cutting - A large cube is painted and then cut into smaller cubes. - Questions usually ask about the number of smaller cubes with 0, 1, 2, or 3 faces painted. ### Solving Open Dice Problems #### Principle of Alternating Faces - In any straight line (horizontal or vertical) of an open die, every alternate face is opposite to each other. #### Example Patterns Consider these common open dice configurations: 1. **Linear (4 faces in a row, 2 attached to sides):** ``` [1] [2][3][4][5] [6] ``` - `2` is opposite `4` - `3` is opposite `5` - `1` is opposite `6` (the remaining two) 2. **Cross Pattern:** ``` [A] [B][C][D] [E] [F] ``` - `B` is opposite `D` - `A` is opposite `E` - `C` is opposite `F` #### Steps to Solve 1. **Identify straight lines:** Look for rows or columns of three or more faces. 2. **Apply alternating rule:** Faces that are one position apart in a straight line are opposite. 3. **Identify remaining faces:** The last two unmatched faces will be opposite each other. 4. **Visualize folding:** Mentally fold the die to confirm adjacencies. Remember, adjacent faces in the open diagram will remain adjacent when folded (unless they become opposite). ### Solving Closed Dice (Single Die) #### 1. Two Positions (One Common Face) - **Rule:** If two positions of the same die show one common face, and the positions of the common face are different, then the faces adjacent to the common face are clockwise/anticlockwise in the same order. - **Method:** 1. Identify the common face. 2. From the common face, move clockwise (or anti-clockwise) in both views. 3. The faces at corresponding positions in the clockwise/anti-clockwise sequence will be opposite each other. 4. The remaining face (not seen) will be opposite the common face. **Example:** View 1: Top=3, Front=1, Right=5 View 2: Top=3, Front=2, Right=4 - Common face: `3` - Clockwise from `3` in View 1: `3 -> 1 -> 5` - Clockwise from `3` in View 2: `3 -> 2 -> 4` - Therefore: `1` is opposite `2`, `5` is opposite `4`. - Remaining face (`6`) is opposite the common face (`3`). #### 2. Two Positions (Two Common Faces) - **Rule:** If two positions of the same die show two common faces, then the remaining non-common faces are opposite to each other. **Example:** View 1: Top=1, Front=2, Right=3 View 2: Top=1, Front=5, Right=2 - Common faces: `1`, `2` - Remaining non-common faces: `3` (from View 1), `5` (from View 2) - Therefore: `3` is opposite `5`. - To find opposites of `1` and `2`, you might need another view or assume it's a standard die if no other info. If it's a standard die, `1` opposite `6`, `2` opposite `5`. But here `2` is adjacent to `5`, so it's a non-standard die. #### 3. Two Positions (No Common Face) - This scenario usually implies a non-standard die, or the problem is designed to be solved by elimination/standard die assumption. - If you assume it's a standard die, then `1` is opposite `6`, `2` opposite `5`, `3` opposite `4`. - If it's a non-standard die, you cannot determine opposites without more information (e.g., a third view or explicit rules). #### 4. Three or More Positions - Use the "Two Positions (One Common Face)" method by comparing any two views at a time. - Try to find views with one common face to easily identify opposite pairs. ### Cube Cutting Formulas Consider a large cube painted on all its faces, then cut into `n x n x n` smaller identical cubes. Here, `n = (Side of large cube) / (Side of small cube)`. 1. **Total number of smaller cubes:** $N = n^3$ 2. **Cubes with 3 faces painted:** - These are the corner cubes. - There are always **8** such cubes (unless `n = 2`. - If `n = 1`, there is only 1 cube, and all 6 faces are painted. - If `n = 0`, it means no cube is cut. ### Advanced Dice Concepts #### Non-Standard Dice - These dice do not follow the rule that opposite faces sum to 7. - Problems will typically provide multiple views from which you can deduce the arrangement. - The "one common face" and "two common faces" rules still apply for determining relative positions. #### Dice from Net (Folding) - Similar to open dice, but sometimes the net might be irregular. - Always identify pairs of opposite faces first using the alternating rule. - Then, choose an arbitrary face and mentally fold the net around it to see adjacencies. #### Missing Face Problems - Given a view of a die with one face missing (e.g., a blank face). - You need to determine what number should be on the blank face, often based on other views or standard die rules. #### Combined Problems - Sometimes questions combine elements, e.g., an open die needs to be folded, then rotated, and then you find the opposite face. - Break down complex problems into simpler steps.