Mathematics: Class 8 (Part 2) — Chapter 6: Algebra Play

Algebra is an indispensable tool for modeling numerical scenarios, justifying mathematical statements, and analyzing puzzles.

6.2 'Think of a Number' Tricks

• Basic Trick: Think of a number (x), double it (2x), add 4 (2x + 4), divide by 2 (x + 2), subtract original. Result is always 2.
• Date Trick: Let Month = M, Day = D. Formula: (((M × 5) + 6) × 4 + 9) × 5 + D = 100M + 165 + D. Subtract 165 from the final answer to reveal the date.

6.3 Number Pyramids

In a number pyramid, each number is the sum of the two directly below it.

• 3-Row Algebraic Model: Bottom row: a, b, c → Middle: a+b, b+c → Top: a + 2b + c.
• Connects to the Virahāňka-Fibonacci sequence where each number is the sum of the two preceding ones.

6.4 Fun with Grids (Calendar Magic)

• 2×2 Calendar Grid: Let top-left be a. Grid values: a, a+1, a+7, a+8.
• Sum Formula: 4a + 16. Given the sum, solve for a to find all dates.

6.5 The Largest Product

To maximize the product of a 2-digit and 1-digit number using three given digits (p < q < r):

• The largest digit (r) must be the multiplier.
• The remaining digits form the multiplicand in decreasing order (qp).
• Optimal Formula: qp × r.

6.6 Decoding Divisibility Tricks

• Difference of Reversed 2-Digit Numbers: (10a + b) - (10b + a) = 9(a - b). Always divisible by 9.
• Sum of Reversed 2-Digit Numbers: (10a + b) + (10b + a) = 11(a + b). Always divisible by 11.
• 3-Digit Cycling Sum: abc + bca + cab is always divisible by 37 and 3.
• Repeating 3-Digit Number: abcabc is divisible by 7, 11, and 13 (since 7 × 11 × 13 = 1001).