Mathematics: Class 7 (Part 1) — Chapter 6: Number Play

Children stand in a line and call out a number based on their relative heights:

• Rule: Each child calls out the number of children standing in front of them who are taller than they are.
• The first person in line always says 0 because there is no one in front of them.
• If a person is the tallest in the group, their number is always 0.
• The largest possible number in a group of $n$ people is $n - 1$.

1. Height Sequences: Arrange stick figures to match given sequences:

• Sequence (a): $0, 1, 1, 2, 4, 1, 5$ corresponds to the height order (from tallest to shortest represented by letters) $\rightarrow F, C, B, G, A, D, E$.

2. Truth Value of Statements:

• "If a person says '0', then they are the tallest in the group." — Only Sometimes True (They could just be the first person in line, or taller than everyone in front of them but not everyone behind them).
• "If a person is the tallest, then their number is '0'." — Always True.
• "The first person's number is '0'." — Always True.
• "If a person is not first or last in line, then they cannot say '0'." — Only Sometimes True.
• "The person who calls out the largest number is the shortest." — Only Sometimes True.
• "What is the largest number possible in a group of 8 people?" — 7.

• Even Numbers: Can be arranged in pairs without any leftovers.
• Odd Numbers: Cannot be arranged in pairs; they are always one more (or one less) than a collection of pairs.

The sum or difference of numbers follows strict parity rules:

• $\text{Even} \pm \text{Even} = \text{Even}$
• $\text{Odd} \pm \text{Odd} = \text{Even}$
• $\text{Even} \pm \text{Odd} = \text{Odd}$
• $\text{Odd} \pm \text{Even} = \text{Odd}$

Generalizations:

• The sum of any number of even numbers is always Even.
• The sum of an even number of odd numbers is Even.
• The sum of an odd number of odd numbers is Odd.

For an $m \times n$ grid, the parity of the total number of small squares is:

• Even if at least one of the dimensions ($m$ or $n$) is even.
• Odd if both dimensions ($m$ and $n$) are odd.

• Always Even Parity: Expressions like $2n$, $100p$, and $48w - 2$.
• Always Odd Parity: Expressions like $2n - 1$, $2m + 1$, and $8a + 3$.
• Variable Parity: Expressions like $3n + 4$, which can be even or odd depending on the value of $n$.
• The $n$-th even number is given by the formula: $\mathbf{2n}$
• The $n$-th odd number is given by the formula: $\mathbf{2n - 1}$

• In a $3 \times 3$ grid filled with numbers $1$ to $9$ without repetition, the sum of all numbers in the grid is: $$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$$
• The sum of all row sums (or all column sums) must always equal $45$.

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

Properties of a $3 \times 3$ Magic Square (using numbers 1–9):

• The magic sum must be $\mathbf{15}$ (since $45 \div 3 = 15$).
• The central number must be $\mathbf{5}$.
• The numbers $1$ and $9$ cannot occur in any corner; they must occupy middle boundary positions.
• There is exactly one unique magic square with numbers $1$ to $9$ (ignoring rotations and reflections):

If $m$ is the number in the center of a $3 \times 3$ magic square of consecutive numbers, the grid can be algebraically represented as:

• The magic sum is always $\mathbf{3m}$.

• Chautīsā Yantra: A $4 \times 4$ magic square found in a 10th-century inscription at the Pārśhvanath Jain temple in Khajuraho, India. Its magic sum is $\mathbf{34}$.
• Lo Shu Square: The oldest recorded magic square (over 2000 years old from ancient China), depicted on the back of a mythical turtle.
• Navagraha Yantra & Kubera Yantra: Traditional Indian grids associated with planetary and cultural significance.

• The sequence was first discovered by Sanskrit and Prakrit linguists (such as Virahāṅka, c. 700 CE) while studying poetic meters.
• Syllables in Indian poetry are classified as short (1 beat) or long (2 beats).
• The number of distinct rhythmic patterns of $n$ beats is given by the $(n)$-th term of the Virahāṅka sequence.
• The rule of formation is that each term is the sum of the two preceding terms: $$F_n = F_{n-1} + F_{n-2}$$

The parities of the terms in the Virahāṅka sequence follow a repeating three-step cycle:

Odd, Even, Odd, Odd, Even, Odd, Odd, Even, ...

• Each letter stands for a unique digit from $0$ to $9$.
• The leading digit of a multi-digit number cannot be $0$.

Solve the cryptarithm:

  U T
+ T A
-----
T A T
    

Solution: $U = 9$, $T = 1$, and $A = 0$.