Mathematics: Class 4 — Chapter 10: Elephants, Tigers, and Leopards

• Target Number: The goal is to reach exactly $10$.
• Allowed Moves: Each player can choose to add either $1$ or $2$ to the running total.
• Starting: Player 1 starts by choosing either $1$ or $2$.
• Turns: Player 2 adds $1$ or $2$ to the previous total. Play continues alternately.
• Winning Condition: The player who reaches exactly $10$ first wins.

Analyze the game backwards to find winning positions:

• If the other player has reached $8$, you can choose $2$ to reach $10$ and win.
• If the other player has reached $7$, you can choose $1$ (reaches $8$) or $2$ (reaches $9$). In either case, the opponent can win on their next turn.
• If the other player has reached $6$, you can choose $1$ to reach $7$, forcing them into a losing position.

• Diagonals: Identical numbers appear along the diagonals running from bottom-left to top-right.
• Symmetry: The table is symmetric across the main diagonal (top-left to bottom-right).
• Even and Odd Rows: Rows starting with an even number alternate even and odd numbers, while rows starting with an odd number alternate odd and even numbers. No row or column contains only even or only odd numbers.

• Red Window Frame ($2 \times 2$ grid):
  • Sum of numbers in each row: The sum of the second row is $2$ more than the sum of the first row.
  • Sum of numbers in each column: The sum of the second column is $2$ more than the sum of the first column.
  • Sum of diagonals: The sum of both diagonals in any $2 \times 2$ window is equal (e.g., $10 + 12 = 11 + 11 = 22$).
• Blue Window Frame ($3 \times 3$ grid): Similar constant difference patterns exist for rows, columns, and diagonals.

Take any $2$-digit number, reverse its digits, and find their sum:

• Example: $27 \rightarrow \text{Reverse is } 72 \rightarrow 27 + 72 = 99$
• Example: $48 \rightarrow \text{Reverse is } 84 \rightarrow 48 + 84 = 132$

Key Observation: The sum of any $2$-digit number and its reverse is always a multiple of $11$.

• To get a sum of $55$, the digits must add up to $5$. Possible numbers: $14, 23, 32, 41, 50$.
• To get a sum of $88$, the digits must add up to $8$. Possible numbers: $17, 26, 35, 44, 53, 62, 71, 80$.

• India is home to $\frac{3}{4}$ of the world's tiger population.
• India is home to $\frac{3}{5}$ of the Asiatic elephant population.

To find the total elephant population in Karnataka ($6049$) and Kerala ($3054$):

$$\begin{array}{r@{\quad}c@{\quad}c@{\quad}c@{\quad}c} & \mathbf{Th} & \mathbf{H} & \mathbf{T} & \mathbf{O} \\ & 6 & 0 & 4 & 9 \\ + & 3 & 0 & 5 & 4 \\ \hline & 9 & 1 & 0 & 3 \\ \hline \end{array}$$

Regrouping Rules:

• $10 \text{ Ones (O)} = 1 \text{ Ten (T)}$
• $10 \text{ Tens (T)} = 1 \text{ Hundred (H)}$
• $10 \text{ Hundreds (H)} = 1 \text{ Thousand (Th)}$

Assam has $5719$ elephants, which is $3965$ more than Meghalaya. To find Meghalaya's elephant population, subtract $3965$ from $5719$:

$$\begin{array}{r@{\quad}c@{\quad}c@{\quad}c@{\quad}c} & \mathbf{Th} & \mathbf{H} & \mathbf{T} & \mathbf{O} \\ & 5 & 7 & 1 & 9 \\ - & 3 & 9 & 6 & 5 \\ \hline & 1 & 7 & 5 & 4 \\ \hline \end{array}$$

Regrouping in Subtraction:

• Borrow $1 \text{ Ten}$ to make $10 \text{ Ones}$ if needed.
• Borrow $1 \text{ Hundred}$ to make $10 \text{ Tens}$ (e.g., $1 \text{ Ten}$ becomes $11 \text{ Tens}$ after borrowing from $7 \text{ Hundreds}$).
• Borrow $1 \text{ Thousand}$ to make $10 \text{ Hundreds}$.

• Subtracting $99$: Subtract $100$, then add $1$.
  $$8787 - 99 = 8787 - 100 + 1 = 8687 + 1 = 8688$$
• Adding Near-Hundreds: To add $104$, add $100$ then add $4$.
  $$4596 + 104 = 4596 + 100 + 4 = 4696 + 4 = 4700$$
• Using Known Relations: If $139 + 175 = 314$, then:
  • $$314 - 175 = 139$$