Practice Questions

Calculate the parity (even or odd) of the result of the following sum: The sum of 13 distinct odd numbers and 20 distinct even numbers.

Describe the rule used by the children in the textbook to call out numbers based on their height arrangement. Explain what the number '0' signifies according to this rule.

A student claims that to find the parity of a product of integers, you only need to know if at least one of the integers is even. Critique this claim and justify your reasoning.

Define an even number.

What is the rule to find the next number in the Virahāṅka-Fibonacci sequence?

Solve the cryptarithm where each letter represents a unique digit. Find the values of A and B.

Define a magic square.

Without performing the multiplication, analyze and determine the parity of the number of small squares in a grid with dimensions $203 \times 457$.

In the Virahāṅka-Fibonacci sequence, the 14th term is 377 and the 15th term is 610. Calculate the 16th term in the sequence.

Identify the parity (even or odd) of the sum of two odd numbers.

Propose one algebraic expression that always generates an even number and another that always generates an odd number, for any positive integer $k$. Demonstrate that each expression works for $k=4$ and $k=7$.

Analyze the expression $8m + 3$. Determine if its value is always even, always odd, or can be either, for any whole number $m$.

Critique the claim: "The difference between any two prime numbers is always even." Propose a corrected version of the claim if necessary.

A child is climbing a staircase with 6 steps. At each move, the child can choose to climb either 1 step or 2 steps. Calculate the total number of different ways the child can reach the top of the 6-step staircase.

Design a height arrangement for 6 children where the sequence of numbers they call out is $0, 1, 0, 2, 1, 0$. The rule is: "each child calls out the number of children in front of them who are taller than them." Represent the children's heights with relative numbers 1 (shortest) to 6 (tallest).

Evaluate the following scenario: A piggy bank contains an odd number of ₹1 coins, an odd number of ₹5 coins, and an odd number of ₹10 coins. Is it possible for the total value of the money to be exactly ₹200? Justify your answer using parity.

What is the general formula used to represent any even number?

Explain why the sum of an odd number of odd numbers is always odd.

List three key observations made about a $3 \times 3$ magic square that uses the numbers 1 to 9.

The first few numbers in the Virahāṅka-Fibonacci sequence are $1, 2, 3, 5, 8, \ldots$. List the next four numbers in the sequence and describe the pattern of their parities (odd/even).

Explain why the expression $2n-1$ always represents an odd number for any positive integer $n$.

Explain the systematic process discussed in the text to determine the magic sum of a $3 \times 3$ magic square using numbers from 1 to 9. Also, explain why the sum of all row sums is 45.

Four children are standing in a line from front to back. Their heights are 155 cm, 140 cm, 160 cm, and 150 cm. Apply the rule 'each child says the number of children in front of them who are taller'. What number does the fourth child in the line say?

A shopkeeper has an even number of 500-gram weights and an odd number of 200-gram weights. Analyze if the total weight can be exactly 3000 grams. Justify your answer using parity.

Analyze the parity of the terms in the Virahāṅka-Fibonacci sequence ($1, 2, 3, 5, 8, \dots$). Describe the repeating pattern of parities and use it to determine the parity of the 30th term in the sequence.

Create a $3 \times 3$ magic square using the nine consecutive numbers from 3 to 11. First, calculate the magic sum, then construct the grid.

The pages of a book are numbered from 1 to 250. Analyze if the sum of all these page numbers is an even number or an odd number. Justify your conclusion using parity rules without calculating the actual sum.

A standard $3 \times 3$ magic square is created using the numbers 1 to 9. A new grid is then formed by applying two sequential operations to each number in the square: first, multiply every number by 3, and then add 5 to each result. Analyze if this new grid is also a magic square. If it is, calculate its new magic sum.

Critique the following statement: "In the 'taller people in front' game with 10 people, if a person says '9', they must be the shortest person in the entire group." Justify your evaluation.

Summarize the rules for the parity of the result for the following operations. Provide one example for each rule. (a) even + odd (b) odd + odd (c) even $\times$ odd (d) odd $\times$ odd (e) even $\times$ even

Evaluate whether a $3 \times 3$ magic square can be constructed using nine distinct positive even numbers. Justify your reasoning without attempting to construct one.

Create a $3 \times 3$ magic square using the nine consecutive odd numbers from 3 to 19. Justify your method for determining the magic sum, and then construct the square.

A 'Tribonacci' sequence is formed where each term is the sum of the three preceding terms. Formulate the first 8 terms of this sequence if it starts with $1, 2, 3$. Then, evaluate and describe the parity pattern (odd/even) you observe in the first 8 terms.

Solve the cryptarithm SO + SO = TOO. Each letter represents a unique digit from 0-9. Justify each step of your logical deduction.

Kishor's puzzle involves placing 5 number cards, all with odd numbers, into 5 boxes so they sum to 30. Explain, using the concept of parity, why this puzzle is impossible to solve.

A proposed method for creating new magic squares is to apply a mathematical operation to every cell of a known magic square. Evaluate this method by performing two separate transformations on a standard 1-9 magic square (the Lo Shu square is provided below).

a) Multiply every number by 3, then subtract 2. b) Square every number.

For each case, justify whether the resulting grid is a new magic square. If it is, formulate a rule for the new magic sum based on the original magic sum ($M_{old} = 15$).

Lo Shu Square: $\begin{vmatrix} 2 & 7 & 6 \ 9 & 5 & 1 \ 4 & 3 & 8 \end{vmatrix}$

Describe how to determine the parity of the total number of small squares in a grid with dimensions $m \times n$ without calculating the product $m \times n$.

Draw a possible arrangement of 5 stick figures of distinct heights that results in the sequence $0, 1, 2, 0, 1$. The sequence is based on the rule where each person calls out the number of people standing in front of them who are taller.

A circular path has 10 lampposts, all initially OFF. A person starts at lamppost 1 and walks around the circle, flipping the switch of every lamppost they pass (1, 2, ..., 10, 1, 2, ...). They make a total of 33 switch flips. Evaluate which lampposts will be ON at the end and justify your reasoning.

The Virahāṅka-Fibonacci sequence is $1, 2, 3, 5, 8, ...$. Justify whether the 30th term in this sequence will be odd or even, without calculating the term.

Solve the cryptarithm A B \times C = D E B, where each letter represents a different digit from 0-9. The letters A and D cannot be 0. Find one possible set of values for A, B, C, D, and E.

Kishor's puzzle (5 odd number cards summing to 30) was impossible. Propose a single, simple change to one of the puzzle's conditions (the number of boxes, the type of cards, or the target sum) that makes it solvable. Provide one example solution for your modified puzzle.

Formulate a new, original rule for the "Numbers Tell Us Things" game. Then, create an arrangement of 6 children (with unique heights 1 to 6) and determine the sequence of numbers they would call out based on your new rule. Justify that your generated sequence is correct for the arrangement.

Create your own valid addition cryptarithm puzzle where two 2-digit numbers sum to a 3-digit number. It must use at least four different letters and have a unique solution. Provide the puzzle, its unique solution, and a brief justification for why the solution is unique.

Summarize the historical discovery of the Virahāṅka-Fibonacci numbers. Name at least two scholars mentioned in the text who studied these numbers and describe the context (poetry) in which they were discovered.