Dice - Tricks & Shortcuts for 2026 - 2027 Placement tests, Job Interviews & Exams

This chapter on "DAS" (dice) is a crucial part of nonverbal reasoning, frequently appearing in placement tests and entrance exams. While it may seem daunting, it's one of the easiest and highest-scoring chapters once the concepts are understood. A standard die has six faces. For any given face, there is one opposite face and four adjacent faces. Adjacent faces can never be opposite faces. This fundamental rule is key to solving all problems. There are two main types of dice: 1. **Standard Dice:** These are like the dice used in games such as Ludo. A key characteristic of a standard die is that the sum of the digits on any two opposite faces always equals seven. 2. **Regular (or Random) Dice:** These are the dice typically featured in exams. Unlike standard dice, there is no fixed pattern for the numbers on their faces; any number can appear anywhere. The purpose of these questions is to test a student's understanding of the underlying principles of dice. Questions about dice generally fall into a few categories: * **Single Die:** Questions based on a single die. * **Two or More Dice:** Questions involving two or three dice, which can be further broken down into scenarios: * **One Face Common:** Two dice are shown, and one face is common between them. * **Two Faces Common:** Two dice are shown, and two faces are common between them. * **No Face Common:** Two dice are shown, and no faces are common between them. * **Open Dice:** A flat, unfolded net of a die is presented, and the task is to determine which folded cube can be formed or which faces are opposite. Let's delve into the methods for solving each type of question. **1. When One Face is Common:** Consider two dice where one face is common. To find the opposite faces, identify the common face. From this common face, rotate clockwise for both dice, listing the numbers encountered. * **Example:** Die 1: Common face (2), then 6, then 4. Die 2: Common face (2), then 1, then 3. Aligning these sequences: 2 - 6 - 4 2 - 1 - 3 From this, we deduce that 6 is opposite to 1, and 4 is opposite to 3. To find the face opposite the common face (2), we look for the number from 1 to 6 that is not visible in either arrangement. In this case, if 1, 2, 3, 4, 6 are visible, then 5 must be the missing number. Therefore, 2 is opposite to 5. **Key takeaway:** The face you don't see is the opposite face of the common face. **2. When Two Faces are Common:** If two faces are common between two dice, the remaining two faces that are not common will automatically be opposite to each other. * **Example:** Die 1: 1, 2, 3 Die 2: 1, 5, 3 Here, 1 and 3 are common faces. The remaining faces are 2 and 5. Thus, 2 is opposite to 5. For the common faces (1 and 3), their opposites are the remaining unseen numbers (4 and 6). We cannot definitively say whether 1 is opposite 4 or 6, or whether 3 is opposite 4 or 6, without additional information. However, we know that {1,3} will be opposite to {4,6}. **Key takeaway:** When two faces are common, the third (uncommon) faces are opposite each other. **3. When No Face is Common:** When no faces are common between two dice, there's a simple pattern: * The top face of the first die is opposite the top face of the second die (Top-Top). * The near face of the first die is opposite the near face of the second die (Near-Near). * The far face of the first die is opposite the far face of the second die (Far-Far). * **Example:** Die 1: Top (1), Near (5), Far (3) Die 2: Top (2), Near (4), Far (6) Therefore: 1 is opposite 2, 5 is opposite 4, and 3 is opposite 6. **Key takeaway:** Remember TT NN FF (Top-Top, Near-Near, Far-Far). **Practice Questions and Applications:** * **Question 1 (Two Common Faces):** Given two positions of a cube marked 1-6. Faces 2 and 3 are common. The third faces are 1 and 6. So, 1 is opposite 6. We need to find the face opposite 3. The remaining numbers are 2, 3, 4, 5. So, 2 and 3 are common faces, and