Cube and Cuboid Problem Solving

Jul 19, 2024

Lecture Notes on Cube and Cuboid in Reasoning

Introduction

  • Topic: Cube and Cuboid in Reasoning
  • 15th Chapter of the course on reasoning practice by Piyush.
  • Previous chapter: Dice. Current focus: Cutting of cubes.
  • Context: Problems related to smaller cubes derived from a bigger cube.

Key Concepts

Types of Smaller Cubes After Cutting a Big Cube

  1. Corner Cubes:
    • 3 surfaces painted
    • Total: 8 cubes
  2. Middle Cubes:
    • 2 surfaces painted
    • Formula: 12 * (n - 2)
  3. Central Cubes:
    • 1 surface painted
    • Formula: 6 * (n - 2)^2
  4. Colorless Cubes (No surface painted):
    • Formula: (n - 2)^3
  5. Total Number of Smaller Cubes:
    • Formula: n^3
    • n is the ratio of bigger cube length to smaller cube length.

Important Formulas & Concepts

  • n: Total smaller cubes from a cube of side length n is n^3.
  • Corner Cubes: 8, each with 3 painted surfaces.
  • Middle Cubes: 12 * (n - 2)
  • Central Cubes: 6 * (n - 2)^2
  • Colorless Cubes: (n - 2)^3
  • Example Calculation for n = 4:
    • Colorless cubes: (4 - 2)^3 = 2^3 = 8
    • Corner Cubes: 8
    • Middle Cubes: 12 * (4 - 2) = 24
    • Central Cubes: 6 * (4 - 2)^2 = 24

Practical Examples and Problem Solving

  • Example Problem 1: Computing colorless cubes when n = 4 and 64 smaller cubes are formed.
    • Solution: (4 - 2)^3 = 8
  • Example Problem 2: Compute smaller cubes with given color conditions.

Quick Rules & Techniques

  • When asked for at least one surface painted:
    • Formula: Total cubes - Colorless cubes
    • Example: 27 - 1 = 26 for n = 3
  • Combination Questions: Analyze and apply specific combinations of surface colorings as per constraints provided.

Problem Types

  1. Basic Color Calculations: Finding cubes with specific surface colorings.
  2. Combination Problems: Multiple surface color constraints (e.g., one face green, one face black).

Summary

  • Revise the primary formulas and concepts regularly.
  • Practice solving problems using both the direct formulas and the basic method where needed.
  • Understand the problem context to apply the right counting and combination techniques.

Next Steps

  • Further practice problems related to the cube and cuboid topic.
  • Upcoming focus on practical applications and advanced scenarios.

Motivation

  • Consistent practice and understanding basic concepts thoroughly is key.
  • Use motivation and personal goals to drive studying efforts.

Next Class: Continue with cubes & cuboids - advanced problems Time: 5:00 PM