Reasoning: How to Solve Cube Cutting and Painting - Number of Small Cubes with 0123 Faces Painted

Introduction "Mastering Cube Cutting and Painting can fetch you 20-30 marks in competitive exams, making it a must-know topic to crack the exam quickly and confidently."

WHAT YOU NEED TO KNOW FIRST To solve Cube Cutting and Painting questions, you need to have a basic understanding of:

1. Direction Chart: A chart that shows the 6 faces of a cube and their corresponding directions (up, down, left, right, front, back).
2. BODMAS: A rule to follow when solving mathematical expressions, which stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction.
3. Sitting Arrangement Conventions: A set of rules that govern how people are arranged in a specific order, such as clockwise or anti-clockwise.

CRYSTAL‑CLEAR METHOD (Step‑by‑Step) To solve Cube Cutting and Painting questions, follow these steps:

1. Identify the type of question: Determine if it's a cutting or painting question.
2. Draw a diagram: Draw a diagram of the cube and mark the faces that are cut or painted.
3. Count the number of small cubes: Count the number of small cubes that are formed after cutting the cube.
4. Count the number of faces painted: Count the number of faces that are painted on each small cube.
5. Apply the formula: Apply the formula to calculate the total number of small cubes with 0, 1, 2, or 3 faces painted.

Quick Demo using a simple example Suppose we have a cube that is cut into 8 small cubes, and 3 faces are painted on each small cube. We can draw a diagram and count the number of small cubes and faces painted.

Total number of small cubes = 8 Total number of faces painted = 3 x 8 = 24

WORKED EXAMPLES

Example 1 – Easy A cube is cut into 27 small cubes, and 2 faces are painted on each small cube. How many small cubes have 0, 1, 2, or 3 faces painted?

Total number of small cubes = 27 Total number of faces painted = 2 x 27 = 54

What we learned: To solve this question, we need to count the number of small cubes and faces painted, and then apply the formula to calculate the total number of small cubes with 0, 1, 2, or 3 faces painted.

Example 2 – Medium A cube is cut into 64 small cubes, and 3 faces are painted on each small cube. However, 16 small cubes have 2 faces painted, and 12 small cubes have 3 faces painted. How many small cubes have 0, 1, 2, or 3 faces painted?

Total number of small cubes = 64 Total number of faces painted = 3 x 64 = 192

What we learned: To solve this question, we need to count the number of small cubes and faces painted, and then apply the formula to calculate the total number of small cubes with 0, 1, 2, or 3 faces painted.

Example 3 – Exam‑Style A cube is cut into 125 small cubes, and 2 faces are painted on each small cube. However, 25 small cubes have 1 face painted, and 15 small cubes have 2 faces painted. How many small cubes have 0, 1, 2, or 3 faces painted?

Total number of small cubes = 125 Total number of faces painted = 2 x 125 = 250

What we learned: To solve this question, we need to count the number of small cubes and faces painted, and then apply the formula to calculate the total number of small cubes with 0, 1, 2, or 3 faces painted.

Common Mistakes

MISTAKE → WHY IT HAPPENS → CORRECT APPROACH
1. Not counting the number of small cubes: Why it happens: Students often forget to count the number of small cubes. Correct approach: Always count the number of small cubes before applying the formula.
2. Not counting the number of faces painted: Why it happens: Students often forget to count the number of faces painted. Correct approach: Always count the number of faces painted before applying the formula.
3. Not applying the formula: Why it happens: Students often forget to apply the formula. Correct approach: Always apply the formula to calculate the total number of small cubes with 0, 1, 2, or 3 faces painted.
4. Not considering the type of question: Why it happens: Students often forget to consider the type of question. Correct approach: Always identify the type of question before solving it.
5. Not drawing a diagram: Why it happens: Students often forget to draw a diagram. Correct approach: Always draw a diagram to visualize the problem.

EXAM TRAPS

Trap → How to Spot it → How to Avoid it
1. Trick question: How to spot it: The question seems too easy or too hard. How to avoid it: Read the question carefully and identify the type of question.
2. Misleading information: How to spot it: The information provided seems incorrect or incomplete. How to avoid it: Verify the information provided and use your knowledge to fill in the gaps.
3. Complex calculations: How to spot it: The calculations seem too complex or time-consuming. How to avoid it: Break down the problem into smaller parts and use shortcuts to simplify the calculations.

TIME‑SAVING SHORTCUTS

1. Elimination trick: If the question asks for the number of small cubes with 0, 1, 2, or 3 faces painted, eliminate the options that are not possible based on the information provided.
2. Diagram hack: Draw a diagram to visualize the problem and identify the number of small cubes and faces painted.
3. Formula shortcut: Use the formula to calculate the total number of small cubes with 0, 1, 2, or 3 faces painted.

1‑MINUTE RECAP "Alright, let's recap the strategy for solving Cube Cutting and Painting questions. First, identify the type of question and draw a diagram to visualize the problem. Then, count the number of small cubes and faces painted, and apply the formula to calculate the total number of small cubes with 0, 1, 2, or 3 faces painted. Don't forget to eliminate impossible options and use shortcuts to simplify the calculations. Remember, practice makes perfect, so make sure to practice these questions to master the topic. Good luck on your exam!