their opposites must be 4 and 5. Without more information, we cannot determine if 3 is opposite 4 or 5. Thus, the answer is "cannot be determined." * **Question 2 (Standard Dice):** Given a standard die, what is opposite 4? Since it's a standard die, the sum of opposite faces is 7. So, 4 + X = 7, which means X = 3. * **Question 3 (Two Common Faces with Colors):** Two positions of a die marked with colors (Red, Blue, Green, Yellow, White, Black). "Yellow" and "Green" are common. The remaining faces are "White" and "Black". Therefore, "White" is opposite "Black". * **Question 4 (Tricky Sum with Two Common Faces):** Six faces marked 1-6. Two positions show 2 and 3 as common faces. This means 1 and 6 are opposite. The question asks for the sum of *possible* numbers opposite 3. Since 2 and 3 are common, their opposites must be the remaining numbers (4 and 5). So, 3 could be opposite 4 or 5. The sum of these possibilities is 4 + 5 = 9. * **Question 5 (Three Positions of a Die):** Find the number opposite 3 from three given positions. * Comparing position 1 and 2: Faces 2 and 3 are common. Thus, 1 is opposite 6. * Comparing position 2 and 3: Faces 2 and 6 are common. Thus, 3 is opposite 4. The answer is 4. This highlights that sometimes not all given information is needed; strategic comparison is key. * **Question 6 (Three Positions with Letters):** Faces marked D, E, F, G, H, I. Find the letter opposite E. * Comparing position 1 and 2: E and F are common. Thus, D is opposite I. * Comparing position 2 and 3: E and I are common. Thus, F is opposite G. The faces are D, E, F, G, H, I. We know D-I and F-G. The only remaining pair is E-H. So, E is opposite H. * **Question 7 (Three Standard Dice - Sum of Opposites):** Given top faces of three standard dice (2, 5, 3). Find the sum of numbers on their opposite faces. * Die 1 (top 2): Opposite is 5 (2+5=7). * Die 2 (top 5): Opposite is 2 (5+2=7). * Die 3 (top 3): Opposite is 4 (3+4=7). The sum of opposite faces is 5 + 2 + 4 = 11. * **Question 8 (Adjacent Faces - No Diagram):** Six faces 1-6. One is adjacent to 2, 3, 5. Which of the following is true? (e.g., 2 is adjacent to 5, 1 is adjacent to 4, 1 is adjacent to 6, 4 is adjacent to 6). If 1 is adjacent to 2, 3, 5, it means 1's opposite face is *not* 2, 3, or 5. The remaining numbers are 4 and 6. So, 1's opposite face is either 4 or 6. If 1 is on the front, 2 on left, 3 on top, 5 on right. Then the bottom face is either 4 or 6, and the back face is the other of 4 or 6. The faces 4 and 6 will always be adjacent to each other because they occupy the remaining two faces that are not opposite 1. For example, if 1 is front, 4 is back, then 6 must be bottom (or vice versa). Back and bottom faces are adjacent. This means "4 is adjacent to 6" is always true. * **Question 9 (One Common Face - Adjacent to One):** Two positions of a die. One common face (2). * Die 1: 2 (common), 1 (clockwise), 3 (clockwise) * Die 2: 2 (common), 6 (clockwise), 4 (clockwise) Opposite pairs: 1-6, 3-4. The remaining number is 5, so 2-5. The question asks what can come on the face adjacent to 1. Since 6 is opposite 1, 6 cannot be adjacent to 1. The options that include 6 are incorrect. The numbers adjacent to 1 are 2, 3, 4, 5. So, 5 and 3 can be adjacent to 1. **Open Dice:** When an open net of a die is given, the trick to finding opposite faces is to "jump one place" along a straight line. * **Arrangement 1 (Linear):** A - B - C - D - E - F (example faces) * Pick A, jump one (skip B), so A is opposite C. * Pick B, jump one (skip C), so B is opposite D. * The remaining faces are E and F, so E is opposite F. **Key takeaway:** Alternate faces in a straight line are opposite. The remaining two faces form the last opposite pair. * **Example (Open Dice):** D-E-F G H I * From D, jump one to F. So D is opposite F. * From E, jump one to G. So E is opposite G. * The remaining faces are H and I. So H is opposite I. When checking which folded cube can be formed, remember that opposite faces can never be adjacent