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Chapter Analysis
Intermediate18 pages • EnglishQuick Summary
The chapter 'Number Play' explores mathematical ideas about even and odd numbers, their properties, and the concept of parity. It delves into patterns and exercises with grids and magic squares, illustrating how sums and arrangements demonstrate number parity concepts. The chapter also introduces students to the Virahāṅka–Fibonacci sequence and its applications, enhancing their understanding of numbers within cultural and historical contexts.
Key Topics
- •Parity and Even Numbers
- •Odd Numbers and Their Properties
- •Magic Squares
- •Virahāṅka–Fibonacci Sequence
- •Cryptarithms and Number Puzzles
- •Grid Patterns in Arithmetic
- •Mathematical Sequences
Learning Objectives
- ✓Understand the concepts of even and odd numbers and their sums.
- ✓Learn how to create and solve magic squares.
- ✓Explore the historical and mathematical significance of the Virahāṅka–Fibonacci sequence.
- ✓Develop problem-solving skills through cryptarithms.
- ✓Investigate number properties through arithmetic expressions.
Questions in Chapter
Using your understanding of the pictorial representation of odd and even numbers, find out the parity of the following sums: (a) Sum of 2 even numbers and 2 odd numbers (e.g., even + even + odd + odd) (b) Sum of 2 odd numbers and 3 even numbers (c) Sum of 5 even numbers (d) Sum of 8 odd numbers.
Page 130
For each of the statements given below, think and identify if it is Always True, Only Sometimes True, or Never True. Share your reasoning.
Page 128
What is the next number in the Virahāṅka–Fibonacci sequence after 55?
Page 141
Solve this cryptarithm: UT + TA = TAT
Page 144
Angaan wants to climb an 8-step staircase. His playful rule is that he can take either 1 step or 2 steps at a time. In how many different ways can he reach the top?
Page 144
Additional Practice Questions
Identify the pattern in the Virahāṅka–Fibonacci sequence and predict the next three terms after 89.
mediumAnswer: The next terms in the sequence are obtained by summing the last two numbers. After 89, the next terms are 144, 233, and 377.
Create a 3x3 magic square using numbers 2-10.
hardAnswer: A magic square consists of numbers arranged such that each row, column, and diagonal sums to the same number. Using numbers 2-10 pragmatically, for this range: 2 7 6, 9 5 1, 4 3 8, each line sums to 15.
Write a formula to find the nth odd number and explain your reasoning.
easyAnswer: The nth odd number can be found using the formula 2n - 1. This is derived from observing that odd numbers follow the pattern: 1, 3, 5, ..., which can be represented by the sequence 2n - 1 where n starts from 1.
Explain why the sequence 1, 3, 5, ..., 2n - 1 represents all odd numbers.
mediumAnswer: An odd number is one more than an even number, and can be expressed as 2n - 1. This sequence increments by 2, which is consistent with the pattern of odd numbers.
Discuss how magic squares demonstrate the idea of parity.
mediumAnswer: In a magic square, the sum of each row, column, and diagonal is uniform, showing that by arranging numbers strategically, we can observe consistent parity properties, such as sums remaining odd or even.
Prove that the sum of any two consecutive numbers is always odd.
easyAnswer: Consecutive numbers are pairs differing by one, comprising one even and one odd number. Since even + odd = odd, their sum is always odd.
Determine if it is possible for two magic squares using different consecutive numbers to have the same magic sum.
hardAnswer: Yes, depending on the configuration of the numbers, different sets of consecutive numbers can achieve the same magic sum due to the uniform distribution and balancing of numbers in the grid.