Key Points

Parity is the property of an integer being either even or odd. Even numbers are divisible by 2 (e.g., 2, 4, 6), while odd numbers are not (e.g., 1, 3, 5).

Any even number can be written in the form $2n$, and any odd number can be written as $2n-1$ or $2n+1$, where $n$ is an integer.

The rules for adding numbers based on their parity are: Even + Even = Even; Odd + Odd = Even; Even + Odd = Odd.

The rules for subtracting numbers based on their parity are: Even - Even = Even; Odd - Odd = Even; Even - Odd = Odd; Odd - Even = Odd.

The rules for multiplying numbers are: Even $\times$ Any Integer = Even; Odd $\times$ Odd = Odd. A product is odd only if all its factors are odd.

A magic square is a square grid of numbers where the sum of the numbers in each row, each column, and both main diagonals is the same. This constant sum is called the magic sum.

For a $3 \times 3$ magic square using the numbers 1 through 9, the magic sum must be 15. This is calculated by dividing the total sum of the numbers (45) by 3.

In a $3 \times 3$ magic square using numbers 1 to 9: the number 5 must be in the center, even numbers (2, 4, 6, 8) must occupy the corners, and the remaining odd numbers (1, 3, 7, 9) occupy the middle-edge positions.

The Virahāṅka-Fibonacci sequence is a series of numbers where each term is the sum of the two preceding terms. The rule is $F_n = F_{n-1} + F_{n-2}$.

Starting with 1 and 2, the sequence is $1, 2, 3, 5, 8, 13, 21, 34, ...$ where $3 = 1+2$, $5 = 2+3$, and so on.

The parity of the terms in the Virahāṅka-Fibonacci sequence follows a repeating 3-term pattern: Odd, Even, Odd. For example, in $1, 2, 3, 5, 8, 13, ...$ the parities are O, E, O, O, E, O, ...

Cryptarithms are mathematical puzzles where letters represent distinct digits from 0 to 9. The objective is to decipher the value of each letter using logic and arithmetic rules.